amalgamation of inverse semigroups and operator algebras

Donsig We focus on three constructions: amalgamated free products of inverse groups, C*-algebras of inverse semigroups, and amalgamated free products of C*-algebras.. Although the order

3218333 2006

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by ProQuest Information and Learning Company

Steven P Haataja, Ph.D.

University of Nebraska, 2006

Advisors: John C Meakin and Allan P Donsig

We focus on three constructions: amalgamated free products of inverse groups, C*-algebras of inverse semigroups, and amalgamated free products of C*-algebras The starting point is an amalgam [S1, S2, U] of inverse semigroups that

semi-is full, i.e., the embeddings of U into S1 and S2 are bijective on the semilattice ofidempotents Although the order structure of the amalgamated free product is well-understood, the structure of the maximal subgroups was somewhat mysterious prior

to this work We use Bass-Serre theory to characterize these maximal subgroupsand determine which graphs of groups arise in this setting We obtain necessaryand sufficient conditions for the amalgamated free product to have trivial subgroups.One surprising consequence is that an amalgamated free product of finite inversesemigroups may be finite

We analyze the structure of the C*-algebra of an inverse monoid S using techniquesdeveloped by Sieben Let E be the semilattice of idempotents of S, and extend theMunn action of S on E to a partial action of S on C∗(E) We prove that C∗(S)

is isomorphic to the partial crossed product of C∗(E) and S using this action Togeneralize our construction to inverse semigroups, we determine the effect on C∗(S)

of attaching an identity to S Our construction simplifies the construction given byPaterson

Finally we consider C∗(S) when the inverse semigroup S is the amalgamated freeproduct of a full amalgam [S1, S2, U] We prove that the C*-functor commutes with

analogous result for the complex algebra of S Using the characterization of maximalsubgroups given above, we identify some amalgamated free products of C*-algebras

by recognizing them as C*-algebras of inverse semigroups Thus, we can identifycertain amalgams whose K-theory was found by McClanahan

ACKNOWLEDGEMENTSFirst I would like to thank everyone who provided much needed assistance in theweeks after my ice-skating accident I will be forever grateful to Jamie Radcliffe,Wendy Hines, Deanna Turk, Terri Moore, Mu-wan Huang, Ian Pierce, David Milan,Bob Ruyle, Martha Gregg, Marilyn Johnson, and my neighbor Jay Penner for allthe times they went out of their way for me during my two months in a wheelchair.

To my parents I can hardly express in words my gratitude for all of the love andsupport they have given me Even when my life took an abrupt turn, they werethere for me, either in person or in spirit I love you

I would like to thank the faculty at Black Hills State University for the enthusiasmfor learning I gained from them They are also primarily responsible for my teachingstyle, although I probably take enjoying teaching further than they would

The faculty at UNL have been wonderful With each different course I took, theirenthusiasm drew me in, providing me with yet another topic I wanted to study indetail My broad mathematical interests are no doubt due to them

I’d also like to thank the members of my committee for their questions, comments,suggestions and nitpicks My co-advisors John Meakin and Allan Donsig were theworst (ahem! best) nitpickers of them all It’s tough to learn two fields; but theymade it happen

Finally I must thank Tim and Natalie Sorenson, the best friends one could imagine.They have selflessly come through for me over the years Tim has gone above andbeyond the call many times, and it was Natalie who gently pushed me back to UNL

to pursue my Ph.D You two are the best

1.1 Semigroups and inverse semigroups 2

1.2 Green’s relations and examples 6

1.3 Categories and groupoids 12

1.4 Free products and amalgamated free products 16

1.5 Complex algebras and operator theory 18

1.6 Basic representation theory 25

1.7 C*-algebras of inverse semigroups 28

1.8 Essential K-theory 31

2 Maximal subgroups of full amalgamated free products 34 2.1 The basic idea of Bass-Serre theory 34

2.2 Amalgamation of groupoids 36

2.3 Structure of the maximal subgroups 38

2.4 Examples and applications 45

3 Constructions of the C*-algebra of an inverse semigroup 54 3.1 The covariance algebra of an inverse monoid 54

3.2 Crossed product decompositions of C∗(S) 62

3.3 Examples 66

4 C*-algebras of inverse semigroups and full amalgams 72 4.1 The complex algebra of a full amalgam 72

4.2 The C*-algebra of a full amalgam of inverse semigroups 75

4.3 Some K-theory calculations 77

Chapter 1

Preliminaries

Throughout N, Z and C denote the non-negative integers, the integers and thecomplex numbers, respectively All semigroups will be discrete and countable unlessotherwise specified The zero of a semigroup will usually be denoted θ Semilatticeswill always be meet semilattices

Function notation will depend upon context Semigroup theory and operatoralgebra theory have conflicting traditions regarding placement of the function symbol.Operator algebraists always place their functions on the left, so the image of x underthe function (operator) A is denoted A(x) or more simply Ax In semigroup theoryfunctions are usually placed on the right, so the image of x under the function ϕ

is denoted xϕ However, when the function corresponds to an action, the functionsymbol is placed on the appropriate side, so left (resp right) actions get the functionplaced on the left (resp right) This allows for efficient symbolism when both leftand right actions are present Hence when discussing inverse semigroups, functionswill generally go on the right, and when discussing operator algebras (including C*-algebras of inverse semigroups) functions will be placed on the left We will remindthe reader of this convention as needed

This chapter provides a brief overview of the mathematics that appears in thisthesis We give essential definitions, a minimal number of standard results, andprovide basic examples that will reappear in later chapters The only new materialoccurs in Section 1.6 There we provide a means of describing representations ofinverse semigroups on Hilbert space in terms of homomorphisms of inverse semigroups

1.1 Semigroups and inverse semigroups

This section contains basic results about semigroups with a focus toward resultsrelevant to inverse semigroups Further discussion about the ideas mentioned herecan be found in ([4], [5]) for semigroups and in ([22], [34]) for inverse semigroups

A semigroup is a set equipped with an associative binary operation A monoid

is a semigroup that possesses an identity Given a semigroup S, one can always find

a smallest monoid that contains S The smallest monoid containing S is usuallydenoted S1 and is defined as follows If S possesses an identity, S1 = S If S doesnot have an identity, then set S1 = S ∪ {1}, where 1 is a symbol not in S Themultiplication in S1 is given by s · t = st if s, t ∈ S and s · 1 = 1 · s = s for any s ∈ S1.Another way to attach an identity to a semigroup S is simply to add an identity

to S, regardless of whether S already has one The resulting set is denoted here by

S• := S∪ {1}, with multiplication given as above for S. 1

An idempotent of S is an element e such that e2 = e The set of idempotents of asemigroup is usually denoted E(S) or more simply as E, if the associated semigroup

is clear from context Associated to every idempotent e in S there is a subgroup

He of S given by He = {x ∈ S : ex = x and xy = yx = e, for some y ∈ S} Thesubgroups He of S, where e ∈ E(S), are the maximal subgroups of S If all themaximal subgroups of S are trivial, then S is said to be combinatorial

An element a of S is regular if there is a b in S such that aba = a If b also satisfiesbab = b, then b is called an inverse of a Note that if aba = a then bab is an inverse of

a, and both ab and ba are idempotents A semigroup is regular if every element has atleast one inverse A semigroup is an inverse semigroup if every element has a uniqueinverse In this case the inverse of an element a ∈ S is written a−1 or occasionally a∗.The latter notation occurs frequently in the operator algebra literature as the inverseoperation is an involution, that is, it satisfies (a−1)−1 = a and (ab)−1 = b−1a−1 forall a, b ∈ S We use the notation a−1 to denote the inverse of the element a ∈ Sthroughout

A small calculation shows that idempotents commute in an inverse semigroup.This leads to a useful characterization of inverse semigroups

Proposition 1.1.1 [4, Theorem 1.17] A semigroup S is inverse if and only if S isregular and any two idempotents of S commute

Thus if S is an inverse semigroup, E(S) forms a (meet) semilattice Also for any

s ∈ S and e ∈ E(S), both ses−1 and s−1es are idempotents.

An inverse semigroup has a natural partial order given by s ≤ t if there is anidempotent e such that s = et Verifying that this is a partial order is a routinecalculation Note that if s ≤ t as above, then s = et = e(tt−1t) = t(t−1et), so it iseasy to see that s ≤ t if and only if there is an f ∈ E(S) such that s = tf In fact,for any s, t ∈ S

s ≤ t ⇔ s = ss−1t ⇔ s = ts−1sExample 1.1.2 Let G be a group with identity ε, let n be a fixed positive integer,and let Bn(G) = {(i, g, j) : 1 ≤ i, j ≤ n, g ∈ G} ∪ {θ} Define a multiplication by

denote the n × n zero matrix, and set Bn(G) = {Ei,j(g) : 1 ≤ i, j ≤ n, g ∈ G} ∪ {0n},with usual matrix multiplication as the product (The product of the entry g ∈ G

by 0 is 0.) It is easy to see that Bn(G) ∼= Bn(G) by mapping (i, g, j) 7→ Ei,j(g) and

θ 7→ 0n We thus will describe nonzero elements of Bn(G) using either ordered triples

or matrices, depending upon circumstances

If G is the trivial group, the matrix notation can be simplified In this case, welet G = {1} and use the notation Ei,j instead of Ei,j(1) The elements of Bn can beviewed as lying in Mn(C)

One must be careful when discussing subobjects of classes of semigroups A semigroup of a semigroup S is a nonempty subset T of S closed under multiplication

sub-An inverse subsemigroup of an inverse semigroup S is a subset T of S closed bothunder multiplication and taking of inverses A subsemigroup of an inverse semigroupneed not be an inverse semigroup A (inverse) submonoid of a monoid M is a (inverse)subsemigroup of M that possesses the identity of M

Semigroups also possess ideals A left ideal of a semigroup S is a nonempty subset

I of S such that SI ⊆ I, where SI = {si : s ∈ S, i ∈ I} A nonempty subset I of S is

a right ideal if IS ⊆ I A (two-sided) ideal I is a subset that is both a left ideal andright ideal For a nonempty subset A of S, the ideal generated by A is the smallestideal of S containing A It is denoted by hAi

A semigroup is said to be simple if it contains no proper ideals If S has a zeroelement, then S is 0-simple if the only nontrivial ideal of S is {θ}

For an element a ∈ S, the principal right (resp left) ideal generated by a is thesmallest right (resp left) ideal of S containing a, namely the set aS1 (resp S1a).The (two-sided) principal ideal generated by a is likewise S1aS1 The notation S1

is employed, as a need not be an element of aS or Sa in general If S is regular orinverse, the relation aba = a ensures that a is an element of aS and Sa

Homomorphisms of semigroups are defined in the usual way: a homomorphism

ϕ : S → T is a set function such that for every s, t ∈ S, (st)ϕ = (sϕ)(tϕ) Theterms injection, surjection, and isomorphism have their usual meanings: one-to-onehomomorphism, onto homomorphism, and bijective homomorphism, respectively If

ϕ : S → T is surjective, then we say that T is a quotient of S and we may write S/ϕinstead of T

It should be noted that semigroup homomorphisms behave quite differently thangroup homomorphisms Every group homomorphism ϕ : G → H determines and

is determined by a normal subgroup of G, specifically the kernel of ϕ There is nosuch analogue in semigroup homomorphisms Instead semigroup homomorphismsare determined by the notion of a congruence A congruence on a semigroup S is

an equivalence relation ∼ on S such that for s, t, u ∈ S, if s ∼ t then su ∼ tu and

us ∼ ut If ∼ is a congruence on S the quotient of S relative to ∼ is written S/ ∼.Here is an example that shows how much congruences can differ from group cosets.Let S be a semigroup such that xy = y for all x, y ∈ S Let ∼ be any equivalencerelation on S For any x, y, z ∈ S, x ∼ y implies xz = z = yz and zx = x ∼ y = zy,

Then S/I is an semigroup, and the quotient homomorphism ϕ : S → S/I is given

by sϕ = θ if s ∈ I and sϕ = s otherwise The semigroup S/I is known as a Reesquotient

An important congruence in inverse semigroup theory is the minimum group gruence, denoted by σ, which is the smallest congruence containing the natural partialorder The quotient S/σ is the maximum group-homomorphic image of S Note that

con-if S has a zero, then S/σ is the trivial group If 1σ−1 := {s ∈ S : sσ = 1} = E(S),then S is said to be E-unitary Equivalently, S is E-unitary if whenever a ≥ e, forsome e ∈ E(S), then a ∈ E(S)

Analogous to the fact that every group can be faithfully represented as a group

of permutations (Cayley’s theorem), every inverse semigroup can be viewed as asemigroup of partial one-to-one functions A function α : X −→ X is partial one-to-one if the domain of α is a subset of X and α is one-to-one on its domain We denotethe domain of α by d(α) and the range of α by r(α) Given a nonempty set X, letSIM(X) denote the set of all partial one-to-one functions on X The composition of

f and g on X, denoted f g has as its domain the largest subset of d(f ) for which thecomposition f g makes sense Specifically the domain of f g is f−1(r(f ) ∩ d(g)) andits range is g(r(f ) ∩ d(g)) This is pictured below, with the maps going from left toright

g f

fg

Figure 1.1:

Observe that the idempotents are precisely the partial identity maps, i.e., theyare the identity maps on subsets of X The analog to Cayley’s theorem is as follows

Theorem 1.1.3 (Vagner-Preston)[4, Theorem 1.20]

Every inverse semigroup S is isomorphic to an inverse subsemigroup of SIM(S)

Much of the basic structure of an inverse semigroup can be determined by studyingfive equivalence relations first discovered by Green ([14], see also [4]) Here are four

of the Green’s relations

Definition 1.2.1 For a, b ∈ S, we define the relations R, L , J and H by

aRb ⇔ aS1 = bS1 aL b ⇔ S1a = S1b aJ b ⇔ S1aS1 = S1bS1 H = L ∩ R

If K is one of the Green’s relations, then the K -classes of S are the equivalenceclasses of S under the relation K The R-class containing the element a ∈ S isdenoted Ra, and the analogous notation is used for the other Green’s relations (Notethat we earlier used the notation He for the maximal subgroup associated to theidempotent e This is because Heis an H -class of S.) Clearly R ⊂ J and L ⊂ J ,

so R ∨ L , the smallest equivalence relation containing both R and L , is a subset

of J The nature of the relation R ∨ L is determined by the following proposition,

“the first fortunate accident in semigroup theory” (original source unknown)

Proposition 1.2.2 [4, Lemma 2.1] The relations R and L commute In otherwords, given a, b ∈ S, there exists an element c ∈ S such that aRcL b if and only ifthere exists an element d ∈ S such that aL dRb

Hence R ◦ L = L ◦ R = R ∨ L Define the fifth Green’s relation D := R ◦ L Because of Proposition 1.2.2 the elements of a D-class can be visualized in an “egg-box” picture This is a rectangular array where R-related elements are placed inthe same row, L -related elements are placed in the same column, and H -relatedelements are placed in the same box Determining the Green’s relations of a givensemigroup is a typical step taken in understanding its structure

The relationships between the five Green’s relations are illustrated in the followingdiagram

D{{{{

{{{{

B B B B

L

C C C

||||

||||

H

In general D ( J , but they are equal in many cases For example, D = J whenever

S is finite An example of an inverse semigroup for which J 6= D appears at theend of Example 1.2.6 A simple semigroup necessarily contains a single J -class If

a semigroup S contains a single D-class, it is said to be bisimple If S has a zero, it

is called 0-bisimple if contains one nonzero D-class

Regular and inverse semigroups can be characterized by the Green’s relations.Proposition 1.2.3 [4, Lemma 2.1] A semigroup is regular if and only if every R-class and every L -class of S contains at least one idempotent A semigroup is inverse

if and only if every R-class and every L -class of S contains precisely one idempotent.Thus every D-class of an inverse semigroup S contains at least one maximalsubgroup If eDf , there is an isomorphism ϕ : He→ Hf given by sending a ∈ He toxax−1 ∈ Hf, where x is any (fixed) element in Rf ∩ Le Thus

Proposition 1.2.4 [4, Theorem 2.20] Any two maximal subgroups of an inversesemigroup S that lie in the same D-class are isomorphic

Thus we can associate a group (up to isomorphism) to any D-class D of an inversesemigroup S, namely any maximal subgroup contained in D This group is called theSch¨utzenberger group of the D-class D

A useful property of products within a D-class is given in the following proposition.Proposition 1.2.5 [4, Theorem 2.17] Let a and b be elements of a semigroup S.Then ab ∈ Ra∩ Lb if and only if there is an idempotent in La∩ Rb

The elements a and b given in Proposition 1.2.5 must be D-related for the ditions to be satisfied It is important to observe that a, b and ab lying in the same

con-D-class does not imply that the H -class containing ab can be determined from the

H -classes of a and b We will see an example of this shortly

The Green’s relations on Bn (see Example 1.1.2) are easy to determine As(i, j)(j, k) = (i, k) and (i, k)E(k, j) = (i, j), we see that the nonzero R-classes of

S are Ri = {(i, j)|1 ≤ j ≤ n}, for 1 ≤ i ≤ n Likewise the L -classes have the form

Lj = {(i, j)|1 ≤ i ≤ n} (1 ≤ j ≤ n) The H -classes are singletons, and the nonzeroelements of Bn lie in a single D-class

Here is the egg-box picture for B3 with its semilattice of idempotents noted bythe dots and line segments:

(1, 1) (1, 2) (1, 3)(2, 1) (2, 2) (2, 3)(3, 1) (3, 2) (3, 3)

0

•

4 4 4 4 4 4 4 4

1 x−mxnRx−pxq if and only if m = p

2 x−mxnL x−pxq if and only if n = q

3 S is bisimple

4 x−mxnH x−pxq if and only if x−mxn = x−pxq, that is, S is combinatorial

5 The idempotents of S are the elements of the form x−nxn, n ∈ N

6 x−mxn≥ x−pxq if and only if both q ≥ n and n − m = q − p

The product of two elements given in canonical form is as follows, where i, j, k, l ∈ N.

The natural partial order runs down the diagonals of the egg-box, e.g., 1 > a−1a >

a−2a2 > a−3a3 > · · · The maximum group-homomorphic image of S is thus Z Also,since S is bisimple, all products lie in the same D-class Thus the product of a and

a−1a2, for example, lies in the same D-class as a and a−1a2, but it does not lie in

Ra∩ La− 1 a 2 because there is no idempotent in Ra∩ La− 1 a 2 (see Proposition 1.2.5)

As noted earlier, some inverse semigroups do not satisfy J 6= D We exhibit oneexample here Color the H -classes of S alternately white and black in a checkerboardpattern, with the H -class of 1 colored white Then the elements in the white colored

H -classes form an inverse semigroup T having one J -class and two D-classes EachD-class of T is isomorphic to the bicyclic, and the natural partial order in T bouncesback and forth between the two D-classes The bicyclic’s appearance inside T is not

an accident: every inverse semigroup for which J 6= D contains an isomorphic copy

of the bicyclic as an inverse subsemigroup This fact is a consequence of [4, Theorem2.54]

Example 1.2.7 We now discuss free inverse semigroups Inverse semigroups form

a class of algebras of type (2, 1) with the operations (·,−1), and are defined bythe identities x(yz) = (xy)z, xx−1x = x, (x−1)−1 = x, (xy)−1 = y−1x−1, and

xx−1yy−1 = yy−1xx−1 Thus by Birkhoff’s theorem (see [3, Theorem II§11.9]) freeinverse semigroups exist We denote the free inverse semigroup on the set X byFIS(X) Similarly inverse monoids form a class of algebras of type (2, 1, 0) with theoperations (·,−1, 1), and are defined by the preceding identities plus 1x = x1 = x

Hence free inverse monoids also exist; let F IM(X) denote the free inverse monoid onthe set X It can be shown that FIM(X) = FIS(X)•, so it suffices to describe freeinverse monoids.

Several construction of free inverse monoids exist in the literature; we provide abrief description of Munn’s construction ([28]) following the notation in ([24]) Let

X be a set, and let X−1 be a set disjoint from X of the same cardinality Formallythe elements in X−1 are denoted {x−1 : x ∈ X} Let F G(X) be the free group onthe set X, and let Γ(X) be the Cayley graph of F G(X) relative to (the generatingset) X The Cayley graph of F G(X) relative to X has as vertices the elements of

F G(X), and for each g ∈ F G(X) and x ∈ X there is an edge from g to gx labeled by

x In drawing the Cayley graph of a group, it is convenient to have circles representthe vertices and label vertices as needed

x

Here is a portion of the Cayley graph of F G({a, b})

aa

aa

b

bb

b

b

b1

Given a word w ∈ (X ∪ X−1)∗ (the set of all finite strings of letters in X ∪ X−1),form a pair (Γ, g) as follows The tree Γ is the subgraph of Γ(X) traversed by walkingthe path labeled by the letters in w This is called the Munn tree of w The element

g is the group-reduced form of w, or equivalently, the label of the shortest path inΓ(x) starting at 1 and ending at the same vertex as the path labeled by w

For example, let X = {a, b} and let w = aa−2ab2b−1abb−1 The reduced form w

is ba, and the tree Γ for w is given in Figure 1.2 below The vertex 1 has an arrow

a b b

The product of two elements (Γ, g) and (∆, h) is given by (Γ, g)(∆, h) = (Γ ∪ (g ·

∆), ρ(gh)), where g · ∆ is the translation of ∆ in Γ(X) by left multiplication by g,and ρ(gh) is the group-reduced form of the product gh The pairs (Γ, g) solve theword problem for FIM(X)

Theorem 1.2.8 ([28, Theorem 2.8]) Let X be a nonempty set Two words w1, w2 ∈(X ∪ X−1)∗ are equal in FIM(X) if and only if the subgraphs of Γ(X) traversed by w1

and w2 are equal and ρ(w1) = ρ(w2)

Thus equality in FIM(X) is determined by equality of the pairs (Γ, g) Manyproperties of F IM(X) can be deduced from this representation of its elements.Theorem 1.2.9 [28, Lemma 3.1 and Theorem 3.2] Let (Γ1, g1), (Γ2, g2) ∈ FIM(X).Then

1 (Γ1, g1) is an idempotent if and only if g1 = 1

2 (Γ1, 1) ≤ (Γ2, 1) if and only if Γ2 is a subgraph of Γ1 in Γ(X)

xx−1 x

x−1 x−1x

x2x−2 x2x−1 x2

xx−1 x−1x2x−1 x−1x2

x−2 x−2x x−2x2

... 0-direct union of a finite number ofsemigroups with zero In Chapters and we consider full amalgams of 0-directunions of Brandt semigroups

We study the structure of various algebras associated... monoids and monoid homomorphisms

InvSgp = inverse semigroups and inverse semigroup homomorphisms

InvMon = inverse monoids and inverse monoid homomorphisms

Gp = groups and group... representation of an inverse semigroup

Definition 1.6.4 A representation of an inverse semigroup S is a homomorphism

of inverse semigroups π : S → PI(P), where P is a set of commuting