jections in a von Neumann algebra are equivalent ifand only ifthey have the same central carrier. Two infinite projections in a factor on a separable space are equivalent. For the fi[r]
Trang 1This is a volume in
PURE AND APPLIED MATHEMATICS
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A list of recent titles in this series is available from the publisher upon request
Trang 2FUNDAMENTALS OF THE THEORY
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Fundamentals of the theory of operator algebras (Pure and applied mathematics ; 100)
Includes bibliographies and indexes
Contents: v 1 Elementary theory - v 2
Trang 4PREFACE
Most of the comments in the preface appearing at the beginning of Volume I are fully applicable to this second volume This is particularly so for
the statement of our primary goal: to teach the subject rather than be
encyclopaedic Some of those comments refer to possible styles of reading and using Volume I The reader who has studied the first volume following the plan that avoids all the material on unbounded operators can continue in this volume, deferring Lemma 6.1.10, Theorem 6.1.1 1, and Theorem 7.2.1’ with its
associated discussion to a later reading This program will take the reader to Section 9.2, where Tomita’s modular theory is developed At that point, an
important individual decision should be made: Is it time to retrieve the
unbounded operator theory or shall the first reading proceed without it? The reader can continue without that material through all sections of Chapters 9
(other than Section 9.2), 10, 11, and 12 (ignoring Subsection 11.2, Tensor
products of unbounded operators, which provides an alternative approach to
the commutant formula for tensor products of von Neumann algebras) However, avoiding Section 9.2 makes a large segment of the post-1970 literature of von Neumann algebras unavailable Depending on the purposes
of the study of these volumes, that might not be a workable restriction Very little of Chapter 13 is accessible without the results of Section 9.2, but Chapter
14 can be read completely
Another shortened path through this volume can be arranged by omitting some of the alternative approaches to results obtained in one way For example, the first subsection of Section 9.2 may be read and the last two omitted on the first reading The last subsection of Section 11.2 may also be omitted It is not recommended that Section 7.3 be omitted on the first reading although it does deal primarily with an alternative approach to the theory of normal states Too many of the results and techniques appearing in that section reappear in the later chapters Of course, all omissions affect the exercises and groups of exercises that can be undertaken
As noted in the preface appearing in Volume I, certain exercises (and groups of exercises) “constitute small (guided) research projects.” Samples of this are: the Banach-Orliz theorem developed in Exercises 1.9.26 and 1.9.34; the theory of compact operators developed in Exercises 2.8.20-2.8.29, 3.5.17,
ix
Trang 5X PREFACE
and 3.5.18; the theory of b(N) developed in Exercises 3.5.5, 3.5.6, and 5.7.14-5.7.21 There are many other such instances To a much greater extent, this process was used in the design of exercises for the present volume; results
on diagonalizing abelian, self-adjoint families of matrices over a von Neu-
mann algebra are developed in Exercises 6.9.14-6.9.35; the algebra of
unbounded operators affiliated with a finite von Neumann algebra is
constructed in Exercises 6.9.53-6.9.55,8.7.32-8.7.35, and 8.7.60 The represen-
tation-independent characterizations of von Neumann algebras appear in
Exercises 7.6.35-7.6.45 and 10.5.85-10.5.87 The Friedrichs extension of a positive symmetric operator affiliated with a von Neumann algebra is
described in Exercises 7.6.52-7.6.55, and this topic is needed in the develop-
ment of the theory of the positive dual and self-dual cones associated with
von Neumann algebras that appears in Exercises 9.5.5 1-9.6.65 A detailed analysis of the intersection with the center of various closures of the convex hull of the unitary conjugates of an operator in a von Neumann algebra is
found in Exercises 8.7.4-8.7.22, and the relation of these results to the theory
of conditional expectations in von Neumann algebras is the substance of the next seven exercises; this analysis is also applied to the development of the theory of (bounded) derivations of von Neumann algebras occurring in
Exercises 8.7.51-8.7.55 and 10.5.76-10.5.79 Portions of the theory of repre-
sentations of the canonical anticommutation relations appear in Exercises
10.5.88-10.5.90, 12.5.39, and 12.5.40 This list could continue much further; there are more than 1100 exercise tasks apportioned among 450 exercises in
this volume The index provides a usable map of the topical relation of exercises through key-word references
Each exercise has been designed, by arrangement in parts and with suitable hints, to be realistically capable of solution by the techniques and skills that will have been acquired in a careful study of the chapters preceding the exercise However, full solutions to all the exercises in a topic grouping may require serious devotion and time Such groupings provide material for special seminars, either in association with a standard course or by them- selves Seminars of that type are an invaluable “hands-on” experience for active students of the subject
Aside from the potential for working seminars that the exercises supply, a
fast-paced, one-semester course could cover Chapters 6-9 The second
semester might cover the remaining chapters of this volume A more leisurely
pace might spread Chapters 6-10 over a one-year course, with an expansive treatment of modular theory (Section 9.2) and a careful review (study) of the unbounded operator theory developed in Sections 2.7 and 5.6 of Volume I Chapters 11-14 could be dealt with in seminars or in an additional semester
course In addition to these course possibilities, both volumes have been written with the possibility of self-study very much in mind
Trang 6PREFACE xi The list of references and the index in this volume contain those of
Volume I Again, the reference list is relatively short, for the reasons mentioned in the preface in Volume I A special comment must be made about the lack of references in the exercise sections Many of the exercises (especially the topic groupings) are drawn from the literature of the subject
In designing the exercises (parts, hints, and formulation), complete, model solutions have been constructed These solutions streamline, simplify, and unify the literature on the topic in almost all cases; on occasion, new results are included References to the literature in the exercise sets could misdirect more than inform the reader It seems expedient to defer references for the exercises to volumes containing the exercises and model solutions; a signifi- cant number of references pertain directly to the solutions We hope that the benefits from the more sensible references in later volumes will outweigh the present lack; our own publications have been one source of topic groupings subject to this policy
Again, individual purposes should play a dominant role in the proportion
of effort the reader places on the text proper and on the exercises In any case,
a good working procedure might be to include a careful scanning of the exercise sets with a reading of the text even if the decision has been made not
to devote significant time to solving exercises
Trang 7CHAPTER 6
We take up the detailed study of von Neumann algebras in this chapter The principal tool for this study is the technique of “comparison” of the projections in a von Neumann algebra W relative to 9 By these means we develop a notion of “equivalence” of projections in 9 (meaning, loosely, “of the same size relative to 9’’) Associated with this equivalence, we have a partial ordering of (the equivalence classes of) projections in 9-with corresponding notions of “finite” and “infinite” projections relative to 9%’
In these terms, we can separate von Neumann algebras into broad types (algebraically non-isomorphic) and show that each such algebra is a direct sum of algebras of the various types (the so-called “type decomposition” of von Neumann algebras) The simplest of the types (“Type I von Neumann algebras”) is analyzed and examples of some of the other types are studied
6.1 Polar decomposition and equivalence
In the discussion following Lemma 2.4.8, we observed that each bounded
operator Ton a Hilbert space .# can be expressed as H + iK, with H and K
self-adjoint operators We referred to H and K as the “real” and “imaginary”
parts of T-noting the analogy between this representation of T and the corresponding representation of a complex number in terms of its real and imaginary parts
If we pursue the analogy between representations (decompositions) of complex numbers and those of linear operators, we are led to consider the possibility of a “polar decomposition ” of operators analogous to the de- composition of a complex number as the product of a positive number (its modulus) and a number of modulus 1
With the function calculus for self-adjoint operators at our disposal, there
is no problem in producing a “polar decomposition” for an invertible operator
T As modulus, both (T*T)’12 and (TT*)’” suggest themselves At first
guess, we might expect the number of modulus 1 in the polar decomposition
of a complex number to correspond to a unitary operator in the case of an operator The non-commutativity of the operator situation introduces a
399
Trang 8400 6 COMPARISON THEORY OF PROJECTIONS
complicating factor Shall we multiply the modulus of Ton the left or right by
the unitary operator (if it is, indeed, to be a unitary operator); and which of
(T*T)’”, (TT*)’12 shall we use as modulus? A small amount ofexperimenta-
tion shows that writing T = U(T*T)”’ (somewhat hopefully), and, then,
“solving” for U as T(T*T)-’’’ produces a unitary operator U (while
T(TT*)- ”’ will not, in general, be unitary-nor would (T*T)-”’T) The
computation involved in this is
( T ( T * T ) - ”’x, T( T* T ) - ‘“x) = ( ( T * T ) - ‘12T*T(T*T)- ‘”x, X)
= (x, x)
If WH is another “polar decomposition” of T (with W unitary and H
positive), then H = W*T so that H2 = H*H = T*WW*T = T*T As
H 2 0, and the positive square root of a positive operator is unique (see Theorem 4.2.6), H = (T*T)’/’ and W = T(T*T)-’” = U Of course,
T* = (T*T)”’U*, while T* has its own polar decomposition, T* =
V*(T**T*)”2 = V * ( T T * ) 1 ’ 2 Thus T = (TT*)’l2V; and this last equality provides a “polar decomposition” for T with the positive operator factor appearing on the left This, incidentally, redresses the balance between the two candidates for “modulus” of T Combining T = U(T*T)’” and T* =
(T*T)’/’U*, we have TT* = U(T*T)U* (so that TT* and T*T are
unitarily equivalent, when T is invertible) Since U(T*T)’12U* is a positive square root of U(T*T)U*, (TT*)’12 = U(T*T)’/’U* But V*(TT*)’/’ =
T* = (T*T)’/’U*, so that UV*(TT*)’l2 = U(T*T)1’2U* = (TT*)”2; and
V = U Thus the same unitary operator appears in the “right” and “left”
polar decomposition of T
For the polar decomposition of the general bounded operator, we must replace the unitary operators of the preceding discussion by operators that map one (closed) subspace of a Hilbert space isometrically onto another and annihilate the orthogonal complement of the first subspace Such operators
are called partial isometries The first subspace is called the initial space of the
partial isometry, and the second subspace (its range) is called its jinal space The projections with these subspaces as ranges are called the initial and Jinal
projecfions, respectively, of the partial isometry
6.1.1 PROPOSITION The operator V acting on the Hilbert space X is a partial isometry ifand only if V*V is a projection E I n this case, E is the initial projection of V, VV* is thejnal projection F of V, and V* is a partial isometry with initial projection F andfinal projection E
Proof Suppose, first, that V is a partial isometry with initial projection
E Then IIVxll = IIVEx + V ( 1 - E)xII = I(VExI1 = llExll 5 llxll; so that
II V (1 I 1 Ifx is a unit vector in the range of E, then 1 = (x, x ) = ( V x , V x ) =
Trang 96 I POLAR DECOMPOSITION AND EQUIVALENCE 40 I
( V * V x , x) From Proposition 2.1.3 (the “Cauchy-Schwarz equality”),
V * V x = x Ify is in the range of I - E, V*Vy = V*(O) = 0 Thus V*V = E
Suppose, now, that V* V is a projection E Then for each x in the range of
E, (x, x) = ( V * Vx, x) = ( V x , V x ) ; while, for y orthogonal to the range of
E, 0 = (V*Vy, y ) = ( V y , Vy) Thus V is isometric on E(R) and 0 on
(I - E)(R) It follows that V is a partial isometry with initial projection E
In addition, V = V E = VV*V, and VV*VV* = VEV* = VV* Thus VV*
is a projection F and FV = V Consequently F ( X ) contains V ( X ) But
F(R) = V V * ( X ) E V ( X ) Hence F is the final projection of V As VV* =
(I/*)* V* = F, we conclude, from the foregoing, that V* is a partial isometry with initial projection F and final projection E
6.1.2 THEOREM (Polar decomposition) If T is a bounded operator on
the Hilbert space #, there is a partial isometry V with initial space the closure
r(T*) of the range of T* and final space r(T) such that T = V(T*T)’” =
(TT*)’/’V I f T = W H with H positive and W a partial isometry whose initial
space is r(H), then H = (T*T)l12 and W = V If neither T nor T* annihilates
a non-zero vector, then V is a unitary operator
Recall from Proposition 2.5.13 that r(T*) = r(T*T) so that
r(T*) = r((T*T)l/’) Since
((T*T)’”x, (T*T)’”x) = (T*Tx, X) = (Tx, T x ) ,
there is a partial isometry V with initial space r(T*) and final space r(T) such
that T = V(T*T)”’ Thus T* = (T*T)”’V* and TT* = VT*TV* Now
If T and T* have (0) as null space, their ranges are dense in X Hence V is
a unitary operator, in this case
Note that (T*T)’l2 and (TT*)l/’ are contained in each C*-algebra
containing T However, V may not lie in such an algebra If T is a positive
operator, V is R(T) With 2I the algebra of multiplications by continuous
functions on L,([O, 1)) (relative to Lebesgue measure) and H multiplication
by a positive function that vanishes on [0, $1, R(H) is a projection different from 0 and I Since ‘?I contains no projections other than 0 and I , the polar
Trang 10402 6 COMPARISON THEORY OF PROJECTIONS
decomposition of H cannot be effected in ‘ill If T is invertible, (T*T)’12 and
U (= T(T*T)-”’) lie in each C*-algebra containing T The critical informa-
tion concerning the possibility of polar decomposition within a C*-algebra is found in the proposition that follows
6.1.3 PROPOSITION If T lies in a von Neumann algebra 9 and U H is the polar decomposition of T, then U and H are in 9
Proof As noted, H = (T*T)’I2 €9, since 9 is, in particular, a C*- algebra containing T If T’EW‘, T’UHx = T’Tx = TT‘x; while UT’Hx =
UHT’x = TT‘x Thus UT‘ and T’U agree on the range of H Since T‘
commutes with H , both the range of H and its orthogonal complement are
stable under T’ As U is 0 on this complement, both UT’ and T’U are 0 there Thus UT’ = T’U and U E 9” = 9
If T is normal, (T*T)’” = (TT*)”’(=H) Thus U H = T = H U (from Theorem 6.1.2) Conversely, from uniqueness of the polar decomposition (‘‘left’’ and “right”), if U H = H U , (T*T)”’ = (TT”)”’ and T*T = TT*
To compare the dimensions of the ranges of two projections E and F
acting on a Hilbert space, we compare the cardinality of orthonormal bases for each of these subspaces Another (equivalent) technique for comparing the dimensions of the ranges of E and F to see if they are the same would be to seek a partial isometry with one as initial projection and the other as final
projection If E and F lie in a von Neumann algebra W and we insist that
our partial isometry lie in 9, we are demanding a stricter comparison of E and F-a comparison relative to 9 The structure of 9 would seem to exert an important influence on the possibility of comparison; and, consequently, the structure this comparison process imposes on the projections of W will reflect the structure of 9
Elaborating this idea leads to the Murray-von Neumann comparison theory of projections in a factor and its extension to a comparison theory of projections in a von Neumann algebra
6.1.4 DEFINITION Two projections E and F are said to be equivalent
relative to a von Neumann algebra W (written, E - F ( 9 ) ) when V*V = E
and VV* = F for some V in 9 H
In view of Proposition 6.1.1, the operator V in 9 is a partial isometry with initial projection E and final projection F Since E = V*V and F = V V * ,
both E and F are in W Most often, the von Neumann algebra W relative to which the equivalence of E and F is being asserted will be clearly indicated by
the context In this case we say that E is equivalent to F and write E - F
Trang 116 I POLAR DECOMPOSITION AND EQUIVALENCE 403
In the proposition that follows, we show that the relation - defined on the projections of 9 is an equivalence relation
6.1.5 PROPOSITION If projections E, F, G in a von Neumann algebra W
satisfy E - F and F - G, then F - E, E - G, and E - E
Since E - F and F - G, there are partial isometries V and Win
9 such that V*V = E , V V * = F, W*W = F, and WW* = G Thus F =
(V*)*V* and E = V*(V*)*; so that F - E As E = E*E = EE*, E is a
partial isometry with initial and final projection E ; and E - E Finally,
6.1.6 PROPOSITION If 92 is a von Neumann algebra and TE 9, then
Proof: From Theorem 6.1.2 and Proposition 6.1.3, T = V(T*T)”’,
V E ~ , and V is a partial isometry with initial projection R(T*) and final
projection R(T) Thus R(T) - R(T*)
R ( T ) - R(T*)
6.1.7 THEOREM (Kaplansky formula) IJE and F are projections in a von Neumann algebra W, then ( E v F - F ) - ( E - E A F)
Proof We note that E v F - F is the range projection of (I - F)E,
while E - E A F is the range projection of E(Z - F ) ( = [ ( I - F)E]*) Once this has been established, the Kaplansky formula follows from Proposition
6.1.6.FromProposition2.5.14,R(E(I - F ) ) = E - E A F;andR((I - F)E)
= I - F - ( I - F ) A (I - E ) = E v F -F(sinceI - ( I - F ) A ( I - E )
= E v F)
6.1.8 PROPOSITION Two projections E and F in a von Neumann algebra
9 have non-zero equivalent subprojections ifand only ifCECF # 0
F, and E , - F , , there is a partial
isometry V in W such that V*V = E , and VV* = F o Since F , I F I CF
and Eo 5 E 5 CE, V = F o V E o = FoCFVCEEo = F o V E o C F C E = 0 Thus
0 = Eo = F ,
Proof If CECF = 0, E , 5 E, F ,
Trang 12404 6 COMPARISON THEORY OF PROJECTIONS
If CECF # 0, then [ @ E ( Z ) ] A [ W F ( Z ) ] # (0), from Proposition 5.5.2
(and Proposition 2.5.3) Thus there are operators A, B in 3 and vectors x, y such that 0 # (AEx, BFy) = (FB*AEx,y), and FTE # 0, where T =
B*A E 9 It follows that R(FTE) and R(ET*F) are non-zero projections in 9
From Proposition 6.1.6, they are equivalent Of course R(FTE) 5 F and
Proof
An extension of the polar decomposition to the case of a closed densely defined linear transformation from one Hilbert space to another forms the basis for the developments in Section 9.2 We describe this extension before passing to a detailed study of the partial ordering of (the equivalence classes
of) projections associated with our equivalence relation The following simple lemma will prove useful to us
6.1.10 LEMMA If A and C are densely deJined preclosed operators and B
is a bounded operator such that A = BC, then A* = C*B*
Proof I f y E 9 ( A * ) , then, for each x in 9 ( A ) ( = g(C)),
( x , A*y) = ( A X , y ) = (BCX, y ) = ( C X , B*y);
so that B*y E 9(C*) and C*B*y = A*y I f y E Q(C*B*), then B*y E 9(C*)
and, for each x in 9 ( C ) (= 9 ( A ) ) ,
( x , C*B*y) = (CX, B*y) = ( B C X , y ) = ( A X , y);
so that y E 9 ( A * ) and A*y = C*B*y
6.1.1 1 THEOREM If T is a closed densely deJined linear transformation from one Hilhert space to another, there is a partial isometry V with initial space
the closure ofthe range of( T*T)' ' andfinal space the closure of the range of T
such that T = V(T*T)112 = (TT*)''2V Restricted to the closures of the ranges of T* and T , respectively, T * T and T T * are unitarily equivalent (and V
implements this equivalence) If T = WH, where H is a positive operator and W
is a partial isometry with initial space the closure of the range of H , then H =
(T*T)'" and W = V If W is a von Neumann algebra, T rj W if and only if
V E W and (T*T)'12 r] 9
Trang 136.2 ORDERING 405
Prooj From Theorem 2.7.8(v), T*T is self-adjoint If x E 9 ( T * T ) , then
x E 9( T ) , T x E 9( T*), and
0 I ( T x , T x ) = ( T*Tx, x)
Thus T*T is positive and has a (unique) positive square root (T*T)'12
(See Proposition 5.6.21 and Remark 5.6.32.) From Remark 2.7.7,9(T*T) is a
core for (T*T)'12 and for T Thus (T*T)'" and T map 9 ( T * T ) onto dense
subsets of their ranges Defining V0(T*T)'I2x to be Tx, for x in 9(T*T), V, extends to a partial isometry V with initial space the closure of the range of
(T*T)'12 and final space the closure of the range of T, since
((T*T)'"x, (T*T)'12x) = (T*Tx, X ) = ( T x , T x )
Moreover, Tx = V(T*T)'"x for each x in 9(T*T)
With x in 9(V(T*T)'I2), choose x , in 9 ( T * T ) such that x, + x and
(T*T)'12x, + (T*T)'I2x Then Tx, = V(T*T)'/'x, -+ V(T*T)'12x Since
T is closed, x E ~ ( T ) and T x = V(T*T)'12x Thus V(T*T)'l2 G T
Conversely, if x E 9( T ) and x, is chosen in 9( T * T ) such that x, + x and
Tx, + Tx, then (T*T)'12x, = V*V(T*T)'12x, = V*Tx, + V*Tx Since
(T*T)'12 is closed, x E 9((T*T)'I2) It follows that T = V(T*T)'12
From Lemma 6.1.10, T* = (T*T)'12V*, so that TT* = VT*TV* Thus the restriction of TT* to the closure of the range of T is unitarily equivalent to
the restriction of T*T to the closure of the range of T*, and Vimplements this
equivalence It follows that (TT*)'12 = V(T*T)'12V*; so that
T = V(T*T)'" = V(T*T)'12V*V = (TT*)'12V
If T = WH with H positive and W a partial isometry with initial space the
closure of the range of H , then, from Lemma 6.1.10, T* = HW* and T*T =
H 2 From Remark 5.6.32, H = (T*T)'12, so that W = V
Let B be a von Neumann algebra and U be a unitary operator in 92'
Then UVU*U(T*T)'(2U* is the polar decomposition of UTU* From
uniqueness of the polar decomposition, T = U T U * if and only if V =
U V U * and (T*T)'12 = U(T*T)'IZU* Thus T r] 9? if and only if V €9 and
Trang 14406 6 COMPARISON THEORY OF PROJECTIONS
6.2.1 DEFINITION If E and F are projections in a von Neumann algebra
@, we say that E is weaker than F (and write E 5 F) when E is equivalent to a
subprojection of F
We shall establish that 5 is a partial ordering on the (equivalence classes of) projections in 9 In the sequel, free use will be made of ail the notational variations and terminology that are associated with an ordering For example,
in the circumstances of Definition 6.2.1, we say that F is stronger than E and write F 2 E (as well as E -<, F) It is worth emphasizing that E < F (E is
strictly weaker than F) is the same as E 5 F and E is not equivalent to F
(written E + F)
6.2.2 PROPOSITION If {E,} and {Fa} are orthogonal families of projections
in a von Neumann algebra 9 such that E, 5 Fa for all a, then C E, 5 C Fa
If E, - Fa for all a, then Fa
Proof SupposeE, - F,foralla;andsuppose Eisapartialisometryin9
with initial projection E, and final projection Fa Defining V to be V, on the
range of E,, for each a, and to be 0 on the range of I - E, where E = C E,
and F = C Fa, V extends (linearly) to a partial isometry with initial pro- jection E and final projection F To see that V is in 9, note that V
coincides with 5, + + on the range of E,, + - + Ean + I - E Since
V,, + - + V,n is a partial isometry in 41, the set of such operators has the uniform bound 1 At the same time, the ranges of the projections E,, + - - - +
Eon + I - E span the Hilbert space From the discussion preceding Remark
2.5.9, V is in the strong-operator closure of W ; and V is in 9
If E, ,< Fa for all a, then E, - G, 5 Fa for all a Thus E - G I F, where
G =
E, -
G,, from what we have proved to this point Hence E 5 F H
6.2.3 PROPOSITION IfE and F are projections in a von Neumann algebra
@ and E - F, then PE - PF for each central projection P in W If E 5 F, then
6.2.4 PROPOSITION If E and F are projections in a von Neumann algebra 9suchthat E 5 F and F 5 E,then E - F
Trang 156.2 ORDERING 407
Proof Let V and W be partial isometries in W such that V*V =
E(=Eo), VV* = F, I F (=F,), W*W = F (=F,), and WW* = El I E
Since V maps the range of E isometrically onto that of F,, V maps the range of
El isometrically onto that of a subprojection F, of F,-algebraically,
(VEl)*VEl = El and VE,(VE1)* = F, Similarly W maps the range of F ,
onto that of a subprojection E , of E l Moreover, V(E - El) is a partial isometry in W with initial projection E - El and final projection F, - F ,
Continuing in this way, by taking successive images under V and W, we
construct two sequences {E,}, {F,} of projections in W such that E =
E , 2 E , 2 E2 2 E 3 2 ., F = Fa 2 F, 2 F2 2 , V maps the range of
E, isometrically onto that of F,+ and W maps the range of F, isometrically onto that of E n + , Thus V maps the range of E m onto that of F,, where
Em = A,, En and Fw = A,, F, In addition En - - F,+, - Fn+2 and
F, - F,+, - - E n + , for n = 0, 1,2, , since V(E, - E n + , ) and W(F, - F,+ 1) are partial isometries in B with initial projections En - En+ 1,
F, - F, + , and final projections F,, - F, + and En + - Em+ , , respectively From Proposition 6.2.2,
6.2.5 PROPOSITION Zf E, F, and G are projections in a von Neumann
algebra W and E 5 F, F 5 G, then E 5 G
such that V* V = E,
V V * = Fa I F, W*W = F, and W W* = Go 5 G In this case, WF, W* is a subprojection G1 of Go in W; and WV is a partial isometry in W with initial projection E and final projection G1 Thus E - G1 I G ; and E 5 G
Proof Suppose V and Ware partial isometries in
Propositions 6.2.4 and 6.2.5 tell us that 5 is a partial ordering of the classes
of equivalent pr0jections.h the usual loose way, we speak of this relation as a partial ordering on the projections
Trang 16408 6 COMPARISON THEORY OF PROJECTIONS
Corollary 6.1.9 and Proposition 6.2.2 can be combined with a maximality (“ measure-theoretic exhaustion ”) argument to show that the equivalence classes of projections in a factor are totally ordered While this result is subsumed in the comparison theorem (Theorem 6.2.7-the more general argument using Proposition 6.1.8 in place of Corollary 6.1.9), it is instructive
to see the argument for the case of factors
6.2.6 PROPOSITION I f E and F are projections in a factor A, either E 5 F
or F 5 E
Proof Let 9 be the family of sets { ( E , , F,)} of ordered pairs (E,, F,)
of equivalent subprojections E,, F, of E, F such that {E,} and {F,} are orthog-
onal families Since ((0,O)) E F is non-empty If we partially order 9 by inclusion, the union of a totally ordered subset is an upper bound for it From Zorn’s lemma, B has a maximal element { ( E , , F,)} If E - C E , and
F - C F, are non-zero, they have equivalent non-zero subprojections Eo and
F , , from Corollary 6.1.9 Adjoining ( E , , F , ) to {(E,, F,)} contradicts the maximality of {<E,, F,)} Thus one of E - C E , and F - C F , is 0 (possibly both are) From Proposition 6.2.2, C E, - C F, Thus either E is equivalent
to a subprojection of F or F is equivalent to a subprojection of E
That is, either E 5 F or F 5 E
The basic ingredient of “total comparability” in factors is found in Corollary 6.1.9, where we learn that non-zero projections E and F have equivalent non-zero subprojections This is established by showing that
F T E # 0 for some T in the factor Roughly speaking, T “compares” a piece
of E with a piece of F While equivalence calls for “comparison” by partial
isometries, the polar decomposition of T supplies this Actually the transition
from comparison by an operator T in A‘ to comparison by the partial
isometry appearing in its polar decomposition is the essence of Proposition
6.1.6; and it is most expedient to use this proposition on F T E Finally, the fact that F T E is non-zero for some Tin A reduces to the observation that the only
projections in A? whose ranges are stable under A? are 0 and I As T takes on
various values in A, the range of TE sweeps out a set of vectors that spans the range of CE (see Proposition 5.5.2) With E non-zero and A! a factor, CE = I, and F cannot annihilate this set unless F = 0 In a von Neumann algebra 9,
if CE # I, it is precisely CE that can “block” comparison of F and E (as in- dicated in Proposition 6.1.8) Formalizing these ideas, we arrive at the comparison theorem, a generalization of Proposition 6.2.6
From the point of view of the comparison theory of projections, it is very useful to treat a von Neumann algebra as a direct sum of factors (although this is not generally valid) For most purposes, three factors will suffice The
sum of the unit projections of the factors in which an operator A of the algebra
Trang 176.2 ORDERING 409
has a non-zero component is C , If E and F are two projections in the algebra,
we compare their component projections in each factor Summing the unit
projections of the factors in which E and F have equivalent components yields the central projection Q of the comparison theorem The sum of the unit projections of the factors in which the component of E is strictly weaker than that of F yields the central projection P of that theorem
6.2.7 THEOREM (Comparison) I f E and F are projections in a von Neumann algebra 9, there are unique orthogonal central projections P and Q maximal with respect to the properties QE - QF, and, if Po is a non-zero central subprojection of P, then Po E < P,F If R , is a non-zero central sub- projection of I - P - Q, then R , F < R , E
We begin by describing the structure of the proof A maximality
argument allows us to locate the central projections Q and P Replacing 9 by
B(I - P - Q), E by ( I - P - Q)E, and F by ( I - P - Q)F, we may now
assume that R , E 5 R, F for a central projection only if R, = 0 Under this
assumption and applying that argument with the roles of E and F reversed,
we construct a maximal central projection R , with the property asserted in the statement of the theorem for I - P - Q We must show that R , = I (or, in
terms of the initial notation, that R , = I - P - Q) If this is not the case, replacing W, E , and F , again, by a(I - R,), ( I - R , ) E , and ( I - R , ) F , we may assume that if R O E 5 R, F or R,F 5 R , E, for a central projection, then
R, = 0 The last stage of the proof consists of proving that this situation leads
to a contradiction That is accomplished by means of the argument of Proposition 6.2.6 (the factor version of the present theorem)-with Prop- osition 6.1.8 in place of Corollary 6.1.9 We proceed to the details
Let {Q,} be an orthogonal family of central projections in W maximal
with respect to the property that Q,E - Q,F for each a From Proposition
6.2.2, C Q,E = QE - QF = C Q,F, where Q = C Q, By maximality of
{Q,}, if P,E - P,F for a central subprojection Po of I - Q, then Po = 0
If Qo is a central projection such that Qo E - Qo F, then (Q, - Qo Q)E -
(Q, - Q o Q ) F , from Proposition 6.2.3 Since Qo - Qo Q is a subprojection Po
of I - Q, Qo - Q o Q = 0, that is, Qo I Q The uniqueness of Q follows
Replacing B, E, and F by W(I - Q), ( I - Q)E, and ( I - Q ) F , we may
assume that Po = 0 if Po is a central projection such that P,E - P,F Let
{ P,} be a maximal orthogonal family of central projections such that P, E <
P,F, and let P be its union-provided there is at least one such P a (Note that
Pa must be non-zero if Pa E < P , F.) Otherwise, let P be 0 From Proposition
6.2.3, Po Pa E 5 Po P,F for each central projection Po Thus, from Proposition
6.2.2, if Po I P,
1 P,P,E = P O P E = P,E s z P,P,F = P,F
Proof
Trang 18410 6 COMPARISON THEORY OF PROJECTIONS
Under our present assumption, if Po E - Po F , then Po = 0 Thus P , E < Po F if0 # Po I P
Again, if Po is a central projection such that P , E < PoF, then
(Po - P0P)E 5 (Po - P0P)F;
and Po - P O P is orthogonal to each P, By maximality of {P,},
( P o - P , P ) E < (Po - PoP)F does not hold Thus (Po - PoP)E - ( P o - P,P)F With the present assumption on 9, Po - P O P = 0 Hence Po I P ; and P is unique
From the preceding, we note that R, = 0 if R, is a central projection in
a(I - P - Q) such that ROE 5 R, F (using our original notation) Replace
B, E, and F by B(I - P - Q), ( I - P - Q)E, and ( I - P - Q)F Applying what we have proved with the roles of E and F reversed, there is a maximal
central projection R , with the property asserted in the statement for I -
P - Q If either R O E S RoF or R,F 5 R O E for a central subprojection R ,
of I - R,, then R, = 0 If R , = I (that is, in the preceding notation, if
R , = I - P - Q), then the proof is complete Assuming that I - R , # 0,
replace 9?, E, and F by B(I - Rl), ( I - R,)E, and (I - R,)F If either
R O E 5 R, F or Ro F 5 R, E, for a central projection R o , then R, = 0 This situation will lead us to a contradiction
Let {(E,, F a ) } be a family of ordered pairs of equivalent subprojections
of E and F maximal with respect to the property that {E,} and {Fa} are orthogonal families Then C E, - C Fa, from Proposition 6.2.2; and
R, E, - Ro C F,, for each central projection R,, from Proposition 6.2.3
Thus if Ro(E - C E,) = 0, R O E - Ro 1 Fa 5 R o F ; and R, = 0 It follows that E - C E, and, similarly, F - C F , have I as central carriers From Proposition 6.1.8, E - C E, and F - C Fa have equivalent non-zero sub-
projections E, and F, Adjoining ( E , , F,) to {(&, Fa)) contradicts its maximality
6.2.8 PROPOSITION If E and F are equivalent projections in a von Neumann algebra 9 acting on a Hilbert space X , then C E = C F
Proof By assumption, V*V = E and VV* = F, for some partial
isometry V in B Thus E = V*FV From Proposition 5.5.2, [BE(%)] and
[92F(X)] are the ranges of C E and C F , respectively Now
[92E(X)] = [ ! W * F V ( X ) ] G [ W F ( X ) ]
Similarly [&?F(X)] c [BE(#)].Thus [ 9 E ( X ) ] = [.%?F(#)],and CE = C,
Trang 196.3 FINITE AND INFINITE PROJECTIONS 41 I
6.2.9 PROPOSITION If E is a cyclic projection in the von Neumann algebra
9, acting on the Hilbert space A?, and F 5 E, then F is cyclic in 9
Proof Since F - E , I E, and, from Proposition 5.5.9, Eo is cyclic, we may assume F - E Let V be a partial isometry in .i;P such that V* V = E and
V V * = F ; and let x be a generating vector for E under 9$’ Then
[*@’VX] = [VW‘X] = V[W’x] = VE(A?) = F(X),
where the equality [ V ~ ’ X ] = V [ W ’ x ] follows from the fact that V is isometric
on [ 9 ’ x ] H
Bibliography: [56]
6.3 Finite and infinite projections
We noted and used the analogy with set theory in proving that E - F
when E 5 F and F 5 E (Proposition 6.2.4) This analogy suggests extending the concepts of finite and infinite to the projection-class ordering
6.3.1 DEFINITION A projection E in a von Neumann algebra 9 is said
to be injinite relative to W when E - E , < E for some projection E, in 9
Otherwise, E is said to bejnite relative to 8 If E is infinite and P E is either 0
or infinite, for each central projection P , E is said to be properly injnite We
say that W is ajinite or properly injinite von Neumann algebra when I is,
respectively, finite or properly infinite
We have avoided the use of the terminology “purely infinite von Neumann algebra” since it appears in the literature with two distinct senses: one to mean what we have defined above as a “properly infinite von Neumann
algebra” and the other to mean what will later be called a “type 111 von
Neumann algebra” (see Definition 6.5.1)
6.3.2 PROPOSITION If E is ajinite projection in the uon Neumann algebra
9, each subprojection o j E isjinite Each minimal projection in W is$nite; and 0 isjnite If E - F and E isjinite, then F isjinite
Proof If Eo in 9 is a subprojection of E and V is a partial isometry in W
with initial projection E, and final projection E l , with El I E,, then
E - E , + V is a partial isometry in &? with initial projection E and final projection E - E , + El ( I E ) As E is finite (in a), E = E - E , + E l ; and
E , = El Thus E , is finite
Trang 20412 6 COMPARISON THEORY OF PROJECTIONS
If E - F, E is finite, F, I F, and F, - F, then there are partial isometries
V and W in 9 with initial projections E and F and final projections F and F,,
respectively In this case, V* W V is a partial isometry with initial projection E
and final projection V*Fo V As V*Fo V 5 E and E is finite, V*Fo V = E
Thus VV*F, VV” = FF,F = F, = VEV* = F ; and F is finite It follows that a projection equivalent to a finite or infinite projection is, respectively, finite or infinite
Since 0 has no proper subprojection, it is finite If G is a minimal projec- tion in a, its only proper subprojection in a is 0; and only 0 is equivalent to 0
(for, if V*V = 0, then V = 0, and VV* = 0) Thus G is finite
In the lemma that follows, we prove the analogue, for infinite projections,
of the possibility of “halving” an infinite set, with each of the halves in one-to- one correspondence with the original set
6.3.3 LEMMA (Halving) If E is a properly infinite projection in a von Neumann algebra R, there is a projection F in R such that F < E and F -
Proof Since E is infinite, E - E, < E If V E 9, V*V = E, and VV* =
E l , then ( E 2 = ) V E I V * < E , and E - El - E , - E , , Continuing in this
way(VE, V* = E3 < E2andE, - E2 - E2 - E,),weconstructacountably
infinite orthogonal family {En - En+ of equivalent non-zero subprojec-
tions of E This family is contained in a maximal such family (Fa)aEA By maximality, we cannot have Fa 5 E - 1 Fa (= E,), for then Fa - F, I E,, and adjoining F, to {Fa} contradicts its maximality From the comparison theorem (Theorem 6.2.7), there is a non-zero central projection P such that
PE, < PF, Since A is an infinite set, there is a subset A, of A such that if
a o ~ A o , A\A, (=A,), A,, and A,\{a,} can each be put into one-to-one correspondence with A From Proposition 6.2.3, PF, - PF, for a and a’ in
To this point, we have proved that if E is a properly infinite projection in a,
there is a non-zero central projection P in a such that PE can be “halved”- that is, there is a subprojection G of PE in&? such that G - PE - G - PE # 0
Trang 216.3 FINITE AND INFINITE PROJECTIONS 413 Let {Qa} be a maximal orthogonal family of non-zero central subprojections
of CE such that each QaE can be halved; and let G, be a subprojection of Q, E
in W such that G, - Q , E - G, - Q, E If CE - C Qo is not 0 then
is properly infinite; and, from what we have proved, there is a non-zero
central subprojection Q , of CE - C Q , such that Q , E can be halved Adjoin- ing Qo to {Q,} contradicts its maximality Thus CE = C Q, Letting F be
C G,, from Proposition 6.2.2 we have,
so that E can be halved
6.3.4 THEOREM If E is a properly injinite projection in the von Neumann
algebra 9, F is a countably decomposable projection in W (in particular, if F is
cyclic in a), and CF 5 CE, then F 5 E
proof IfO= CFCE(=CF),thenF = O ; a n d F 5 E.IfF # O,CECF # 0;
and E, F have equivalent non-zero subprojections E,, F,, respectively By
use of the halving lemma, we construct a countably infinite orthogonal
family {En} of subprojections of E with sum E such that each E, - E (Halve
E as E l + F , ; then halve F, as E , + F,, and so on Now replace E , by
E - En.) Let {F,} be a maximal orthogonal family of subprojections of
F with each F, equivalent to F, (As a consequence of the countable decom- posability of F, the family {F,} can be indexed by integers.) Since {F,) is maximal, the relation F , 5 F - C F, cannot hold (for, otherwise, a “copy”
of F, in F - C F, adjoined to {F,} would contradict that maximality) From
the comparison theorem (Theorem 6.2.7), there is a central projection P such that P(F - C F,) < PF, (and, in particular, PF # 0) Since F, - F , -
Eo 5 P(F - C F,) 5 P E , and PF, 5 PE,.,, from Proposition 6.2.3
Thus
0 # PF = P(F - 1 F,) + C PF, 5 1 P E , = PE,
from Proposition 6.2.2 We have proved that there is a central projection P
such that 0 # PF 5 PE, under the given hypotheses
Let {Pa} be a maximal orthogonal family of non-zero central subpro- jections of CF such that P,F 5 P,E If C , - Pa # 0, from what we have just proved, there is a non-zero central subprojection Po of C , - C Pa such
that P,(C, - C PJF = P,F ,< Po E Adjoining Po to {Pa) contradicts its maximality Thus C, = Pa From Proposition 6.2.2,
1 P,F == CFF = F 5 1 P , E I E
Trang 22414 6 COMPARISON THEORY OF PROJECTIONS
6.3.5 COROLLARY Two properly injinite, countably decomposable pro-
jections in a von Neumann algebra are equivalent ifand only ifthey have the same central carrier Two infinite projections in a factor on a separable space are equivalent
For the final assertion of this corollary, note that, in a factor, an infinite projection is properly infinite
6.3.6 LEMMA Zf(P,} is afamily of central projections in a von Neumann algebra W and E is a projection in &? such that Pa E is finite for each a, then PE is finite, where P = v, P a
Proof If F - P E and F < P E , then 0 # PE - F I P If (PE - F)P,
= 0 for all a, then PE - F annihilates the range of P( = v Pa); and we would have 0 = ( P E - F)P = PE - F, contrary to assumption Thus (PE - F)Pa
# 0 for some a; and P,F < PaPE = PaE From Proposition 6.2.3, PaF -
PaPE = P a E ; so that P,E is infinite in 9-contrary to hypothesis Thus PE
is finite
6.3.7 PROPOSITION If E is an infinite projection in the von Neumann
algebra W , there is a (unique) central projection P in 9 such that P I C PE
is properly infinite, and ( I - P)E isfinite If E is properly infinite and F - E, then F is properly injinite
Let {Qa} be a maximal orthogonal family of central projections
in 9 such that QaE is finite for each a Then, from Lemma 6.3.6, QE is finite,
where Q = Q a Moreover, PE is properly infinite, by maximality of {Qa},
where P = I - Q; and P is unique
If E is properly infinite, F - E, and P is a central projection such that
PF # 0, then PF - PE # 0, from Proposition 6.2.3 Since E is properly
infinite, PE is infinite; and, from Proposition 6.3.2, PF is infinite Thus F is
properly infinite
Proof
6.3.8 THEOREM If E and F arefinite projections in the von Neumann algebra 9, then E v F isjnite in d
Proof Since E v F - F - E - E A F (from the Kaplansky formula-
Theorem 6.1.7) and E - E A F is finite (see Proposition 6.3.2), E v F - F is
finite (again, from Proposition 6.3.2) As E v F = F + E v F - F , it will suffice to show that the sum (union) of two orthogonal finite projections in 9
is finite
We assume that EF = 0 Suppose E + F is infinite From Proposition
6.3.7, there is a central projection P such that P(E + F ) is properly infinite
Trang 236.3 FINITE AND INFINITE PROJECTIONS 41 5 From Proposition 6.3.2, PE and PF are finite We may assume, therefore, that
E + F is properly infinite From the halving lemma (Lemma 6.3.3), there is a
subprojection G of E + F such that G - E + F - G (= G’) - E + F From
the comparison theorem (Theorem 6.2.7), there is a central projection Q such that Q(G A E ) 5 Q(G‘ A F ) and ( I - Q)(G’ A F ) 5 ( I - Q)(G A E) One
at least, of Q(E + F ) and ( I - Q)(E + F ) is not zero If, say, Q(E + F ) # 0,
then Q(E + F ) is infinite; while QE and QF are finite and orthogonal More-
over,QG - QG’ - Q(E + F);and
G - G A E - E v G - E and E v G - E and G’ A F are orthogonal
subprojections of F To see this last, note that a vector in the range of G‘ A F is orthogonal to both the range of G and of E-hence, to the range
of E v G; while E v G 5 E + F,so that E v G - E I F From Proposition 6.3.2, it follows that G is finite (for G 5 F and F is finite) But G - E + F and
E + F was assumed to be infinite Thus E v F is finite
It is instructive to review the preceding argument assuming that W is a factor In this case, all reference to Q disappears; and we can pass from the introduction of G to the argument of the last paragraph of the proof
6.3.9 LEMMA If {E,),,w and {Fb}h.B are infinite, orthogonal families of
non-zero projections in a uon Neumann algebra 2, each E, is cyclic, and the
unions E and F of {Ea} and { F b } are such that F 5 E; then K‘ 5 K, where K and
K’ are the cardinal numbers of A and B, respectively If F - E and each Fb is cyclic, then K = K‘
Prooj Let V be a partial isometry in W such that V*V = F and VV* =
E , I E Then { V F b V * ] is an orthogonal family of non-zero projections with
union Eo and cardinal number K’ We may assume that F I E
If x, is a unit generating vector for E, under B’ and Fbx, = 0, then (0) =
[9’Fbx,] = [F,.%”x,].WiththeassumptionF I E,F,x, # Oforsomeain A;
otherwise Fb annihilates the range of each E,, hence of E, and Fb = 0 con- trary to hypothesis Thus, if ya = { h : h E B, F b X , # 01, B = uucw y,
Now, CheBIIF:bX,lIZ 5 ll.xul12 = 1 ; so that F b X , # 0 for at most a countable
number of elements h of B, and .Yi has cardinal number at most KO As
B = U u s a Y, h” I K KO = K H
Trang 24416 6 COMPARISON THEORY OF PROJECTIONS
6.3.10 PROPOSITION If G is a j n i t e projection in a von Neumann algebra
9 and CG is countably decomposable relative to the center %? of 9, then G is the sum of a countable number of cyclic projections in 9 In particular, G is count-
ably decomposable
Proof Let Go be a non-zero cyclic subprojection of G Let {G,} be an orthogonal family of subprojections of G maximal with respect to the property that each G, - Go The family {C,} is finite; for otherwise it could be put into one-to-one correspondence with a proper subfamily Using Proposition 6.2.2,
G would be equivalent to a proper subprojection-contradicting the hy-
pothesis that G is finite
If Go - G, I G - C G,, adjoining G I to { G,} contradicts the maximality
of {C,} Thus Go $ G - C G, Using the comparison theorem (Theorem
6.2.7), there is a (non-zero) central subprojection P of C , such that
P(G - 1 G,) < PGo
From Proposition 6.2.9, P(G - C G,) and each PG, are cyclic Thus PG is the sum of a finite number of cyclic projections
maximal with respect to the property that PbG is the sum of a finite number of cyclic projections If Po = CG - C P b and Po # 0, from the preceding, Po
contains a non-zero central subprojection P such that PG is the sum of a
finite number of cyclic projections By maximality of { P b } , Po = 0 Since CG is
countably decomposable in %?, {Pbj is a countable family As G = 1 PbG,
each PbG is the sum of a finite number of cyclic projections, and { P b ) is a countable family; G is the sum of a countable number of cyclic projections From Proposition 5.5.19, G is countably decomposable in 9 H
6.3.1 1 THEOREM (Generalized invariance of dimension) Zj G is a finite projection in a von Neumann algebra 9, {E,),sA and { F b ) , , , are orthog- onalfamilies of subprojections of a projection E in 9? maximal with respect to the property that E, - Fb - G for each a and b ; then A and B have the same cardinal number
Proof If G 6 E, E has no subprojection equivalent to G ; and both A and B are empty On the other hand, if G 5 E, then, by maximality, both A and IB have at least one element Assuming this, we can replace G (our “test” projection) by any E , and, so, assume that G I E Since two projections in
E B E are equivalent (relative to E B E ) if and only if they are equivalent in 92,
we may work in EWE and assume that E = 1 Since C, = C,, = C, (from Proposition 6.2.8), (E,) and {FJ are maximal families in BCG Again, working in RCG, we may assume that C, = CEa = CFb = 1
Trang 256.3 FINITE AND INFINITE PROJECTIONS 417
From Proposition 6.3.7, there is a central projection Q such that Q is finite
and I - Q is either 0 or properly infinite
(i) Suppose Q # 0 We show that A (and B) are finite Suppose the
contrary Then the orthogonal family {QE,},, of non-zero, equivalent
subprojections of Q can be put into one-to-one correspondence with a proper
subfamily Using Proposition 6.2.2, Q is, then, equivalent to a proper sub-
projection of itself-contradicting the fact that Q is finite Thus A (and B) are finite
the contrary In particular, then, I - Q # 0 and I - Q is properly infinite From Theorem 6.3.8, ( I - Q)E is finite; since E, - G, G is finite, E = xos A E,,
and A is a finite set If P is a non-zero central subprojection of I - Q,
P(=P(Z - Q)) is infinite, PE is finite, and P = P(I - E ) + PE Thus, from Theorem 6.3.8, P(I - E ) (= P(I - Q)(I - E)) is infinite; and ( I - Q ) ( I - E )
is properly infinite We are assuming that ( I - Q)G $ ( I - Q)(I - E )
(to reach a contradiction) From the comparison theorem, there is a (non- zero) central subprojection P of I - Q such that P(Z - Q)(I - E ) < PG
But PG is finite and P(I - Q)(I - E ) ( = P ( I - E)) is infinite, contradicting Proposition 6.3.2 Thus ( I - Q)G 5 ( I - Q)(I - E), and ( I - Q)G - El I
We note, next, that (QE,JoEA is maximal (as an orthogonal family of
subprojections of Q ) with respect to the property that each QE, - QG If this
is not the case, there is a subprojection E , of Q(I - E ) equivalent to QG From Proposition 6.2.2, G = QG + ( I - Q)G - E , + El I Q(I - E ) +
( I - Q)(I - E) = I - E Adjoining E , + El to {E,} contradicts the maxi-
mality of {E,}, so that {QE,),.,js maximal
It follows that QG $ Q(I - E ) ; and there is a central subprojection P of
Q such that P(I - E ) < PG If there are fewer elements in A than in B,
{PE,, P(I - E)} can be put in one-to-one correspondence with a subset of from Proposition 6.2.2, P is equivalent to a proper subprojection of itself
But P is a subprojection of Q and Q is finite Thus A cannot have fewer ele-
ments than B; and, symmetrically, B cannot have fewer elements than A
We assume, now, that Q = 0 Thus I is properly infinite We proved
in (i), with I - Q in place of I , that G 5 I - E, when A is a finite set, But then
(E,} is not maximal Thus A and B are infinite sets Let K and Ec' be their (respective) cardinal numbers
Since {E,} is maximal, there is a central projection Psuch that P(I - E ) <
PG Employing Proposition 6.2.2, P - PE, since PE, - PG for all a and
{PE,} is an infinite orthogonal Family Thus PF 5 PE, where F = x b E B Fb
If Po is a non-zero central subprqjection of P cyclic in the center of 9, then
( I - Q)(I - E)
( P F b } AS PE, - PFb, P(I - E ) < PG PFb, and P = PE -k P(I - E),
(ii)
Trang 26418 6 COMPARISON THEORY OF PROJECTIONS
PoG is the sum of a countable number n of cyclic projections in 9, from Proposition 6.3.10 As PoG - PoE,, each PoE, is the sum of n orthogonal
cyclic projections (from Proposition 6.2.9) The family of all these cyclic
projections in all P o E , has sum PoE and cardinal number K ( = n K ) As
yo F 5 Po E and { P o F,} has cardinal number K', from Lemma 6.3.9, K' < K
Symmetrically N I K'; and h' = K'
If 9 is B ( Z ) in the preceding theorem, and G is a minimal projection (see
Proposition 6.3.2), then {E,} and { F b } are familiesof minimal projections with sum I They correspond to orthonormal bases for %' The fact that these families have the same cardinal number establishes, again, that the dimension
of a Hilbert space is an invariant, independent of the orthonormal basis used in calculating it (see Theorem 2.2.10) The theorem draws its name from this application
6.3.12 PROPOSITION I f E is a properly infinite projection in a von Neumann algebra 9 and G is a j n i t e projection in @such that C , = C E , then
E is the sum of cr.family {G,} ofprojections equivalent to G ifeither of the follow-
ing c.onditions is satisjid:
(i) &? is a factor;
(ii)
Proof Let { E,} be an orthogonal family of subprojections of E maximal
with respect to the property that each E, - G As in the proof of Theorem
6.3.1 1, there is some (non-zero) central subprojection P of C , (= C E = CEn)
such that P(E - x E,) < PG and C PE, - PE Let I/ be a partial isometry
in W with initial projection x PE, and final projection PE Then { VPE, V*} is
an orthogonal family of projections with sum P E, each equivalent to P G
If B is a factor, P = I Taking VE,V* for G,, our assertion follows
If E is countably decomposable, {PE,} is a countably infinite family We relabel it {PE,) Let {Pb} be an orthogonal family of non-zero central sub-
projections of CG maximal with respect to the property that PbE is the sum ofacountablefamily {Gbn} ofprojectionsequivalent to PbG IfC, - C Pb # 0,
then, from the foregoing, there is a non-zero central subprojection Po of
C , - Pb such that Po E is the sum of a countable family of projections
equivalent to PoG Adjoining Po to { f h } contradicts the maximality of { f h )
Thus C, = C P, From Proposition 6.2.2, x, G,, ( = G , ) - 1, PbG =
C,G = G Moreover, G , = E
E is countably decomposable in 2
Conditions (i) and (ii) are curiously different restrictions One is a
cardinality restriction on E, and the other is a restriction on the center of
W With E properly infinite, it will take an infinite number of copies of G to
Trang 276.4 ABELIAN PROJECTIONS 419
sum to E over each non-zero central portion PE of E As P varies over an orthogonal family, the cardinal number of copies necessary may vary Condition (i) says that P cannot vary Condition (ii) says that as P varies only KO copies of G are needed to sum to PE
general von Neumann algebra The abelian projections provide such a
“translation” of minimal projections In terms of direct sums of factors, the abelian projections are those whose component projections in each factor are either Oor a minimal projection With this (heuristic) description in mind, the statements of the results of this section become transparent For working purposes, we must devise a general (“global”) characterization of abelian projections
6.4.1 DEFINITION A projection E in a von Neumann algebra W is said
to be an abelian projection in .I when E 9 E is abelian
6.4.2 PROPOSITION Each subprojection of an abelian projection in a uon Neumann aigebra 9 is the product of the abelian projection and a central
projection A projection in d is abelian if and onlj) if it is minimal in the class of
projections in A? with the same central carrier Each abelian projection in 92 is finite I f % is the center of W and E is an abelian projection in 9, then E 9 E = %E
Suppose E is abelian in 9 From Proposition 5.5.6, %E is the
center of EWE Since E B E is abelian, %E = E 9 E I f F is a subprojection in 2
of E, then F = EFE E E 2 E = %E Thus F = Co E = C C E = CE, where
C (= CoCE) E % and CC, = C Now F = C E = F 2 = C-E, SO that
(C - C2)E = 0
Since C - C 2 E % G W’, 0 = (C - C’)CE = C - C 2 , from Theorem 5.5.4
Thus C is a (normal) idempotent in %, so that C is a central projection It
follows that each subprojection in 9 of an abelian projection E has the form
PE with P a central projection If CF = C , and F = PE, then CE = CF I P ;
Proof
O 7
Trang 28420 6 COMPARISON THEORY OF PROJECTIONS
and PE = E = F Thus E is minimal in the class of projections in W with
central carrier C E , and, in particular, E is finite
Suppose, now, that E is minimal in the class of projections in W with
central carrier C E If G is a subprojection in 3 of E, then G I CG E If G <
C,E, then G + ( I - C,)E < E and G + ( I - C,)E has central carrier
C,-contradicting our present assumption Thus G = CGE It follows that each projection in EWE is in V E Since EWE is a von Neumann algebra (Corollary 5.5.7), it is generated by its projections (Theorem 5.2.2); and
EWE c %E Thus E is abelian
It follows that each non-zero abelian projection in a factor is a minimal projection in that factor
6.4.3 PROPOSITION A projection E is a minimal projection in a von Neumann algebra W acting on a Hilbert space X , ifand only ifEWE consists of scalar multiples of E If E is a minimal projection in 9, E is an abelian projection
in 9, CE is a minimal projection in the center V of B, BCE is a factor, and
B'E = B(E(2))
Proof If E is a projection in B and EWE consists of scalar multiples of E,
then, if F is a subprojection in 3 of E, F = EFE E EWE; so that F = aE As F
is a projection, F = E or F = 0; and E is a minimal projection in B
If E is a minimal projection in 9, then each projection in EWE is either E
or 0 Since EWE is a von Neumann algebra, it is generated by its projections Thus EWE consists of scalar multiples of E From Proposition 5.5.6, EWE is the commutant of B'E in B(E(3f')) Thus W'E = B ( E ( X ) ) From Proposition
5.5.5, W'CE is * isomorphic to B'E, so that B'C, is a factor From Proposition 5.5.6, again, B'CE has WC, as commutant on 9(CE(X)) Thus WCE is a factor Each central subprojection P of C E is a central projection in BCE Since WCE
is a factor, P is either 0 or C E Hence C E is a minimal projection in V As
E B E consists of scalar multiples of E, it is abelian, so that E is an abelian
projection in B H
The preceding argument establishes that a projection Q in the center %? of a
von Neumann algebra 3 is a minimal projection in V if BQ is a factor Con- versely, if Q is a minimal projection in %, 9 Q is a factor since each central projection in W Q is a subprojection of Q in % (hence, equal either to 0 or Q)
6.4.4 PROPOSITION I f E is a minimal projection in a von Neumann algebra 9 acting on a Hilbert space X and x is a unit vector in E ( X ) , then [ a x ]
is the range of a minimal projection E' in 99' and EE' is the one-dimensional
projection G in B ( 2 ) with x in its range
Trang 296.4 ABELIAN PROJECTIONS 42 I
Proof Since E is a minimal projection in 9, from Proposition 6.4.3,
9 ' E = B ( E ( Z ) ) From Proposition 5.5.5, the mapping T'CE 4 T'E is a
* isomorphism of B'CE onto &?'E Since G is a minimal projection in 9 ' E
(=B(E(%))), G = G'E, where G' is a minimal projection in .%?ICE As G'x = x, [Xx] G G'(%) Thus E' I G'and E' E .&"CE It follows that E' = G'
Thus E' is a minimal projection in %'ICE, hence in B'; and E'E = G
6.4.5 PROPOSITION If {Ea)atA is a family of abelian projections in a uon Neumann algebra 9 and {CEO> is un orthogonal family, then E, is an abelian
projection in 9
Proof Let E hex,,, E,.Then EAE = Co.a.tA E,AE, = CaEA E,AE,,
since E , AE, = E,CEa ACE., Ear = E, AE, CEa CEO = 0, if a # a', where the infinite sums are understood in the sense of strong-operator convergence over
the net of finite subsets of A Now,
E , AE, E, BE, = Ear BE,, E , AE,,
for all a, a' in A and A , B in 9, since both products are 0 if a # a', and E,.%E,
is abelian Thus EAE and EBE commute, and E is an abelian projection in 2
6.4.6 PROPOSITION Zf E and F are projections in a uon Neumann algebra
9, E is abelian, and
(i) E - F, then F is aheliun;
(ii) CE I C F , then E 5 F ;
(iii) C E = C, and F is abelian, then E - F
Proof (i) If V, A , and B are in 9 and V*V = E, VV* = F, then
since E B E is abelian Thus F B F is abelian ; and F is an abelian projection in B
If E $ F, then, from the comparison theorem, there is some (non-
zero) central subprojection P of CE such that PF < PE In this case, 0 #
PF - E l < PE, contradicting Proposition 6.4.2, since PE is an abelian pro- jection and CE1 = C p F = P = CpE Thus E 5 F
(iii) If CE = CF and F is abelian, then E - F, since E 5 F and F 6 E,
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6.4.8 PROPOSITION If E is an abelian projection in a von Neumann algebra .%’ and F is a projection in JR such that F 5 C E , then F is a sum of abelian projections
C E ,
C,E is a non-zero abelian projection with central carrier C, From Prop- osition 6.4.6, C,E - F , I F , and F , is a (non-zero) abelian projection Let
{ F , } be a maximal orthogonal family of non-zero abelian subprojections of F
Since F - C F , is a subprojection of C E , it is either 0 or dominates a non-zero abelian projection By maximality of {F,}, 0 = F - C Fa, and F is a sum of abelian projections H
Proof If F = 0, there is nothing to prove Assume F # 0 AS CF
Bibliography: [47,48]
6.5 Type decomposition
With the results of the preceding sections at our disposal, several ways of distinguishing among von Neumann algebras, algebraically, suggest them- selves Have they abelian projections? Have they non-zero finite projections? Have they infinite projections? The various combinations of affirmative and negative replies to these questions lead us to the “type” description of von Neumann algebras
6.5.1 DEFIN~T~ON A von Neumann algebra W is said to be of type I if it
has an abelian projection with central carrier I -of type I, if I is the sum of n
equivalent abelian projections If 2 has no non-zero abelian projections but
has a finite projection with central carrier I , then !9? is said to be of type II-
of type 11, if I is finite-of type 11, if I is properly infinite If W has no non- zero finite projections, 9 is said to be of type 111 H
A number of simple facts, related to the above definition, follow easily from the techniques developed in the earlier sections of this chapter If a von Neumann algebra W is of type I (or I,, or 11, or II,, or II,, or 111) the same is
true of W P for each non-zero central projection P in 9 If 9 is of type I,, it is also of type I; and, of course, if 9 is of type either 11, or 11, , it is also of type 11 However, a von Neumann algebra cannot be of more than one of the types 1, 11,111, nor can it be of both types 11, and 11, Slightly less obvious is the fact (established at the end of the proof of the following theorem) that if a von Neumann algebra is of both types I, and I,, then m = n
6.5.2 THEOREM (Type decomposition) If W is a von Neumann algebra acting on a Hilbert space there are (mutually orthogonal) centrul projections
P,, n not exceeding dim 2, P,,, P , _ , and P , , with sum I , niuximal with respect
Trang 316.5 TYPE DECOMPOSITION 423
to the properties that WP, is of tvpe I , or PI, = 0, 9 P C , is of type 11, or P,, = 0,
9 P C m is of type 11, or PCm = 0, and W P , is of tjipe 111 or P , = 0
Proof Let {E,} be a family of abelian projections maximal with respect
to the property that {CEa} is orthogonal From Propositions 6.4.5 and 5.5.3,
C E, is an abelian projection with central carrier CEy (= Pd) Thus, either
Pd = 0 or acting on P , ( Y ) , is a type I von Neumann algebra By maximality of {E,}, I - Pd has no non-zero abelian subprojections
From Proposition 6.3.7, there is a central subprojection P,, of I - Pd such that P,, is finite and I - Pd - P,, is either 0 or properly infinite Since .%Pel
has no non-zero abelian projections and P,, is finite, either P,, = 0 or 9 P c 1 is
a type 11, von Neumann algebra
Let {G,} be a family of finite subprojections of I - Pd - P,, maximal
with respect to the property that {CG8 1 is an orthogonal family From Lemma
6.3.6 and Proposition 5.5.3, C G, is a finite projection with central carrier
I: CGc (=P,,) By maximality of {Gcj, I - P, - P,, - PCm ( = P m ) has no non-zero finite subprojections in 9, so that either P , = 0 or B P m is a von
Neumann algebra of type 111 As I - P, - P,, is either 0 or properly infinite,
P,_ is either 0 or properly infinite If P,_ # 0, since P,, is the central carrier of
the finite projection C G, and PCm has no non-zero abelian subprojections,
BP,_ is of type 11,
It remains to show that Pd is the sum of a family { P,} of central projections
such that P, is the sum of n equivalent abelian projections Let {Q,} be an
orthogonal family of central subprojections of P, each of which is the sum of n
equivalent abelian projections E a j , j = 1,2, , n, where n is some cardinal
number not exceeding dim -X In addition let { Q , ) be maximal with respect
to the property of being orthogonal Since E,, - E,? and C j E,, = Q,,
CE,,, = CEaL = = Q,, from Proposition 5.5.3 Thus xo Eaj ( = E j ) is an abelian projection with central carrier 2, Q, (= P,), from Proposition 6.4.5 It follows, from Proposition 6.4.6(iii), that E , - E, - As P , = xy= E j ,
either P, is 0 or B P , is of type I,
Since 0 is an abelian projection and is the sum of n equivalent abelian projections, the preceding discussion envisages, as it must, the possibility that each Q, is 0 In this case P, = 0 The essence of the argument appears now
We show that Pd = ~ , l s d i m P I , If 0 # Pd - V P, ( = P), then, for each n,
P will fail to have a single non-zero central subprojection that is the sum of n equivalent abelian projections, by maximality of the family {Q,) used to define
P, Since P , is C , for some abelian projection E, P is the central carrier of the
abelian projection P E (= F) Let { F b ) be a maximal orthogonal family of
subprojections of P equivalent to F By maximality of { F b } , F 5 P - C F b ;
and, from the comparison theorem, there is a non-zero central subprojection
Po of P such that Po(P - Fb) = Po - PoFb - Fo < POF From
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Proposition 6.4.2, F , = CFoP,F = C,F; and F , is abelian If F , = 0,
then Po = Po F b , and {Po F b } is a family of equivalent abelian projections,
from Propositions 6.2.3 and 6.4.6(i) (since F b - F , for all b) If F , # 0, then
osition 6.2.8; and from Proposition 6.4.6(i), CFo is the sum CFo - CFoFb +
C CFoF, of equivalent abelian projections In any event, if P # 0, P has a non-zero central subprojection that is the sum of equivalent abelian sub-
projections-contradicting the maximality of {Q,} (for some n not exceeding
dim H) Thus P = 0, and P , = v P,
We note that P,P, = 0 if n # ni; so that v P, = c P , = Pd From the
preceding discussion, if 0 # P = P, P,, then P is the sum both of n and of m
equivalent abelian projections Each of these abelian projections is finite,
from Proposition 6.4.2, so that Theorem 6.3.1 1 applies, and n = m
Type I von Neumann algebras are sometimes called discrete von
Neumann algebras to indicate the fact that the identity can be decomposed
as a sum of central projections (the P , of Theorem 6.5.2) each of which is the
“discrete” sum of projections minimal with the given central projection as central carrier The type 11 von Neumann algebras, by contrast, are described
as continuous (In some of the literature, this description is applied to von Neumann algebras of type I11 as well.) This terminology is the basis for the notation P d , P,,, PCm In case P , = 0, the von Neumann algebra is sometimes
referred to as semijnite
PO - c PO F b = CF,(P, - c PO F b ) = CF, - CF, F b - F O 7 from Prop-
W
6.5.3 COROLLARY A factor ,I is either of type I,, or 11,, or 11,, or 111
I t is of type I ifit has a minimal projection-of type I, ifZ is the sum of n minimal projections If A# has no minimal projections but has a non-zerofinite projection,
it is of type 11-of type 11, if1 isfinite-of type 11, ifZ is infinite I f A has no non-zero Jinite projections, it is of type 111
6.5.4 REMARK Note that factors of type I, ( n finite) and type 11, are finite von Neumann algebras, while factors of the other types are
properly infinite von Neumann algebras (see Definition 6.3.1) We shall note
in Theorem 6.6.1 that factors of type I,, are * isomorphic to a(&?), where S
has dimension n, so that such factors have finite linear dimension when n is
finite On the other hand, the fact that a factor of type 11, has no minimal projections allows us to choose an infinite orthogonal family of non-zero projections in it It follows that each factor of type 11, has infinite linear dimension
The corollary that follows extends, to general von Neumann algebras and
abelian projections, the fact that each projection in B(X) is a sum of (mini-
mal) projections with one-dimensional ranges
Trang 336.5 TYPE DECOMPOSITION 425 6.5.5 COROLLARY If E and Eo are projections in a von Neumann algebra
9, CE = CEO, and Eo is abelian in 9, then there is (1 family {Qj} of central
projections in 9 with sum C E such that Q j E is the sum ofj equivalent abelian projections If 9 C E o is of type I,, then 1 j 5 n I n particular, ifEo is a minimal projection in W, then E is the sum of j equivalent minimal projections in W
From Proposition 6.4.8, E is a sum of abelian projections in
9-each of which is abelian in the von Neumann algebra E 9 E (= B0)
Arguing as in the first paragraph of the proof of Theorem 6.5.2, we conclude
that W o is of type I Applying Theorem 6.5.2 to a,, let p j be the central
projection in So such that either a0pj is of type Ij or p j = 0 Now p j is a (central) projection in Zo, and 9; = Z E , from Corollary 5.5.7 From
Proposition 5.5.5, the mapping T‘E -+ T‘CE is a * isomorphism of W’E onto
9‘CE Let Qj be the image of p j under this mapping Then Q j is a central
projection in 9 ‘ C E , hence in w’ and in 9 Moreover, QjE = pjjfor, in general, T‘CEE = T’E) By construction, either ai is 0 or QiE = Pi = F ,
+ + F j , where F,, , F j are equivalent (non-zero) abelian projections
in EBE As each abelian projection in EWE is abelian in 2 and equivalence
in E.9E persists in 9, { Q j ) serves as the required family of central projections
in 9 (Note for this that xi p j = E, so that Cj Qj = C,.)
If.%?cE(= 9CEo)isoftypeI,,thenC, = El + + E,,whereE,, , E,
are equivalent abelian projections in a C E From Proposition 6.4.6(iii), QjE, - F , Generalized invariance of dimension (Theorem 6.3.1 1) applied
to Qj and to the families {QjE,, , QjE,}, { F , , , F j , , F,} (this latter
family being an extension of { F l , , F j } to a maximal orthogonal family of
subprojections of Q j equivalent to F , ) tells us that n = m, so that j I n
(Recall, for this, that each abelian projection is finite, from Proposition 6.4.2.)
If E , is minimal in 9, then &’CEO( = 9C,) is a factor of type I,, for some n,
from Proposition 6.4.3 and Corollary 6.5.3 From the preceding (or from Proposition 6-43), E is a sum of (non-zero) abelian projections in B C E
Since 9 C E is a factor, each of these abelian projections is minimal in 9 C E ,
hence in 9 H
Proof:
We proved, in Lemma 6.3.3, that a properly infinite projection can be
“halved.” This result can be extended to assert divisibility into any number
of equivalent subprojections Keeping in mind the situation of projections in
B(%), we should not expect to be able to halve a projection whose range has
dimension 5 More generally, if W is a von Neumann algebra of type I, with n
odd, the identity in 9 cannot be halved The possibility of finer and finer subdivision of projections combined with the fact that we can pass to strong- operator limits suggests that (finite) divisibility can always be effected in a von Neumann algebra with no central portion of type I This assertion is the substance of the lemma that follows
Trang 34426 6 COMPARISON THEORY OF PROJECTIONS
6.5.6 LEMMA If E is a projection in a von Neumann algebra with no central portion of type I (equivalently, with no non-zero abelian projections), for
each positive integer n, there are n equivalent (orthogonal) projections with sum E
Since E is not abelian, there is a proper subprojection F of E in S
such that CF = C E (from Proposition 6.4.2) With Q the central carrier of
E - F in 9, it follows that QF has central carrier Q From Proposition 6.1.8,
Q F and E - F have non-zero equivalent subprojections F , and F , (and, of course F1F2 = 0) Now F , is not abelian in 3, so that, from what we have just
established, F , has two equivalent orthogonal non-zero subprojections M ,
and M 2 The equivalence of F , and F 2 provides us with two equivalent or-
thogonal non-zero subprojections, N , and N 2 , of F 2 , where M , - M 2 -
N 1 - N 2 Continuing in this way we produce 2” (and, hence, n) equivalent orthogonal non-zero subprojections of E
Let Y be the family of sets {.Pi, , F#} of n elements, where each 5
is an orthogonal family {EY)}aEA of subprqjections of E, each family is
indexed by A, E‘f’ - - E;’ for each a in A, and u3= 5 is an orthogonal family We define a partial ordering I of Y so that {Fir ,F,} I
{Pi, , %;} precisely when the indexing of the sets of the second family extends the indexing of those of the first family (which entails, in particular, that G %J for all j) Let {Fi, , Fn} be a maximal element of Y relative
to this ordering; and let E j be the union of the projections in 5 From Proposition 6.2.2 { E l , , En} is an equivalent orthogonal family of sub- projections of E If E - cy= , E, # 0, then it is not abelian in 9 From the
preceding paragraph, we can find an orthogonal family { F l , , F,,} of
equivalent non-zero subprojections of E - C;= E j , in that case Adjoining
F j to 4, we construct a set in Y properly larger than {P,, , F“} (relative
to the given partial ordering on 9) This would contradict the maximality of {.Fl, , F,,} so that E = x:j”= Ei
6.6.1 THEOREM If Ji is a type I,, factor, then A is * isomorphic to B(X), where X has dimension n
Trang 356 6 TYPE 1 ALGEBRAS 42 7
Proof: According to Corollary 6.5.3, I is the sum of n minimal projec-
tions in A If E is a minimal projection in A and x is a unit vector in its
range, [.XX] (= 2) is the range of a minimal projection E' in JK and A E ' is
B ( 2 ) from Propositions 6.4.3 and 6.4.4 Since 4' is a factor, ,A" is a factor and C, = I From Proposition 5.5.5, %HE' ( = B ( X ) ) is * isomorphic to
A(= A C E , ) As n minimal projections in A have sum I , there are n one- dimensional projections (corresponding to them) in &?(*) that have sum 1
Thus X is n-dimensional
In Section 2.6, Matrix representations, we discussed the concept of n x n matrices [ z b l o b E D whose entries are bounded operators z b on a Hilbert
space 2 We noted that these matrices acted (in the usual matrix fashion) as
linear operators on the direct sum 0 Xb( = .#) of n copies x b of 2 -the
n-fold direct sum of X with itself-and that some of the operators so ob- tained were bounded Each operator in g(#) arises in this way-that is, has
a matrix representation If n is finite, we saw that each such matrix cor- responds to a bounded operator, but that this is not the case if n is infinite
Suppose now that R is a von Neumann algebra acting on 2 If M Q %' is
the set of operators in B(#) whose matrix representations have each entry
in W and 9 Q I , is the set of operators in n Q W with all diagonal entries equal to the same operator in 9 and all other entries 0, we shall note that both sets are von Neumann algebras If cp is a * isomorphism of W with a von Neumann algebra 5 acting on a Hilbert space .Xi then n Q cp and cp Q I ,
are * isomorphisms of n Q W with n 0 Y and W Q I , with 5 Q I , , respec- tively, where (n Q V ) ( [ x b ] ) = [cp(T,,,)] for an operator in n Q 9 and
cp Q I , is the restriction of n Q cp to 9 Q I ,
6.6.2 LEMMA If 92 is u con Neumann algebru acting on a Hilbert space
2 and 9 is the n:fold direct sum of .g with itself, then n Q 92 and 9 0 I ,
acting on 2 are von Neumann algebras, W Q I , is * isomorphic to 9, and
(9?' @ I,)' = n Q 3 If .T is a von Neumann algebra acting on a Hilbert space
.W' irnd cp is a * isomorphism of.@ onto ,K then n Q cp is a * isomorphism Of'
n Q 9 unto n Q 5
Proof: The rules of (infinite) matrix multiplication, established in
Section 2.6 for the matrix representations of operators in B(.#), make it
apparent that
(i) (9' Q 1,)' = n Q B ;
(ii) (n @ 9')' = 4 @ I , ;
(iii) B Q I , is * isomorphic to 9
It follows from (i) that n Q W acting on 2 is a von Neumann algebra, and
from (ii) that @ @ I , is a von Neumann algebra
Trang 36428 6 COMPARISON THEORY OF PROJECTIONS
It is equally apparent, from the discussion in Section 2.6, that n Q cp is a
* isomorphism of n Q 9 onto n Q F when n is finite-so that n Q cp is an isometry, from Theorem 4.1.8(iii), in this case If n is infinite, this comment applies to the C*-(von Neumann) algebra of matrices whose entries outside
a given finite diagonal block are 0 From Proposition 2.6.13, [ q ( q b ) ] is a bounded operator on ,x the n-fold direct sum of X with itself, if [ K b ] E
n Q 9, since each finite diagonal block of [ V ( x , b ) ] has the same bound as that of the corresponding finite diagonal block in [ T a ] Thus [cp( T o , b ) ] is in
n Q F and has the same bound as [ K , b ] , so that n 0 cp is an isometric (adjoint-preserving) linear mapping of n Q W onto n Q F that carries the unit of n Q 9 onto that of n Q 5 Since, for a self-adjoint operator If,
0 I H I 21 if and only if 111 - HI( I 1, n Q cp maps an element in n Q 9
onto a positive element in n @ 5 if and only if that element is positive (That
is, n Q cp is an order isomorphism of n Q W onto n @ F.)
To conclude that (n 0 cp)(AB) = (n @ cp)(A)(n Q cp)(B) for each A and B
in n 0 9-and, hence, that n @ cp is a * isomorphism-we encounter the question of whether V ( x b o B q , b S b , c ) and x b e , f p ( q , b ) ( P ( S b , c ) are the same,
where A = [T,,,] and B = [ s o , , ] Both sums are strong-operator con- vergent The fact that cp is a * isomorphism does, indeed, imply this equality, for we shall prove (Corollary 7.1.16) that such mappings are strong-operator
continuous on bounded sets At this occasion, an easy ad hoc argument will
allow us to draw the desired conclusion
If x b E B q , b S b , c has only a finite number of non-zero terms, then
q ( x b T , b S b c ) = x b c p ( T , b ) c p ( s b , ) , since cp is an isomorphism Thus
(n 0 v)(W = (n 0 cp)(A)(n 0 cp)(B),
if either A has only a finite number of columns with non-zero entries or B has at most a finite number of rows with non-zero entries If E,, is the pro-
jection in n Q W with matrix [R,,,], where R b , b = I if b is in the finite subset
B, of B and R,,b = 0 for all other entries, then {E,,,} and {(n Q cp)(Eb)} are strong-operator convergent to I over the net of finite subsets of B Now
{A*E,,A} is a monotone increasing net with least upper bound A*A
Since n Q cp is an order isomorphism and
(n 0 v)(A*EQ3,4 = (n 0 cp)(A*)(n 0 cp)(E,,A)
= (n 0 cp)(A*)(n 0 cp)(E,,)(n 0 cp)(A),
{(n @ cp)(A*E,,A)} is a monotone increasing net with least upper bound
(n Q cp)(A*A) and (n Q cp)(A*)(n Q cp)(A) Thus
(n 0 cp)(A*A) = (n 0 cp)(A*)(n 0 cp)(A)
Substituting, successively, H + iK, H , K , and H + K for A, where H and K are self-adjoint operators in n Q 9, and combining, we conclude that
Trang 37For this purpose, we single out a system of n x n matrix units in &?, Such a system is a family ( E a , J a b E B of operators in -4’ such that & b E c , d = 0 if
b # c and Ea,,E,,d = where B has cardinal number n and z b e B Eb.*
is strong-operator convergent to I If, in addition, Ez,b = Eb,a, we say that the system is self-adjoint In any case, = Eb.b When the system is self-
adjoint, {Eb,bf is an orthogonal family of projections with sum I ; and each
is a partial isometry (with initial projection E b , b and final projection
In matrix terms, Ea, corresponds to the matrix with all entries 0 except
in position (a, b)-where the entry is I
6.6.3 LEMMA If ( E a & b E B is a self-adjoint system ofn x n matrix units for a von Neumunn algebra W acting on a Hilhert space X and 5 is the sub- algebra of &? consisting of those elements commuting with {En.,}, then, for each
T in W, CcEB E,,.TEb is strong-operutor convergent to an element K b of
Y, [ T b I a , b e B (= q ( T ) ) E n Q .z and cp is a * isomorphism of A’ onto n @ K
Moreover, Ea,a%?Ea,a is * isomorphic to T j o r each a
is a von Neumann algebra, since {En, b } is a self-adjoint family, and that SF is the intersection of
W with this commutant Thus F is a von Neumann algebra If B, is a finite
subset of B and x is a vector in H,
Proof: Note, first, that the commutant of
CEBO
It follows that the net {CceBo Ec.a TE,,,} of finite partial sums of
1 E c a T E b , c
C E B
has 11 TI1 as uniform bound and is Cauchy convergent in the strong-operator
topology From Proposition 2.5.1 1, it is strong-operator convergent to an
operator K , b in W Since multiplication by an operator is strong-operator COIltinUOUS On a(%), E c d K , b = Ec.a T E b d = x b E c d Thus T , b E
Trang 38430 6 COMPARISON THEORY OF PROJECTIONS
If B, is a finite subset of B with no elements, E,, = x b E B o Eb,b, and
qB,(EBoTEB,) = [T,,bla,bEBO, then cpB, is a * isomorphism of EBoWEB, onto no @ F To see this, note that EBoTEBo = Ca,bEBo Ea,,TEb., and EBC,&'E,, is a von Neumann algebra acting on E B O ( X ) If EBoTE,,, = 0,
then 0 = Ec,aEa,.E,,TE,,Eb~bEb~c = EC,.TEb,,, for all a and b in B, and c
in B Thus 0 = CcEB Ec,aTEb,c = G , b , for all a and b in 5,; and cpB, is well defined Linearity of cpB, is evident Moreover, Tu*.b = ZcEB T*Eb.,
= xcEB(Ec.bTEa,c)* Since CccB EC.bTEa,, converges to Tb,a in the weak-
as well as in the strong-operator topology, and the adjoint operation is weak-operator continuous on B ( f ) , (T*),, 6 = ( Th.n)* Thus q,, preserves
adjoints Now, (TE,,S)a.b = ZcsB Ec,a T ( x d E B o Ed.d)SEh.c; while the entry
in the a, b position of the product of [c,b],&B0 and [So.b]u.b~B~ is
F i n a l l y , i f A E F a n d a , b E B o , t h e n ~ c E B Ec,a(AEa.b)Eb,c = xCEB AE,,, = A,
so that cpso(AEa,b) has A at the a, b position and 0 at all other positions Thus qBo is a * homomorphism of EB{,9EB, onto no 0 F If Ta,b = 0 when
consisting of those matrices with 0 in the a, b position unless a and b are both
in B, Changing notation slightly, let pen now denote this restriction of cp, and ,YBo denote the subalgebra of n @ 3 Of course, ,F,, is a von Neumann algebra * isomorphic to no @ X
B) such that If .Y and y are vectors in X and E > 0 there is a finite set B,(
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Thus, with T and S in the unit ball of A‘,
5 c I((T - S ) E b , c x , Ea.cJ’>I + 2 1 I I E c c x I I I I E c , c Y I I
C E B O c 9 B o
It follows that the mapping assigning T,, to T is weak-operator continuous
on the unit ball of 9
Let $ ( = x b t B 0 .rV,) be the n-fold direct sum of .F with itself, and let
Fb be the projection of .# on r x b Since Q is isometric and U b s B ~ ~ ( $ 1
spans an everywhere-dense linear subspace of 2, weak-operator continuity
of cp on (W), will be established if we show that the mapping
-* ( d T ) F b x , F u J ’ )
is weak-operator continuous on (%’),, for all a, b in B and x, y in 2 This follows from the preceding paragraph, since (cp( T ) F b x , F,JJ) = (K.bx,, yo),
for some x o 7 yo in ~X (See the discussion following Theorem 5.1.2.) Now
~(4’) contains 3&,, for each finite subset B, of B, and the union of these is weak-operator dense in n 0 .K We conclude, with the aid of the Kaplansky density theorem, that q ( d ) = n 0 ST
It follows that cp is a * isomorphism from the fact that each qBo is and the fact that cp is weak-operator continuous on the unit ball of 9 Note that
cp(E,,,) is the matrix with I as the (a, h) entry and 0 as all other entries In
particular cp(E, ) is the projection whose matrix has I as the (a, a) entry and
0 as all other entries Of course, cp(E,,,)cp(.4’)cp(E,,,) ( = V ( E ~ , ~ & ’ E ~ , , ) ) is isomorphic to Y, so that E,,,,.JAE,,, is isomorphic to 5 H
6.6.4 LEMMA I f (Eb)beB is afuniify ofequiuatent projections with sum f
in u uon Neumann algebra 9, is a partial isometry in & with initial pro-
,jection E,,, unrljnul projectiori E,, trnd Ebo.h = €,,, then (E,, b , EZ.bo)o, b e is CI
.se!fadjoint sysrein qf’iiiatrix units,Jbr .JA
Proof: Note that E a b o E ~ , b o = E u b o E b o = E a b o Let E a b be E a b o E b * b o
Then ‘z h = ‘ b , hn ‘z bn = ‘ b a and ‘ a a = ‘ a bo ‘z bo = ‘u’ Thus c a e B ‘ 0 a
= I Note, too, that
Ea.bEc.d = E ~ b o E b n b E ~ b , r E b , d = Eu.boEbo.bEbE,Ec.boEbo.d = ‘7
unless b = c Finally,
Ea,c E c , h = Eu bc, Er bo ‘c bo ‘t bn = ‘ u bn Eb, El ho = ‘ a bo ‘t bo = ‘ a b ’
Thus (Ea,b)a,bcB is a self-adjoint system of matrix units for A’
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By representing the operators in B ( 2 ) as matrices of scalars relative to an
orthonormal basis for 2 (see Section 2.6, Matrix representations), Theorem
6.6.1 can be reformulated to state that a factor of type I,, is * isomorphic to
n 0 C Remarking that U2 is * isomorphic to the center of A, our factor, we arrive at a statement that applies, as well, to a von Neumann algebra of type
1,
6.6.5 THEOREM If 9 is a von Neumann algebra of type I, with center W and E is an abelian projection with central carrier I , then W is * isomorphic to both n Q %? and n Q (EWE)
Proof Since 9 is of type I,, there is a family ( & , ) b E B of equivalent abelian projections in W such that B has cardinal number n and Eb = I
Choose bo in B, let Ebo,bo be Ebo, and let E,,bo be a partial isometry in W with
initial projection E,, and final projection E, for each a different from b, Let E,,b be E,,boE~,bo From Lemma 6.6.4, (E,,b)o, is a self-adjoint system
of n x n matrix units for 9 From Lemma 6.6.3, A? is isomorphic to n 0 .%
where F is the algebra of elements in W commuting with {E,.b}
We show that F = % Clearly % E 3 Suppose A E 5 For each B in 9,
B = x , , b s B E,B&, It Will suffice to show that AE,BEb = E,BEbA Now,
AE, BEb = AE,, bEb Eb , BE, = a!?,* b Eb AEbEb ,BEb
= E,.bEbEb BEbEbAEb = E,BEbA,
since EbWEb is abelian and A commutes with {Ea,b}
Since CE = I , the mapping A' + A'E of 9 onto R E is a * isomorphism, from Proposition 5.5.5 Thus % and WE are * isomorphic Since E is abelian
in 9, EWE = %E From Lemma 6.6.2, n Q % and n @ (EWE) are * iso- morphic
If 9I is a finite-dimensional C*-algebra, its center W is * isomorphic to a finite-dimensional C ( X ) Thus % is the linear span of projections Q minimal
in W If '+a acts on .X: \u being finite-dimensional is weak-operator closed Thus 9IQ is a (finite-dimensional) factor (from the comments following Proposition 6.4.3) These considerations yield the result that follows
6.6.6 PROPOSITION Each ,finire-dimensional C*-algebra is a j n i t e direct
sum qJfhctors oj' (@ire) type I
Bibliography: [47,48, 561