Operator algebras and applications the abel symposium 2015

1 Introduction Rigid C-tensor categories arise as representation categories of compact groups andcompact quantum groups and also as part of the standard invariant of a finite indexsubfac

The Abel Symposium 2015

ABEL SYMPOSIA

Edited by the Norwegian Mathematical Society

More information about this series athttp://www.springer.com/series/7462

Sergey Neshveyev • Christian Skau

Toke M Carlsen

Department of Science and Technology

University of the Faroe Islands

Tórshavn, Faroe Islands

Nadia S LarsenDepartment of MathematicsUniversity of Oslo

Oslo, NorwaySergey Neshveyev

Department of Mathematics

University of Oslo

Oslo, Norway

Christian SkauDepartment of Mathematical SciencesNorwegian University of Scienceand Technology

Library of Congress Control Number: 2016945020

Mathematics Subject Classification (2010): 46Lxx, 37Bxx, 19Kxx

© Springer International Publishing Switzerland 2016

This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.

The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

Printed on acid-free paper

This Springer imprint is published by Springer Nature

The registered company is Springer International Publishing AG Switzerland

The Norwegian government established the Abel Prize in mathematics in 2002, andthe first prize was awarded in 2003 In addition to honoring the great Norwegianmathematician Niels Henrik Abel by awarding an international prize for outstandingscientific work in the field of mathematics, the prize shall contribute toward raisingthe status of mathematics in society and stimulate the interest for science amongschool children and students In keeping with this objective, the Niels Henrik AbelBoard has decided to finance annual Abel Symposia The topic of the symposiamay be selected broadly in the area of pure and applied mathematics The symposiashould be at the highest international level and serve to build bridges betweenthe national and international research communities The Norwegian MathematicalSociety is responsible for the events It has also been decided that the contributionsfrom these symposia should be presented in a series of proceedings, and SpringerVerlag has enthusiastically agreed to publish the series The Niels Henrik AbelBoard is confident that the series will be a valuable contribution to the mathematicalliterature.

Chair of the Niels Henrik Abel Board Helge Holden

v

Målet for vår vitenskap er på den ene side å oppnå nye resultater, og på den annen side å sammenfatte og belyse tidligere resultater sett fra et høyere ståsted.

Sophus Lie1

The Abel Symposium 2015 focused on operator algebras and the wide ramificationsthe field has spawned Operator algebras form a branch of mathematics that datesback to the work of John von Neumann in the 1930s Operator algebras wereproposed as a framework for quantum mechanics, with the observables replaced byself-adjoint operators on Hilbert spaces and classical algebras of functions replaced

by algebras of operators Spectacular breakthroughs by the Fields medalists AlainConnes and Vaughan Jones marked the beginning of an impressive development,

in the course of which operator algebras established important ties with other areas

of mathematics, such as geometry, K-theory, number theory, quantum field theory,dynamical systems, and ergodic theory

The first Abel Symposium, held in 2004, also focused on operator algebras It

is interesting to see the development and the remarkable advances that have beenmade in this field in the years since, which strikingly illustrate the vitality of thefield

The Abel Symposium 2015 took place on the ship Finnmarken, part of theCoastal Express line (the Norwegian Hurtigruten), which offered a spectacularvenue The ship left Bergen on August 7 and arrived at its final destination, Harstad

in the Lofoten Islands, on August 11 The scenery the participants saw on the waynorth was marvelous; for example, the ship sailed into both the Geirangerfjord andTrollfjord

There were altogether 26 talks given at the symposium In keeping with theorganizers’ goals, there was no single main theme for the symposium, but rather

a variety of themes, all highlighting the richness of the subject It is perhapsappropriate to draw attention to one of the themes of the talks, which is theclassification program for nuclear C-algebras In fact, a truly major breakthrough

1 “The goal of our science is on the one hand to obtain new results, and on the other hand to summarize and illuminate earlier results as seen from a higher vantage point.” Sophus Lie

vii

in this area occurred just a few weeks before the Abel Symposium 2015—amazingtiming! Some of the protagonists in this effort—one that has stretched over morethan 25 years and has involved many researchers—gave talks on this very topic atthe symposium The survey article by Wilhelm Winter in this proceedings volumeoffers a panoramic view of the developments in the classification program leading

up to the breakthrough mentioned above

Alain Connes and Vaughan Jones were also among the participants, and theygave talks on topics ranging, respectively, from gravity and the standard model inphysics to subfactors, knot theory, and the Thompson group, thus illustrating thebroad ramifications of operator algebras in modern mathematics

Ola Bratteli and Uffe Haagerup, two main contributors to the theory of operatoralgebras, tragically passed away in the months before the symposium Their legacywas commemorated and honored in a talk by Erling Størmer One of the articles

in this volume is by Uffe Haagerup, and its publication was made possible with thehelp of three of Haagerup’s colleagues from the University of Copenhagen, to whom

he had privately communicated the results shortly before his untimely passing.The articles in this volume are organized alphabetically rather than thematically.Some are research articles that present new results, others are surveys that cover thedevelopment of a specific line of research, and yet others offer a combination ofsurvey and research These contributions offer a multifaceted portrait of beautifulmathematics that both newcomers to the field of operator algebras and seasonedresearchers alike will appreciate

April 2016

C-Tensor Categories and Subfactors for Totally Disconnected

Groups 1

Yuki Arano and Stefaan Vaes

Decomposable Approximations Revisited 45

Nathanial P Brown, José R Carrión, and Stuart White

Exotic Crossed Products 61

Alcides Buss, Siegfried Echterhoff, and Rufus Willett

On Hong and Szyma ´nski’s Description

of the Primitive-Ideal Space of a Graph Algebra 109

Toke M Carlsen and Aidan Sims

Commutator Inequalities via Schur Products 127

Equilibrium States on Graph Algebras 171

Astrid an Huef and Iain Raeburn

Semigroup C-Algebras 185

Xin Li

Topological Full Groups of Étale Groupoids 197

Hiroki Matui

Towards a Classification of Compact Quantum Groups of Lie Type 225

Sergey Neshveyev and Makoto Yamashita

ix

A Homology Theory for Smale Spaces: A Summary 259

C -Tensor Categories and Subfactors for Totally Disconnected Groups

Yuki Arano and Stefaan Vaes

Abstract We associate a rigid C-tensor category C to a totally disconnected locally compact group G and a compact open subgroup K < G We characterize

when C has the Haagerup property or property (T), and when C is weakly amenable When G is compactly generated, we prove that C is essentially equivalent

to the planar algebra associated by Jones and Burstein to a group acting on a locallyfinite bipartite graph We then concretely realizeC as the category of bimodules

generated by a hyperfinite subfactor

1 Introduction

Rigid C-tensor categories arise as representation categories of compact groups andcompact quantum groups and also as (part of) the standard invariant of a finite indexsubfactor They can be viewed as a discrete group like structure and this analogy haslead to a lot of recent results with a flavor of geometric group theory, see [9,17,18,

25,26]

In this paper, we introduce a rigid C-tensor categoryC canonically associated with a totally disconnected locally compact group G and a compact open subgroup

K < G Up to Morita equivalence, C does not depend on the choice of K The

tensor categoryC can be described in several equivalent ways, see Sect.2 Here,

we mention that the representation category of K is a full subcategory of C and that

the “quotient” of the fusion algebra ofC by Rep K is the Hecke algebra of finitely supported functions on KnG =K equipped with the convolution product.

When G is compactly generated, we explain how the C-tensor categoryC is

related to the planar algebraP (i.e standard invariant of a subfactor) associated in

[5,11] with a locally finite bipartite graphG and a closed subgroup G < Aut.G / At

© Springer International Publishing Switzerland 2016

T.M Carlsen et al (eds.), Operator Algebras and Applications, Abel Symposia 12,

DOI 10.1007/978-3-319-39286-8_1

1

the same time, we prove that these planar algebrasP can be realized by a hyperfinite

subfactor

Given a finite index subfactor N  M, the notions of amenability, Haagerup property and property (T) for its standard invariant G N ;M were introduced by Popa

in [23,24] in terms of the associated symmetric enveloping algebra T  S (see [21,

23]) and shown to only depend onG N ;M Denoting byC the tensor category of M-bimodules generated by the subfactor, these properties were then formulated in

M-[26] intrinsically in terms ofC , and in particular directly in terms of G N ;M We recall

these definitions and equivalent formulations in Sect.4 Similarly, weak amenability

and the corresponding Cowling-Haagerup constant for the standard invariantG N ;M

of a subfactor N  M were first defined in terms of the symmetric enveloping

inclusion in [3] and then intrinsically for rigid C-tensor categories in [26], seeSect.5 Reinterpreting [1,6], it was proved in [26] that the representation category of

SUq.2/ (and thus, the Temperley-Lieb-Jones standard invariant) is weakly amenableand has the Haagerup property, while the representation category of SUq.3/ hasproperty (T)

For the C-tensor categoriesC that we associate to a totally disconnected group

G, we characterize when C has the Haagerup property or property (T) and when C

is weakly amenable We give several examples and counterexamples, in particular

illustrating that the Haagerup property/weak amenability of G is not sufficient for

C to have the Haagerup property or to be weakly amenable Even more so, when

C is the category associated with G D SL.2; Q p /, then the subcategory Rep K with

K D SL.2; Zp / has the relative property (T) When G D SL.n; Q p / with n  3,

the tensor categoryC has property (T), but we also give examples of property (T) groups G such that C does not have property (T).

Our main technical tool is Ocneanu’s tube algebra [19] associated with any rigid

C-tensor category, see Sect.3 When C is the C-tensor category of a totally

disconnected group G, we prove that the tube algebra is isomorphic with a canonical dense -subalgebra of C0.G/ ÌAdG, where G acts on G by conjugation We can

therefore express the above mentioned approximation and rigidity properties of thetensor category C in terms of G and the dynamics of the action G ÕAd G by

conjugation

In this paper, all locally compact groups are assumed to be second countable

We call totally disconnected group every second countable, locally compact, totallydisconnected group

2 C-Tensor Categories of Totally Disconnected Groups

Throughout this section, fix a totally disconnected group G For all compact open subgroups K1; K2 < G, we define

C1W the category of K1-K2-L1.G/-modules, i.e Hilbert spaces H equipped with

commuting unitary representations..k1//k12K1 and ..k2//k22K2 and with

a normal -representation˘ W L1.G/ ! B.H / that are equivariant with respect to the left translation action K1 Õ G and the right translation action

K2Õ G;

C2W the category of K1-L1.G=K2/-modules, i.e Hilbert spaces H equipped with

a unitary representation ..k1//k12K1 and a normal -representation˘ W

L1.G=K2/ ! B.H / that are covariant with respect to the left translation action K1Õ G=K2;

C3W the category of G-L1.G=K1/-L1.G=K2/-modules, i.e Hilbert spaces H

equipped with a unitary representation..g// g2G and with an L1.G=K1

/-L1.G=K2/-bimodule structure that are equivariant with respect to the left

translation action of G on G=K1and G=K2;

and with morphisms given by bounded operators that intertwine the given structure

Let K3 < G also be a compact open subgroup We define the tensor product

H ˝ K2K of a K1-K2-L1.G/-module H and a K2-K3-L1.G/-module K as the

Hilbert space

H ˝ K2K D f 2 H ˝ K j .k2/ ˝ .k2// D  for all k22 K2gequipped with the unitary representations..k1/ ˝ 1/k12K1 and .1 ˝ .k3//k32K3

and with the representation.˘H ˝ ˘K / ı  of L1.G/, where we denote by  W

L1.G/ ! L1.G/ ˝ L1.G/ the comultiplication given by .F//.g; h/ D F.gh/ for all g ; h 2 G.

The tensor product of a G-L1.G=K1/-L1.G=K2/-module H and a

G-L1.G=K2/-L1.G=K3/-module K is denoted as H ˝ L1.G=K2 / K and defined

as the Hilbert space

with the unitary representation.H g/ ˝  K g// g2G and with the L1.G=K1

/-L1.G=K3/-bimodule structure given by the left action of 1gK1˝ 1 for gK12 G=K1

and the right action of1 ˝ 1hK3for hK32 G=K3.

We say that objectsH are of finite rank

C1W ifH K2 WD f 2 H j .k2/ D  for all k22 K2g is finite dimensional; as

we will see in the proof of Proposition2.2, this is equivalent with requiringthat K1H is finite dimensional;

C2W ifH is finite dimensional;

C3W if1eK1H is finite dimensional; as we will see in the proof of Proposition2.2,this is equivalent with requiring thatH  1 eK2 is finite dimensional

Altogether, we get that C1 andC3 are C-2-categories In both cases, the cells are the compact open subgroups of G For all compact open subgroups

0-K1; K2< G, the 1-cells are the categories C i K1; K2/ defined above and C i K1; K2/

C i K2; K3/ ! C i K1; K3/ is given by the tensor product operation that we justintroduced Restricting to finite rank objects, we get rigid C-2-categories

Another typical example of a C-2-category is given by Hilbert bimodules over

II1factors: the0-cells are II1 factors, the1-cells are the categories BimodM1-M2 of

Hilbert M1-M2-bimodules and BimodM1-M2 BimodM2-M3 ! BimodM1-M3 is given

by the Connes tensor product Again, restricting to finite index bimodules, we get arigid C-2-category

Remark 2.1 The standard invariant of an extremal finite index subfactor N  M can

be viewed as follows as a rigid C-2-category There are only two 0-cells, namely

N and M; the 1-cells are the N-N, N-M, M-N and M-M-bimodules generated by the

subfactor; and we are given a favorite and generating1-cell from N to M, namely the N-M-bimodule L2.M/.

Abstractly, a rigid C-2-category C with only two 0-cells (say C and ),irreducible tensor units inCCCandC, and a given generating objectH 2 CC

is exactly the same as a standard -lattice in the sense of Popa [22, Definitions

1.1 and 2.1] Indeed, for every n  0, define H C;n as the n-fold alternating tensor

product ofH and H starting with H Similarly, define H ;nby starting withH

For0  j, define A 0j D End.H C;j / When 0  i  j < 1, define A ij  A 0j as

A ijWD 1i ˝ End.H.1/i ;ji / viewed as a subalgebra of A 0j D End.H C;j/ by writing

H C;j D H C;i H.1/i ;ji The standard solutions for the conjugate equations (see

Sect.3) give rise to canonical projections eC 2 End.H H / and e 2 End.H H /

an equivalence of C-2-categories preserving the generators Conversely given astandard -lattice G , by [22, Theorem 3.1], there exists an extremal subfactor

N  M whose standard invariant is G and we can define C as the C-2-category of

the subfactor N  M, generated by the N-M-bimodule L2.M/ as in the beginning of

this remark One can also defineC directly in terms of G (see e.g [14, Section 4.1]for a planar algebra version of this construction)

Thus, also subfactor planar algebras in the sense of [12] are “the same” as rigid

C-2-categories with two 0-cells and such a given generating object H 2 CC.For more background on rigid C-tensor categories, we refer to [16]

Proposition 2.2 The C-2-categories C1 and C3 are naturally equivalent In particular, fixing K1 D K2 D K, we get the naturally equivalent rigid C-tensor categories C 1;f K < G/ and C 3;f K < G/ Up to Morita equivalence,1these do not depend on the choice of compact open subgroup K < G.

Proof Using the left and right translation operatorsgandg on L2.G/, one checks

that the following formulae define natural equivalences and their inverses betweenthe categoriesC1,C2andC3

• C1 ! C2 W H 7! H K2, whereH K2 is the space of right K2-invariant vectors

and where the K1-L1.G=K2/-module structure on H K2 is given by restrictingthe corresponding structure onH

• C3 ! C2 W H 7! 1 eK1 H and where the K1-L1.G=K2/-module structure on

1eK1 H is given by restricting the corresponding structure on H

• C2! C3W H 7! L2.G/ ˝ K1H given by

f 2 L2.G/ ˝ H j  k1˝ .k1// D  for all k12 K1g

and where the G-L1.G=K1/-L1.G=K2/-module structure is given by the resentation g ˝ 1/g2G , multiplication with F ˝ 1 for F 2 L1.G=K1/ andmultiplication with.id ˝ ˘/.F/ for F 2 L1.G=K2/

rep-By definition, if H 2 C1 has finite rank, the Hilbert space H K2 is finitedimensional Conversely, ifK 2 C2andK is a finite dimensional Hilbert space,

then the corresponding objectH 2 C1 has the property that both K1H and H K2

are finite dimensional Therefore,H 2 C1 has finite rank if and only if K1H is a

finite dimensional Hilbert space A similar reasoning holds for objects inC3.

It is straightforward to check that the resulting equivalenceC1 $ C3preserves

tensor products, so that we have indeed an equivalence between the C-2-categories

C1andC3.

To prove the final statement in the proposition, it suffices to observe that for

all compact open subgroups K1; K2 < G, we have that L2.K1K2/ is a nonzero

finite rank K1-K2-L1.G/-module and that L2.G=.K1\ K2// is a nonzero finite rank

1 In the sense of [ 15 , Section 4], where the terminology weak Morita equivalence is used; see also [ 25 , Definition 7.3] and [ 18 , Section 3].

G-L1.G=K1/-L1.G=K2/-module, so that C i ;f K1 < G/ and C i ;f K2 < G/ are

The rigid C-2-categories C1 andC2 can as follows be fully faithfully embedded

in the category of bimodules over the hyperfinite II1 factor We construct thisembedding in an extremal way in the sense of subfactors (cf Corollary2.4)

To do so, given a totally disconnected group G, we fix a continuous action GÕ˛

P of G on the hyperfinite II1 factor P that is strictly outer in the sense of [27,

Definition 2.1]: the relative commutant P0\ P Ì G equals C1 Moreover, we should

choose this action in such a way that Tr ı˛g D .g/1=2Tr for all g 2 G (where

 is the modular function on G) and such that there exists a projection p 2 P of

finite trace with the property that ˛k p/ D p whenever k belongs to a compact subgroup of G Such an action indeed exists: write P D R0˝ R1where R0is a copy

of the hyperfinite II1factor and R1is a copy of the hyperfinite II1factor Choose acontinuous trace scaling actionRC

0 Õ˛ 1 R1 By [27, Corollary 5.2], we can choose

a strictly outer action GÕ˛ 0 R0 We then define˛gD ˛0/g˝ ˛1/.g/1=2and we

take p D 1˝p1, where p12 R1is any projection of finite trace Whenever k belongs

to a compact subgroup of G, we have .k/ D 1 and thus ˛ k p/ D p.

Whenever K1; K2< G are compact open subgroups of G, we write

ŒK1 W K2 D ŒK1W K1\ K2 ŒK2 W K1\ K21:Fixing a left Haar measure on G, we have ŒK1W K2 D .K1/ .K2/1 Therefore,

we have thatŒK W gKg1 D .g/ for all compact open subgroups K < G and all

g 2 G.

Theorem 2.3 Let G be a totally disconnected group and choose a strictly outer

action GÕ˛ P on the hyperfinite II1factor P and a projection p 2 P as above For every compact open subgroup K < G, write R.K/ D pPp/ K Then each R K/ is a copy of the hyperfinite II1factor.

To every K1-K2-L1.G/-module H , we associate the Hilbert R.K1/-R.K2 module K given by (2.1) below Then H 7! K is a fully faithful 2-functor Also,

/-bi-H has finite rank if and only if K is a finite index bimodule In that case,

dimR K1 /.K / D ŒK1W K21=2 dim

C1.H / and

dimR.K2/.K / D ŒK2W K11=2 dim

C1.H / ; where dim C1.H / is the categorical dimension of H 2 C1.

Proof Given a K1-K2-L1.G/-module H , turn H ˝ L2.P/ into a Hilbert P Ì K1

/-.P Ì K2/-bimodule via

u k   ˝ b/  u r D .k/.r/ ˝ ˛1

r b/ for all k 2 K1; r 2 K2;  2 H ; b 2 L2.P/;

a    d D ˘ ˝ id/˛.a/  1 ˝ d/ for all a; d 2 P;  2 H ˝ L2.P/;

where˛ W P ! L1.G/ ˝ P is given by ˛.a//.g/ D ˛1.a/.

Whenever K < G is a compact open subgroup, we define the projection p K 2

L G/ given by

p K D .K/1Z

K

k dk:

We also write e K D pp K viewed as a projection in P Ì K Since P  P Ì K  P Ì G,

we have that P0\ P Ì K/ D C1, so that P Ì K is a factor So, P Ì K is a copy of

the hyperfinite II1factor and e K 2 P Ì K is a projection of finite trace We identify

R K/ D e K PÌK/e Kthrough the bijective -isomorphism pPp/ K ! e K PÌK/e KW

a 7! ap K In particular, R K/ is a copy of the hyperfinite II1factor.

So, for every K1-K2-L1.G/-module H , we can define the R.K1/-R.K2bimodule

/-K D e K1 H ˝ L2.P//  e K2: (2.1)

We claim that EndR K1/R.K2 /.K / D End C1.H / naturally More concretely, we

have to prove that

End.PÌK1/.PÌK2/.H ˝ L2.P// D End C1.H / ˝ 1 ; (2.2)where EndC1.H / consists of all bounded operators on H that commute with

.K1/, .K2/ and ˘.L1.G// To prove (2.2), it is sufficient to show that

EndPP H ˝ L2.P// D ˘.L1.G//0˝ 1 : (2.3)Note that the left hand side of (2.3) equals.˘ ˝ id/˛.P/0\ B.H / ˝ P Assume that T 2 ˘ ˝ id/˛.P/0\ B.H / ˝ P In the same was as in [27, Proposition 2.7], it

follows that T 2 ˘.L1.G//0\1 For completeness, we provide a detailed argument

Define the unitary W 2 L1.G/ ˝ L.G/ given by W.g/ D  g We view both T and ˘ ˝ id/.W/ as elements in B.H / ˝ P Ì G/ For all a 2 P, we have

.˘ ˝ id/.W/ T ˘ ˝ id/.W/.1 ˝ a/ D ˘ ˝ id/.W/ T ˘ ˝ id/˛.a/ ˘ ˝ id/.W/

D 1 ˝ a/ ˘ ˝ id/.W/ T ˘ ˝ id/.W/:Since the action˛ is strictly outer, we conclude that ˘ ˝id/.W/ T ˘ ˝id/.W/D

S ˝ 1 for some S 2 B.H / So,

T D ˘ ˝ id/.W/.S ˝ 1/ ˘ ˝ id/.W/ : The left hand side belongs to B.H / ˝ P, while the right hand side belongs to B.H / ˝ L.G/, and both are viewed inside B.H / ˝ P Ì G/ Since P \ L.G/ D C1,

we conclude that T D T0˝ 1 for some T0 2 B.H / and that

T0˝ 1 D ˘ ˝ id/.W/.S ˝ 1/ ˘ ˝ id/.W/ :

Defining the normal -homomorphism W L.G/ ! L.G/˝L.G/ given by  g/ D

g˝ g for all g 2 G, we apply id ˝ and conclude that

It follows that T0commutes with˘.L1.G// and (2.2) is proven

It is easy to check thatH 7! K naturally preserves tensor products So, we have

found a fully faithful2-functor from C1to the C-2-category of Hilbert bimodulesover hyperfinite II1factors

To compute dimR.K2/.K /, observe that for all k 2 K1, r 2 K2 and g 2 G, we

have˛kgr p/ D ˛ kg p/ D ˛ g.˛g1kg p// D ˛ g p/ Therefore, as a right P Ì K2module, we have

g p/, where the Hilbert space L g WD ˘.1K1gK2/.K1H / comes with

the unitary representation..r// r2K2and where the right.P Ì K2/-module structure

.id ˝ ˛r /.V g / D V g .r/ ˝ 1/ for all r 2 K2 :

Then left multiplication with V gintertwines the right.P Ì K2/-module structure onthe Hilbert spaceL g ˝ L2 p g P/ with the right P Ì K2/-module structure given by

. ˝ b/  du r/ D  ˝ ˛1

r bd/ for all  2 L g ; b 2 L2 p g P/; d 2 P; r 2 K2:Therefore,

So, we have proved that

To make the connection with the categorical dimension ofH , it is useful to view

H as the image of a G-L1.G=K1/-L1.G=K2/-module H0 under the equivalence

of Proposition2.2 This means that we can viewH as the space of L2-functions

 W G ! H0with the property that.g/ 2 1 eK1  H0 1gK2 for a.e g 2 G The

L1.G/-module structure of H is given by pointwise multiplication, while the K1

K2-module structure onH is given by

.k    r/.g/ D .k/.k1gr1/ for all k 2 K1; r 2 K2; g 2 G :

With this picture, it is easy to see that

˘.1K1gK2/.H K2/ Š 1eK1 H0 1K1gK2:The map 7! Q with Q.g/ D .g/.g/ is an isomorphism between H and the space of L2-functions 0with the property that g1K1H01eK2for

a.e g 2 G The L1.G/-module structure is still given by pointwise multiplication, while the K1-K2-module structure is now given by

Also note that for every g 2 G, we have

g/2D ŒK2 W K1 dim.1K2g1K1 H0 1eK2/ dim.1eK1 H0 1K1gK2/ :Thus, whenever g/ ¤ 0, we have that g/  ŒK2 W K11=2 Since

dimR.K2/.K / D X

g2K1nG=K2

g/ ;

we conclude that there are only finitely many double cosets g 2 K1nG=K2for which

1K2g1K1 H0 1eK2is nonzero and for each of them, it is a finite dimensional Hilbertspace This implies thatH0 1eK2is finite dimensional, so thatH0has finite rank

We have proved thatH 7! K is a fully faithful 2-functor from C 1;f to thefinite index bimodules over hyperfinite II1 factors Moreover, for given compact

open subgroups K1; K2 < G, the ratio between dim R K1 /.K / and dim R.K2/.K /

equalsŒK1 W K2 for all finite rank K1-K2-L1.G/-modules H Since the functor

is fully faithful, this then also holds for all R.K1/-R.K2/-subbimodules of K It

follows that the categorical dimension ofK equals

ŒK2W K11=2dim

R K1 /.K / D ŒK1W K21=2 dim

R.K2 /.K / :

Since the functor is fully faithful, the categorical dimensions ofH 2 C 1;f and

K 2 Bimod f coincide, so that

ŒK2W K11=2 dim

R K1 /.K / D dim C1.H / D ŒK1W K21=2 dim

R.K2 /.K / :

(2.6)u

Corollary 2.4 Let G be a totally disconnected group with compact open subgroups

K˙ < G and assume that H is a finite rank G-L1.G=KC/-L1.G=K/-module Denote by C D CCC; CC; CC; C/ the C-2-category of G-L1.G=K˙/-

L1.G=K˙/-modules (with 0-cells KC and K) generated by the alternating tensor products of H and its adjoint.

Combining Proposition 2.2 and Theorem 2.3 , we find an extremal hyperfinite subfactor N  M whose standard invariant, viewed as the C-2-category of N-N, N-

M, M-N and M-M-bimodules generated by the N-M-bimodule L2.M/, is equivalent with C ; H / (cf Remark 2.1 ).

Proof A combination of Proposition2.2and Theorem2.3provides the finite index

R.KC/-R.K/-bimodule K associated with H Take nonzero projections p˙ 2

R.K˙/ such that writing N D pCR.KC/pC and M D pR.K/p, we have thatdimM pC K  p/ D 1 We can then view N  M in such a way that L2.M/ Š

pC  K  p as N-M-bimodules The C-2-category of N-N, N-M, N and

M-M-bimodules generated by the N-M-bimodule L2.M/ is by construction equivalent

with the rigid C-2-category of R.K˙/-R.K˙/-bimodules generated by K Since

the2-functor in Theorem2.3is fully faithful, this C-2-category is equivalent with

C and this equivalence maps the N-M-bimodule L2.M/ to H 2 CC uFrom Corollary2.4, we get the following result

Proposition 2.5 Let P be the subfactor planar algebra of [ 5 , 11 ] associated with

a connected locally finite bipartite graph G , with edge set E and source and target maps s W E ! VC, t W E ! V, together with2 a closed subgroup G < Aut.G / acting transitively on VCas well as on V Fix verticesv˙ 2 V˙and write K˙ DStabv˙.

There exists an extremal hyperfinite subfactor N  M whose standard invariant

is isomorphic with P We have ŒM W N D ı2where

#fe 2 E j s.e/ D w; t.e/ D vg ŒStab w W Stab v1=2:

2 Note that in [ 5 ], also a weight function W VCtV! R C

0 scaled by the action of G is part of the

construction But only when we take to be a multiple of the function v 7! ŒStab v W Stab v C  1=2,

we actually obtain a subfactor planar algebra, contrary to what is claimed in [ 5 , Proposition 4.1].

Moreover, P can be described as the rigid C-2-category C 3;f G; K˙; K˙/ of all finite rank G-L1.G=K˙/-L1.G=K˙/-modules together with the generating object

`2.E / 2 C 3;f G; KC; K/ (cf Remark 2.1 ).

Proof We are given G Õ E and G Õ VC, G Õ Vsuch that the source and target

maps s; t are G-equivariant and such that G acts transitively on VCand on V Put

K˙ D Stab v˙ and note that K˙ < G are compact open subgroups We identify G=K˙ D V˙ via the map gK˙ 7! g  v˙ In this way,H WD `2.E / naturally becomes a finite rank G-L1.G=KC/-L1.G=K/-module Denote by C the C-2-

category of G-L1.G=K˙/-L1.G=K˙/-modules generated by the alternating tensorproducts ofH and its adjoint.

In the2-category C3, the n-fold tensor product H ˝ H ˝    equals `2.E C;n/,whereE C;nis the set of paths in the graphG starting at an even vertex and having length n Similarly, the n-fold tensor product H ˝ H ˝    equals `2.E ;n/, where

E ;n is the set of paths of length n starting at an odd vertex So by construction,

under the equivalence of Remark 2.1, C together with its generator H 2 CC

corresponds exactly to the planar algebraP constructed in [5,11]

By Corollary2.4, we get that.C ; H / is the standard invariant of an extremal hyperfinite subfactor N  M In particular, ŒM W N D ı2 withı D dimC3.H /.

Combining (2.6) with (2.4), and using that

#fe 2 E j s.e/ D w; t.e/ D vg ŒStab w W Stab v1=2:

Combining (2.6) with (2.5), we similarly get that

ı D X

w2V

#fe 2 E j s.e/ D vC; t.e/ D wg ŒStab w W Stab vC1=2:

To conclude the proof of the proposition, it remains to show thatC is equal to

the C-2-category of all finite rank G-L1.G=K˙/-L1.G=K˙/-modules For the

G-L1.G=KC/-L1.G=K/-modules, this amounts to proving that all irreducible

representations of KC\ Kappear in

`2.paths starting at vCand ending atv/ :

Since the graph is connected, the action of KC\ Kon this set of paths is faithfuland the result follows The other cases are proved in the same way u

Remark 2.6 Note that the subfactors N  M in Proposition 2.5are irreducible precisely when G acts transitively on the set of edges and there are no multiple edges This means that the totally disconnected group G is generated by the compact open subgroups K˙< G and that we can identify E D G=.KC\ K/, V˙D G=K˙

with the natural source and target maps G =.KC\ K/ ! G=K˙ The irreducible

subfactor N  M then has integer index given by ŒM W N D ŒKCW KC\ K ŒKW

KC\ K

We finally note that the rigid C-tensor categoriesC 1;f K < G/ and C 3;f K < G/

also arise in a different way as categories of bimodules over a II1 factor in the

C4W the category of 1. /-modules, i.e Hilbert spaces H equipped with

two commuting unitary representations of 1. /that are covariant with respect to the left and right translation actions

 ;

C5W the category of 1

covariant with respect to the left translation action

with morphisms again given by bounded operators that intertwine the givenstructure

To define the tensor product of two objects inC4, it is useful to viewH 2 C4as afamily of Hilbert spaces.H g/g2 together with unitary operators.k/ W H g ! H kg

and.k/ W H g ! H gk1 for all k 2

product of two 1. /-modules H and K is then defined as

.H ˝ K / g Š M

h2

.H i h/ ˝ K i h/1g/ ;

but this isomorphism depends on the choice of the section

As in Proposition 2.2, C4 and C5 are equivalent C-categories, where theequivalence and its inverse are defined as follows

• C4! C5W H 7! K , with

K g D˚

.h/h2g ˇˇ h 2 H h; hk1 D .k/ h for all h 2 g 

again, this isomorphism depends on a choice of section

• C5 ! C4 W K 7! H , with H g D K g and the obvious 1. /-modulestructure

We say that an objectH 2 C5 has finite rank if H is a finite dimensional

Hilbert space This is equivalent to requiring that all Hilbert spacesH g are finitedimensional and that there are only finitely many double cosets g

is nonzero Similarly, we say that an objectH 2 C4 has finite rank if all HilbertspacesH gare finite dimensional and if there are only finitely many double cosets

gis nonzero Note here that an algebraic variant of the category

of finite rank objects inC4was already introduced in [29]

In this way, we have defined the rigid C-tensor categoryC 4;f

of the finite rank objects inC4 Note that, in a different context, this rigid C-tensor

need not be an equivalence of categories: an objectH 2 C5

with an object in the range of this functor if and only if the representation of

H is of the form k 7! ..k// for a (necessarily unique) continuous representation

 of K on H

Composing with the equivalence of categories in Proposition2.2, we have foundthe fully faithful C-tensor functor  W C3.K < G/ ! C4

finite rank objects to finite rank objects By construction, maps the G-L1

.G=K/-L1.G=K/-module L2.G=K/ ˝ L2.G=K/ (with G-action given by  g˝ g/g2Gand

obvious left and right L1 1. /-module `2. /.Next, given the outer action Õ˛ P, we write N D P

Consider the category Bimod.N/ of Hilbert N-N-bimodules We define the naturalfully faithful C-tensor functorC4

K D L2.P/ ˝ H and where the N-N-bimodule structure on K is given by

.au k /  b ˝ /  du r / D a˛ k b/˛ kh d/ ˝ .k/.r1/

for all a h By construction, this functor mapsthe 1. /-module `2. / to the N-N-bimodule L2.M/.

Denoting byC the tensor category of finite index N-N-bimodules generated by the finite index N-subbimodules of L2.M/, it follows that C is naturally monoidally

equivalent to the tensor subcategoryC0ofC 3;f K < G/ generated by the finite rank subobjects of L2.G=K/ ˝ L2.G=K/ So, it remains to prove that C0D C 3;f K < G/ Taking the n-th tensor power of L2.G=K/ ˝ L2.G=K/ and applying the equivalence

between the categories C 3;f K < G/ and C 2;f K < G/, it suffices to show that every irreducible K-L1.G=K/-module appears in one of the K-L1.G=K/-modules

L2.G=K/ ˝    ˝ L2.G=K/ with diagonal G-action and action of L1.G=K/ on the

last tensor factor Reducing with the projections1gK, this amounts to proving that

for every g 2 G, every irreducible representation of the compact group K \ gKg1appears in a tensor power of L2.G=K/ Because K < G is a Schlichting completion,

we have thatT

h2G hKh1D feg so that the desired conclusion follows. u

3 The Tube Algebra of C .K < G/

Recall from [19] the following construction of the tube -algebra of a rigid Ctensor categoryC (see also [9, Section 3] where the terminology annular algebra

-is used, and see as well [25, Section 3.3]) Whenever I is a full3family of objects in

C , one defines as follows the -algebra A with underlying vector space

Here and in what follows, we denote the tensor product inC by concatenation and

we denote by.ˇ; / the space of morphisms from  to ˇ By definition, all ˇ; /are finite dimensional Banach spaces Using the categorical traces Trˇ and Tr on.ˇ; ˇ/, resp .; /, we turn ˇ; / into a Hilbert space with scalar product

hV; Wi D Trˇ.VW/ D Tr.WV/ :For everyˇ 2 C , the categorical trace Trˇ is defined by using a standard solutionfor the conjugate equations forˇ, i.e morphisms sˇ 2 ˇˇ; "/ and tˇ 2 ˇˇ; "/satisfying

is the categorical dimension ofˇ

We will also make use of the partial traces

Trˇ˝id W ˇ˛; ˇ/ ! ˛; / W Trˇ˝id/.V/ D t

ˇ˝ 1/.1 ˝ V/.tˇ˝ 1/ :WheneverK is a Hilbert space, we denote by onb.K / any choice of orthonor-

mal basis inK The product in A is then defined as follows: for V 2 i˛; ˛j/ and

W 2 j0ˇ; ˇ; k/, the product V W equals 0 when j ¤ j0and when j D j0, it is equal to

for all V 2 i˛; ˛j/.

The -algebraA has a natural positive faithful trace Tr and for V 2 i˛; ˛j/, we

have that Tr.V/ D 0 when i ¤ j or ˛ ¤ ", while Tr.V/ D Tri V/ when i D j and

˛ D ", so that V 2 i; i/.

Up to strong Morita equivalence, the tube -algebraA does not depend on the choice of the full family I of objects in C , see [18, Theorem 3.2] and [25, Section7.2] Also note that for an arbitrary object˛ 2 C and i; j 2 I, we can associate with

V 2 i˛; ˛j/ the element in A given by

Formally allowing for infinite direct sums in C , one defines the C-tensorcategory of ind-objects inC Later in this section, we will only consider the rigid

C-tensor categoryC of finite rank G-L1.G=K/-L1.G=K/-modules for a given totally disconnected group G with compact open subgroup K < G In that case,

the ind-category precisely4is the C-tensor category of all G-L1.G=K/-L1

.G=K/-modules Whenever K1; K2 are ind-objects, we denote by .K1; K2/ the vector

space of finitely supported morphisms, where a morphism V W K2 ! K1 is said

to be finitely supported if there exist projections p iofK ionto a finite dimensionalsubobject (i.e an object inC ) such that V D p1V D Vp2

We say that an ind-objectH0inC is full if every irreducible object i 2 Irr.C / is

isomorphic with a subobject ofH0 We define the tube -algebra ofC with respect

to a full ind-objectH0as the vector space

˛2Irr.C /

.H0˛; ˛H0/

on which the -algebra structure is defined in the same way as above Note that

.H0; H0/ naturally is a -subalgebra of A , given by taking ˛ D " in the above

description ofA In particular, every projection of p of H0on a finite dimensionalsubobject ofH0can be viewed as a projection p 2 A These projections serve as

local units: for every finite subsetF  A , there exists such a projection p satisfying

p  V D V  p for all V 2 F.

Whenever p" is the projection ofH0 onto a copy of the trivial object ", we

identify p" A  p"with the fusion -algebraCŒC  of C , i.e the -algebra with

vector space basis Irr.C /, product given by the fusion rules and -operation given

by the adjoint object

To every full family I of objects in C , we can associate the full ind-object H0by

taking the direct sum of all i 2 I The tube -algebra of C associated with I is then

naturally a -subalgebra of the tube -algebra ofC associated with H0 If everyirreducible object ofC appears with finite multiplicity in H0, then this inclusion is

an equality and both tube -algebras are naturally isomorphic

For the rest of this section, we fix a totally disconnected group G and a compact open subgroup K < G We denote by C the rigid C-tensor category of all finite rank

G-L1.G=K/-L1.G=K/-modules, which we denoted as C 3;f K < G/ in Sect.2 Wedetermine the tube -algebraA of C with respect to the following full ind-object.

4 Using Proposition 2.2, every G-L1.G=K/-L1.G=K/-module is a direct sum of finite rank modules because every K-L1.G=K/-module is a direct sum of finite dimensional modules,

which follows because every unitary representation of a compact group is a direct sum of finite dimensional representations.

H0D L2.G  G=K/ with F  /.g; hK/ D F.gK/ .g; hK/ ;   F/.g; hK/ D .g; hK/ F.ghK/ and .x//.g; hK/ D .x1g ; hK/

(3.1)

for all 2 L2.G  G=K/, F 2 L1.G=K/, x; g 2 G, hK 2 G=K Note that every

irreducible object ofC appears with finite multiplicity in H0

We denote by.Ad g/ g2G the action of G on G by conjugation: Ad g/.h/ D ghg1 In the rest of this paper, we will make use of the associated full and reduced

C-algebras

C0.G/ Ì f

AdG and C0.G/ Ì r

AdG ;

as well as the von Neumann algebra L1.G/ ÌAdG We fix the left Haar measure

 on G such that .K/ D 1 We equip L1.G/ ÌAdG with the canonical normal

semifinite faithful trace Tr given by

half way between the left and the right Haar measure of G.

We consider the dense -algebra Pol.L1.G/ ÌAdG/ defined as

Pol.L1.G/ ÌAdG/ D spanf1U u x p L j U  G compact open subset ; x 2 G ;

L < G compact open subgroupg

We now identify the tube -algebra ofC with Pol.L1.G/ ÌAdG/ For every

x 2 G and every irreducible representation  W K \ xKx1 ! U K /, we denote

byH ; x/ 2 Irr.C / the irreducible G-L1.G=K/-L1.G=K/-module such that  is isomorphic with the representation of K \ xKx1on1xK  H ; x/  1 eK Note thatthis gives us the identification

Irr.C / D f.; x/ j x 2 KnG=K ;  2 Irr.K \ xKx1/g : (3.4)

We denote by the character of, i.e the locally constant function with support

K \ xKx1and.k/ D Tr..k// for all k 2 K \ xKx1.

Theorem 3.1 The G-L1.G=K/-L1.G=K/-module H0 introduced in (3.1) is full.

There is a natural -anti-isomorphism  of the associated tube -algebra A onto the -algebra Pol.L1.G/ ÌAdG / The -anti-isomorphism  is trace preserving Denoting by p"the projection in A that corresponds to the unique copy of the trivial object " in H0and identifying p" A  p"with the fusion -algebra of C , we have that  p"/ D 1K p K and that the restriction of  to CŒC  is given by

AdG ! B.K / is any nondegenerate -representation.

As follows, we associate with a unitary half braiding5on ind-C Whenever H

is a G-L1.G=K/-L1.G=K/-module, we consider a new G-L1.G=K/-L1

.G=K/-module with underlying Hilbert spaceK ˝ H and structure maps

K ˝H g/ D g/ ˝  H g/ ;

K ˝H F/ D ˝  H /.F/ ;

K ˝H F/ D 1 ˝  H F/ ; for all g 2 G, F 2 L1.G=K/, with .F/.g; hK/ D F.ghK/.

We similarly turnH ˝ K into a G-L1.G=K/-L1.G=K/-module with structure

Defining the unitary U 2 M.C0.G/ ˝ K.H // given by U.x/ D  H x/ for all

x 2 G and denoting by ˙ W K ˝ H ! H ˝ K the flip map, one checks that

5 Formally, a unitary half braiding is an object in the Drinfeld center of ind-C More concretely,

a unitary half braiding consists of an underlying ind-objectK1 together with natural unitary

isomorphismsH K1 !K1H for all objectsH We refer to [ 17 , Section 2.1] for further details.

the unitary˙ ˝ id/.U/ is an isomorphism between the G-L1.G=K/-L1

.G=K/-modulesK ˝ H and H ˝ K So, defining

K1WD K ˝ L2.G=K/ Š L2.G=K/ ˝ K ;

we have found the G-L1.G=K/-L1.G=K/-module K1 with the property that for

every G-L1.G=K/-L1.G=K/-module H , there is a natural unitary isomorphism

H W H K1 ! K1H :

Here and in what follows, we denote by concatenation the tensor product in the

category of G-L1.G=K/-L1.G=K/-modules So,  is a unitary half braiding for

ind-C

Using the ind-objectH0 defined in (3.1) and recalling thatK1H0 D K ˝ H0

as Hilbert spaces, we define the Hilbert space

K2D K ˝ H0; "/

and we consider the tube -algebraA associated with H0 Using standard solutionsfor the conjugate equations, there is a natural linear bijection

V 2 H0H ; H H0/ 7! eV 2 H H0; H0H /

between finitely supported morphisms

By [25, Proposition 3.14], using the partial categorical trace TrH˝id ˝ id, theunitary half braiding gives rise to a nondegenerate -anti-homomorphism  W

A ! B.K2/ given by

.V/ D Tr H ˝id ˝ id/.

H ˝ 1/.1 ˝ eV/. ˝ 1/ (3.6)for allH 2 C ,  2 K2and all finitely supported V 2 H0H ; H H0/

We now compute the expression in (3.6) more concretely Whenever h 2 G and

K0 < K is an open subgroup such that hK0h1  K, we define the finite rank

G-L1.G=K/-L1.G=K/-module L2.G=K0/h with underlying Hilbert space L2.G=K0/and structure maps

.x  /.gK0/ D .x1gK

0/ ; F1   F2/.gK0/ D F1.gK/ .gK0/ F2.gh1K/ : Note that there is a natural isomorphism L2.G=K0/h Š L2.G=K0/h1 Letting

K0 tend to feg, the direct limit of L2.G=K0/h1 becomes L2.G/ h1 Since H0 DL

h2G =K L2.G/ h1, we identify

H0D M

h2G =K

L2.G/ h

and we view L2.G=K0/h  H0 whenever h 2 G and K0 < K \ h1Kh is an open

subgroup

The Hilbert spaceK2equals the space of K-invariant vectors in1eK  K ˝H0/ 

1eK In this way, the space of K-invariant vectors in1eK  K ˝L2.G=K0/h/1eKurally is a subspace ofK2 But this last space of K-invariant vectors can be unitarily

nat-identified with 1Kh1p hK0h1/K by sending the vector 02 1Kh1p hK0h1/K

whenever K2< K is a small enough open subgroup such that  H k/ is the identity

onH  1 eK for all k 2 K2 Note that because H has finite rank, such an open subgroup K2exists Also, there are only finitely many x 2 G=K such that 1 xK H 1 eK

is nonzero Therefore, the sum appearing in (3.7) is finite

Applying this to the regular representation C0.G/ Ì f

AdG ! B.L2.G  G//, we

see that (3.7) provides a -anti-homomorphisms from A to Pol.L1.G/ ÌAdG/.

Then, a direct computation gives that is trace preserving, using the trace Tr on

L1.G/ ÌAdG defined in (3.2) In particular, is injective

We now prove that is surjective Fix elements g; h; ˛ 2 G satisfying ˛g D h˛ Choose any open subgroup K0 < K such that gK0g1, ˛K0˛1 and K

1 WD

h1˛K0˛1h are all subgroups of K Put H D L2.G=K0/˛ and note that H ,

L2.G=K0/g and L2.G=K1/h are well defined objects in C For every k 2 K, we

consider the vectors

L2.G=K0/g H 1eKis the partial isometry given by

1k ˛gK0˝ 1k ˛K07! .˛/1=2.h/1=21kh ˛K0˝ 1khK1 for all k 2 K:

A direct computation gives that.V/ is equal to a nonzero multiple of

1˛K0˛1h1u˛p gK0g1: (3.8)From (3.7), we also get that maps H0; H0/  A onto Pol.L1.KnG/ Ì K/,

defined as the linear span of all

1Kx u k p L

with x 2 G, k 2 K and L < K an open subgroup In combination with (3.8), itfollows that is surjective

Finally, by restricting (3.7) to the cases where g D h D e and K0D K1D K, we

We recall from [26] the notion of a completely positive (cp) multiplier on a rigid

C-tensor categoryC By [26, Proposition 3.6], to every function' W Irr.C / ! C

is associated a system of linear maps

˛'1jˇ1;˛2jˇ2W ˛1ˇ1; ˛2ˇ2/ ! ˛1ˇ1; ˛2ˇ2/ for all ˛i; ˇi 2 C (3.9)satisfying

Definition 3.2 ([ 26 , Definition 3.4]) LetC be a rigid C-tensor category.

• A cp-multiplier on C is a function ' W Irr.C / ! C such that the maps ˛jˇ;˛jˇ'

on.˛ˇ; ˛ˇ/ are completely positive for all ˛; ˇ 2 C

• A cp-multiplier' W Irr.C / ! C is said to be c0if the function' W Irr.C / ! C

tends to zero at infinity

• A cb-multiplier on C is a function ' W Irr.C / ! C such that

k'kcbWD sup

˛i;ˇi 2Ck ˛'1jˇ1;˛2jˇ2kcb< 1 :

A function' W Irr.C / ! C gives rise to the following linear functional !' W

A ! C on the tube algebra A of C with respect to any full family of objects

containing once the trivial object":

!' W A ! C W !'.V/ D

(

d.˛/ '.˛/ if V D1˛2 "˛; ˛"/,

0 if V 2 i˛; ˛j/ with i ¤ " or j ¤ ".

By [9, Theorem 6.6], the function' W Irr.C / ! C is a cp-multiplier in the sense of

Definition3.2if and only if!' is positive onA in the sense that !'.V  V#/  0

for all V 2 A In Proposition5.1, we prove a characterization of cb-multipliers interms of completely bounded multipliers of the tube -algebra

From Theorem3.1, we then get the following result We again denote byC be

the rigid C-tensor category of finite rank G-L1.G=K/-L1.G=K/-modules and we

identify Irr.C / as in (3.4) with the set of pairs.; x/ where x 2 KnG=K and  is an irreducible representation of the compact group K \ xKx1 In order to identify the

c0cp-multipliers onC , we introduce the following definition.

Definition 3.3 We say that a complex measure on G (i.e an element of C0.G/)

the complex measure x defined by x F/ D !.Fu x / for all F 2 C0.G/ is c0

and if the function G ! Cr G/ W x 7!  x/ tends to zero at infinity, i.e.limx!1k x/k D 0

Proposition 3.4 The formula

'.; x/ D ! p K dim./1u x p K/ (3.10)

gives a bijection between the cp-multipliers ' on Irr.C / and the positive functionals

! on the C-algebra q.C0.G/ Ì f G/q, where q D 1 K p K

The cp-multiplier ' is c0if and only if the positive functional ! is c0in the sense

Proof Note that the G-L1.G=K/-L1.G=K/-module H0 in (3.1) contains exactly

once the trivial module The first part of the proposition is then a direct consequence

of Theorem3.1and the above mentioned characterization [9] of cp-multipliers as

positive functionals on the tube -algebra The isomorphisms for C u C / and C r C /

follow in the same way

Fix a positive functional! on q.C0.G/Ì f

AdG /q with corresponding cp-multiplier ' W Irr.C / ! C given by (3.10) We extend! to C0.G/ Ì f

AdG by !.T/ D !.qTq/ For every x 2 G, define x 2 C0.G/given by x F/ D !.Fu x / for all F 2 C0.G/.

Note that x is supported on K \ xKx1and that ... (3.1) contains exactly

once the trivial module The first part of the proposition is then a direct consequence

of Theorem3. 1and the above mentioned characterization [9] of... transitively on the set of edges and there are no multiple edges This means that the totally disconnected group G is generated by the compact open subgroups K˙< G and that we... equivalence, the tube -algebraA does not depend on the choice of the full family I of objects in C , see [18, Theorem 3.2] and [25, Section7.2] Also note that for an arbitrary object˛ C and