Operator algebras and applications the abel symposium 2015

Abel Symposia 12

Toke M. Carlsen
Nadia S. Larsen
Sergey Neshveyev
Christian Skau Editors

Operator
Algebras and
Applications
The Abel Symposium 2015


ABEL SYMPOSIA
Edited by the Norwegian Mathematical Society

More information about this series at />

Participants at the Abel Symposium 2015.
Photo taken by Andrew Toms


Toke M. Carlsen • Nadia S. Larsen •
Sergey Neshveyev • Christian Skau
Editors

Operator Algebras and
Applications
The Abel Symposium 2015

123

Editors
Toke M. Carlsen
Department of Science and Technology
University of the Faroe Islands
Tórshavn, Faroe Islands

Christian Skau
Department of Mathematical Sciences
Norwegian University of Science
and Technology
Trondheim, Norway

Sergey Neshveyev
Department of Mathematics
University of Oslo
Oslo, Norway

ISSN 2193-2808
Abel Symposia
ISBN 978-3-319-39284-4
DOI 10.1007/978-3-319-39286-8

Nadia S. Larsen
Department of Mathematics
University of Oslo
Oslo, Norway

ISSN 2197-8549

(electronic)

ISBN 978-3-319-39286-8

(eBook)

Library of Congress Control Number: 2016945020
Mathematics Subject Classification (2010): 46Lxx, 37Bxx, 19Kxx
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Foreword

The Norwegian government established the Abel Prize in mathematics in 2002, and

the first prize was awarded in 2003. In addition to honoring the great Norwegian
mathematician Niels Henrik Abel by awarding an international prize for outstanding
scientific work in the field of mathematics, the prize shall contribute toward raising
the status of mathematics in society and stimulate the interest for science among
school children and students. In keeping with this objective, the Niels Henrik Abel
Board has decided to finance annual Abel Symposia. The topic of the symposia
may be selected broadly in the area of pure and applied mathematics. The symposia
should be at the highest international level and serve to build bridges between
the national and international research communities. The Norwegian Mathematical
Society is responsible for the events. It has also been decided that the contributions
from these symposia should be presented in a series of proceedings, and Springer
Verlag has enthusiastically agreed to publish the series. The Niels Henrik Abel
Board is confident that the series will be a valuable contribution to the mathematical
literature.
Chair of the Niels Henrik Abel Board

Helge Holden

v



Preface

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 ramifications
the field has spawned. Operator algebras form a branch of mathematics that dates
back to the work of John von Neumann in the 1930s. Operator algebras were
proposed as a framework for quantum mechanics, with the observables replaced by
self-adjoint operators on Hilbert spaces and classical algebras of functions replaced
by algebras of operators. Spectacular breakthroughs by the Fields medalists Alain
Connes 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 been
made in this field in the years since, which strikingly illustrate the vitality of the
field.
The Abel Symposium 2015 took place on the ship Finnmarken, part of the
Coastal Express line (the Norwegian Hurtigruten), which offered a spectacular
venue. 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 way
north was marvelous; for example, the ship sailed into both the Geirangerfjord and
Trollfjord.
There were altogether 26 talks given at the symposium. In keeping with the
organizers’ 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 perhaps
appropriate to draw attention to one of the themes of the talks, which is the
classification 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

viii

Preface

in this area occurred just a few weeks before the Abel Symposium 2015—amazing
timing! Some of the protagonists in this effort—one that has stretched over more
than 25 years and has involved many researchers—gave talks on this very topic at
the symposium. The survey article by Wilhelm Winter in this proceedings volume
offers 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 they
gave talks on topics ranging, respectively, from gravity and the standard model in
physics to subfactors, knot theory, and the Thompson group, thus illustrating the
broad ramifications of operator algebras in modern mathematics.
Ola Bratteli and Uffe Haagerup, two main contributors to the theory of operator
algebras, tragically passed away in the months before the symposium. Their legacy
was 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 the
help 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 the
development of a specific line of research, and yet others offer a combination of
survey and research. These contributions offer a multifaceted portrait of beautiful
mathematics that both newcomers to the field of operator algebras and seasoned
researchers alike will appreciate.
Tórshavn, Faroe Islands
Oslo, Norway

Oslo, Norway
Trondheim, Norway
April 2016

Toke M. Carlsen
Nadia S. Larsen
Sergey Neshveyev
Christian Skau


Contents

C -Tensor Categories and Subfactors for Totally Disconnected
Groups .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Yuki Arano and Stefaan Vaes

1

Decomposable Approximations Revisited . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Nathanial P. Brown, José R. Carrión, and Stuart White

45

Exotic Crossed Products.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Alcides Buss, Siegfried Echterhoff, and Rufus Willett

61

´
On Hong and Szymanski’s

Description
of the Primitive-Ideal Space of a Graph Algebra . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 109
Toke M. Carlsen and Aidan Sims
Commutator Inequalities via Schur Products . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 127
Erik Christensen
C -Algebras Associated with Algebraic Actions.. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 145
Joachim Cuntz
A New Look at C -Simplicity and the Unique Trace Property
of a Group .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 161
Uffe Haagerup
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


x

Contents

A Homology Theory for Smale Spaces: A Summary. . . . .. . . . . . . . . . . . . . . . . . . . 259
Ian F. Putnam
On the Positive Eigenvalues and Eigenvectors of a Non-negative
Matrix . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 271

Klaus Thomsen
Classification of Graph Algebras: A Selective Survey . . .. . . . . . . . . . . . . . . . . . . . 297
Mark Tomforde
QDQ vs. UCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 321
Wilhelm Winter


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 locally
finite bipartite graph. We then concretely realize C as the category of bimodules
generated by a hyperfinite subfactor.

1 Introduction
Rigid C -tensor categories arise as representation categories of compact groups and
compact quantum groups and also as (part of) the standard invariant of a finite index
subfactor. They can be viewed as a discrete group like structure and this analogy has
lead 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 category C 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 category C 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 of C 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 category C is
related to the planar algebra P (i.e. standard invariant of a subfactor) associated in
[5, 11] with a locally finite bipartite graph G and a closed subgroup G < Aut.G /. At

Y. Arano
Graduate School of Mathematics, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, Japan
e-mail:
S. Vaes ( )
KU Leuven, Department of Mathematics, Leuven, Belgium
e-mail:
© 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


2

Y. Arano and S. Vaes

the same time, we prove that these planar algebras P 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 GN;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 on GN;M . Denoting by C the tensor category of MM-bimodules generated by the subfactor, these properties were then formulated in

[26] intrinsically in terms of C , and in particular directly in terms of GN;M . We recall
these definitions and equivalent formulations in Sect. 4. Similarly, weak amenability
and the corresponding Cowling-Haagerup constant for the standard invariant GN;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], see
Sect. 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 amenable
and has the Haagerup property, while the representation category of SUq .3/ has
property (T).
For the C -tensor categories C 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; Qp /, then the subcategory Rep K with
K D SL.2; Zp / has the relative property (T). When G D SL.n; Qp / with n
3,
the tensor category C 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/ ÌAd G, where G acts on G by conjugation. We can
therefore express the above mentioned approximation and rigidity properties of the
tensor 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, totally
disconnected 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
C1 W

the category of K1 -K2 -L1 .G/-modules, i.e. Hilbert spaces H equipped with
commuting unitary representations . .k1 //k1 2K1 and . .k2 //k2 2K2 and with


C -Tensor Categories and Subfactors for Totally Disconnected Groups

C2 W

C3 W

3

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
K2 Õ G;
the category of K1 -L1 .G=K2 /-modules, i.e. Hilbert spaces H equipped with
a unitary representation . .k1 //k1 2K1 and a normal -representation ˘ W
L1 .G=K2 / ! B.H / that are covariant with respect to the left translation
action K1 Õ G=K2 ;
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=K1 and 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 ˝K2 K of a K1 -K2 -L1 .G/-module H and a K2 -K3 -L1 .G/-module K as the
Hilbert space
H ˝K2 K D f 2 H ˝ K j . .k2 / ˝ .k2 // D

for all k2 2 K2 g

equipped with the unitary representations . .k1 / ˝ 1/k1 2K1 and .1 ˝ .k3 //k3 2K3
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 G1
L .G=K2 /-L1 .G=K3 /-module K is denoted as H ˝L1 .G=K2 / K and defined
as the Hilbert space
H ˝L1 .G=K2 / K
D f 2 H ˝ K j .1gK2 ˝ 1/ D .1 ˝ 1gK2 /
M
H 1gK2 ˝ 1gK2 K
D

for all gK2 2 G=K2 g

g2G=K2

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 gK1 2 G=K1
and the right action of 1 ˝ 1hK3 for hK3 2 G=K3 .
We say that objects H are of finite rank
C1 W
C2 W
C3 W

if HK2 WD f 2 H j .k2 / D for all k2 2 K2 g is finite dimensional; as
we will see in the proof of Proposition 2.2, this is equivalent with requiring
that K1 H is finite dimensional;
if H is finite dimensional;
if 1eK1 H is finite dimensional; as we will see in the proof of Proposition 2.2,
this is equivalent with requiring that H 1eK2 is finite dimensional.


4

Y. Arano and S. Vaes

Altogether, we get that C1 and C3 are C -2-categories. In both cases, the 0cells are the compact open subgroups of G. For all compact open subgroups
K1 ; K2 < G, the 1-cells are the categories Ci .K1 ; K2 / defined above and Ci .K1 ; K2 /
Ci .K2 ; K3 / ! Ci .K1 ; K3 / is given by the tensor product operation that we just
introduced. 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
II1 factors: the 0-cells are II1 factors, the 1-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 a
rigid 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 generating 1-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 in CCC and C , and a given generating object H 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 HC;n as the n-fold alternating tensor
product of H and H starting with H . Similarly, define H ;n by starting with H .
For 0 Ä j, define A0j D End.HC;j /. When 0 Ä i Ä j < 1, define Aij
A0j as
Aij WD 1i ˝ End.H. 1/i ;j i / viewed as a subalgebra of A0j D End.HC;j / by writing
HC;j D HC;i H. 1/i ;j i . 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 /
given by
eC D d.H / 1 sH sH

and e D d.H / 1 tH tH ;

and thus to a representation of the Jones projections ej 2 Akl (for k < j < l).
Finally, if we equip all Aij with the normalized categorical trace, we have defined a
standard -lattice in the sense of [22, Definitions 1.1 and 2.1]. Given two rigid C 2-categories with fixed generating objects as above, it is straightforward to check
that the associated standard -lattices are isomorphic if and only if there exists
an equivalence of C -2-categories preserving the generators. Conversely given a
standard -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 define C 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].


C -Tensor Categories and Subfactors for Totally Disconnected Groups

5

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 C1;f .K < G/ and C3;f .K < G/. Up to Morita equivalence,1 these do not
depend on the choice of compact open subgroup K < G.
Proof Using the left and right translation operators g and g on L2 .G/, one checks
that the following formulae define natural equivalences and their inverses between
the categories C1 , C2 and C3 .
• C1 ! C2 W H 7! HK2 , where HK2 is the space of right K2 -invariant vectors
and where the K1 -L1 .G=K2 /-module structure on HK2 is given by restricting
the corresponding structure on H .
• C2 ! C1 W H 7! H ˝L1 .G=K2 / L2 .G/ given by
f 2 H ˝ L2 .G/ j .1gK2 ˝ 1/ D .1 ˝ 1gK2 /

for all g 2 Gg
M
D
1gK2 H ˝ L2 .gK2 /
g2G=K2

and where the K1 -K2 -L1 .G/-module structure is given by . H .k1 / ˝ k1 /k1 2K1 ,
.1 ˝ k2 /k2 2K2 and multiplication with 1 ˝ F when F 2 L1 .G/.
• C3 ! C2 W H 7! 1eK1 H and where the K1 -L1 .G=K2 /-module structure on
1eK1 H is given by restricting the corresponding structure on H .
• C2 ! C3 W H 7! L2 .G/ ˝K1 H given by
f 2 L2 .G/ ˝ H j .

k1

˝ .k1 // D

for all k1 2 K1 g

and where the G-L1 .G=K1 /-L1 .G=K2 /-module structure is given by the representation . g ˝ 1/g2G , multiplication with F ˝ 1 for F 2 L1 .G=K1 / and
multiplication with .id ˝ ˘ / .F/ for F 2 L1 .G=K2 /.
By definition, if H 2 C1 has finite rank, the Hilbert space HK2 is finite
dimensional. Conversely, if K 2 C2 and K is a finite dimensional Hilbert space,
then the corresponding object H 2 C1 has the property that both K1 H and HK2
are finite dimensional. Therefore, H 2 C1 has finite rank if and only if K1 H is a
finite dimensional Hilbert space. A similar reasoning holds for objects in C3 .
It is straightforward to check that the resulting equivalence C1 $ C3 preserves
tensor products, so that we have indeed an equivalence between the C -2-categories
C1 and C3 .
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 .K1 K2 / 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].


6

Y. Arano and S. Vaes

G-L1 .G=K1 /-L1 .G=K2 /-module, so that Ci;f .K1 < G/ and Ci;f .K2 < G/ are
Morita equivalent for i D 1; 3.
u
t

The rigid C -2-categories C1 and C2 can as follows be fully faithfully embedded
in the category of bimodules over the hyperfinite II1 factor. We construct this
embedding in an extremal way in the sense of subfactors (cf. Corollary 2.4).
To do so, given a totally disconnected group G, we fix a continuous action G Õ˛
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=2 Tr 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 ˝ R1 where R0 is a copy
of the hyperfinite II1 factor and R1 is a copy of the hyperfinite II1 factor. Choose a
˛1
continuous trace scaling action RC
R1 . By [27, Corollary 5.2], we can choose
0 Õ
˛0
a strictly outer action G Õ R0 . We then define ˛g D .˛0 /g ˝ .˛1 / .g/ 1=2 and we
take p D 1 ˝ p1 , where p1 2 R1 is 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 ŒK1 W K1 \ K2  ŒK2 W K1 \ K2 

1

:

Fixing a left Haar measure on G, we have ŒK1 W K2  D .K1 / .K2 / 1 . Therefore,
we have that ŒK W gKg 1  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 G Õ˛ P on the hyperfinite II1 factor 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 II1 factor.
To every K1 -K2 -L1 .G/-module H , we associate the Hilbert R.K1 /-R.K2 /-bimodule K given by (2.1) below. Then H 7! K is a fully faithful 2-functor. Also,
H has finite rank if and only if K is a finite index bimodule. In that case,
dimR.K1/ .K / D ŒK1 W K2 1=2 dimC1 .H /
dim

R.K2 / .K

and

/ D ŒK2 W K1 1=2 dimC1 .H / ;

where dimC1 .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
uk . ˝ b/ ur D .k/ .r/
a

˝ ˛r 1 .b/

d D .˘ ˝ id/˛.a/ .1 ˝ d/

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

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


C -Tensor Categories and Subfactors for Totally Disconnected Groups

7

Whenever K < G is a compact open subgroup, we define the projection pK 2
L.G/ given by
pK D .K/

1

Z
k

dk :

K

We also write eK D ppK 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 II1 factor and eK 2 P Ì K is a projection of finite trace. We identify
R.K/ D eK .PÌK/eK through the bijective -isomorphism . pPp/K ! eK .PÌK/eK W
a 7! apK . In particular, R.K/ is a copy of the hyperfinite II1 factor.
So, for every K1 -K2 -L1 .G/-module H , we can define the R.K1 /-R.K2 /bimodule
K D eK1 .H ˝ L2 .P// eK2 :
We claim that EndR.K1 /
have to prove that
End.PÌK1 /

R.K2 / .K

(2.1)

/ D EndC1 .H / naturally. More concretely, we

.PÌK2 / .H

˝ L2 .P// D EndC1 .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
EndP

P .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,


8

Y. Arano and S. Vaes

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
T0 ˝ 1 ˝ 1 D .˘ ˝ id/.W/13 .˘ ˝ id/.W/12 .S ˝ 1/ .˘ ˝ id/.W/12 .˘ ˝ id/.W/13
D .˘ ˝ id/.W/13 .T0 ˝ 1 ˝ 1/ .˘ ˝ id/.W/13 :

It follows that T0 commutes with ˘.L1 .G// and (2.2) is proven.
It is easy to check that H 7! K naturally preserves tensor products. So, we have
found a fully faithful 2-functor from C1 to the C -2-category of Hilbert bimodules
over hyperfinite II1 factors.
To compute dim R.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 .˛g 1 kg . p// D ˛g . p/. Therefore, as a right .P Ì K2 /module, we have
M
Lg ˝ L2 . pg P/ ;
eK1 .H ˝ L2 .P// Š
g2K1 nG=K2

where pg D ˛g 1 . p/, where the Hilbert space Lg WD ˘.1K1 gK2 /. K1 H / comes with
the unitary representation . .r//r2K2 and where the right .P Ì K2 /-module structure
on Lg ˝ L2 . pg P/ is given by
. ˝ b/ .dur / D .r/

˝ ˛r 1 .bd/ for all

2 Lg ; b 2 L2 . pg P/; d 2 P; r 2 K2 :

Since pg Ppg Ì K2 D pg .P Ì K2 /pg is a factor (actually, K2 Õ pg Ppg is a socalled minimal action), it follows from [28, Theorem 12] that there exists a unitary
Vg 2 B.Lg / ˝ pg Ppg satisfying
.id ˝ ˛r /.Vg / D Vg . .r/ ˝ 1/

for all r 2 K2 :

Then left multiplication with Vg intertwines the right .P Ì K2 /-module structure on
the Hilbert space Lg ˝ L2 . pg P/ with the right .P Ì K2 /-module structure given by
. ˝ b/ .dur / D

˝ ˛r 1 .bd/ for all

2 Lg ; b 2 L2 . pg P/; d 2 P; r 2 K2 :

Therefore,
dim

R.K2 /

Lg ˝ L2 . pg P/

eK2 D dim.Lg / dim
D dim.Lg /

. pPp/K2

L2 . pg PK2 p/

Tr. pg /
D dim.Lg / .g/1=2 :
Tr. p/


C -Tensor Categories and Subfactors for Totally Disconnected Groups

9

So, we have proved that
dim

R.K2 / .K

X

/D

.g/1=2 :

dim ˘.1K1 gK2 /. K1 H /

g2K1 nG=K2

We similarly get that
X

dimR.K1 / .K / D

dim ˘.1K1 gK2 /.HK2 /

.g/

1=2

:

g2K1 nG=K2

To make the connection with the categorical dimension of H , it is useful to view
H as the image of a G-L1 .G=K1 /-L1 .G=K2 /-module H 0 under the equivalence
of Proposition 2.2. This means that we can view H as the space of L2 -functions
W G ! H 0 with the property that .g/ 2 1eK1 H 0 1gK2 for a.e. g 2 G. The
1
L .G/-module structure of H is given by pointwise multiplication, while the K1 K2 -module structure on H is given by
.k

r/.g/ D .k/ .k 1 gr 1 /

for all k 2 K1 ; r 2 K2 ; g 2 G :

With this picture, it is easy to see that
˘.1K1 gK2 /.HK2 / Š 1eK1 H 0 1K1 gK2 :
The map 7! Q with Q .g/ D .g/ .g/ is an isomorphism between H and the
space of L2 -functions Á W G ! H 0 with the property that Á.g/ 2 1g 1 K1 H 0 1eK2 for
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

.k Á r/.g/ D .r/ Á.k 1 gr 1 / :
In this way, we get that
˘.1K1 gK2 /. K1 H / Š 1K2 g

1K

1

H 0 1eK2 :

It thus follows that
dim

R.K2 / .K

/D

X

dim.1K2 g

1K

1

H 0 1eK2 / .g/1=2

and

(2.4)

g2K1 nG=K2

dimR.K1 / .K / D

X

g2K1 nG=K2

dim.1eK1 H 0 1K1 gK2 / .g/

1=2

:

(2.5)


10

Y. Arano and S. Vaes

Also note that for every g 2 G, we have
dim.1K2 g

1K

1

H 0 1eK2 / D ŒK2 W K2 \ g 1 K1 g dim.1g

1K

1

H 0 1eK2 /

D ŒK2 W K2 \ g 1 K1 g dim.1eK1 H 0 1gK2 /
D

ŒK2 W K2 \ g 1 K1 g
dim.1eK1 H 0 1K1 gK2 /
ŒK1 W K1 \ gK2 g 1 

D ŒK2 W K1 

.g/

1

dim.1eK1 H 0 1K1 gK2 / :

It follows that
dim

R.K2 / .K

X

/ D ŒK2 W K1 

dim.1eK1 H 0 1K1 gK2 / .g/

1=2

g2K1 nG=K2

D ŒK2 W K1  dimR.K1 / .K / :
If H has finite rank, also H 0 has finite rank so that H 0 1eK2 and 1eK1 H 0 are
finite dimensional Hilbert spaces. It then follows that K is a finite index bimodule.
Conversely, assume that K has finite index. For every g 2 G, write
Ä.g/ WD dim.1K2 g

1K

H 0 1eK2 / .g/1=2

1

D ŒK2 W K1  dim.1eK1 H 0 1K1 gK2 / .g/

1=2

:

So,
Ä.g/2 D ŒK2 W K1  dim.1K2 g

1K

1

H 0 1eK2 / dim.1eK1 H 0 1K1 gK2 / :

Thus, whenever Ä.g/ ¤ 0, we have that Ä.g/
dim

R.K2 / .K

/D

ŒK2 W K1 1=2 . Since
X

Ä.g/ ;

g2K1 nG=K2

we conclude that there are only finitely many double cosets g 2 K1 nG=K2 for which
1K2 g 1 K1 H 0 1eK2 is nonzero and for each of them, it is a finite dimensional Hilbert
space. This implies that H 0 1eK2 is finite dimensional, so that H 0 has finite rank.
We have proved that H 7! K is a fully faithful 2-functor from C1;f to the
finite index bimodules over hyperfinite II1 factors. Moreover, for given compact
open subgroups K1 ; K2 < G, the ratio between dimR.K1 / .K / and dim R.K2 / .K /
equals ŒK1 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 of K equals
ŒK2 W K1 1=2 dimR.K1 / .K / D ŒK1 W K2 1=2 dim

R.K2 / .K

/:


C -Tensor Categories and Subfactors for Totally Disconnected Groups

11

Since the functor is fully faithful, the categorical dimensions of H 2 C1;f and
K 2 Bimodf coincide, so that
ŒK2 W K1 1=2 dimR.K1 / .K / D dimC1 .H / D ŒK1 W K2 1=2 dim

R.K2 / .K

/:
(2.6)
t
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 ; C C ; 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, NM, 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 Proposition 2.2 and Theorem 2.3 provides the finite index
R.KC /-R.K /-bimodule K associated with H . Take nonzero projections p˙ 2
R.K˙ / such that writing N D pC R.KC /pC and M D p R.K /p , we have that
dim M . 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, M-N and MM-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
the 2-functor in Theorem 2.3 is fully faithful, this C -2-category is equivalent with
C and this equivalence maps the N-M-bimodule L2 .M/ to H 2 CC .
t
u
From Corollary 2.4, 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 VC as well as on V . Fix vertices v˙ 2 V˙ and write K˙ D
Stab v˙ .
There exists an extremal hyperfinite subfactor N M whose standard invariant
is isomorphic with P. We have ŒM W N D ı 2 where
ıD

X

#fe 2 E j s.e/ D vC ; t.e/ D wg ŒStab w W Stab vC 1=2

w2V

D

X

#fe 2 E j s.e/ D w; t.e/ D v g ŒStab w W Stab v 1=2 :

w2VC

Note that in [5], also a weight function W VC tV ! RC
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 vC 1=2 ,
we actually obtain a subfactor planar algebra, contrary to what is claimed in [5, Proposition 4.1].

2


12

Y. Arano and S. Vaes

Moreover, P can be described as the rigid C -2-category C3;f .G; K˙ ; K˙ / of all
finite rank G-L1 .G=K˙ /-L1 .G=K˙ /-modules together with the generating object
`2 .E / 2 C3;f .G; KC ; K / (cf. Remark 2.1).
Proof We are given G Õ E and G Õ VC , G Õ V such that the source and target
maps s; t are G-equivariant and such that G acts transitively on VC and 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 -2category of G-L1 .G=K˙ /-L1 .G=K˙ /-modules generated by the alternating tensor
products of H and its adjoint.
In the 2-category C3 , the n-fold tensor product H ˝ H ˝
equals `2 .EC;n /,
where EC;n is the set of paths in the graph G 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 algebra P constructed in [5, 11].

By Corollary 2.4, we get that .C ; H / is the standard invariant of an extremal
hyperfinite subfactor N
M. In particular, ŒM W N D ı 2 with ı D dimC3 .H /.
Combining (2.6) with (2.4), and using that
1=2

.g/

D ŒgKC g

1

W KC 1=2 D ŒStab.g vC / W KC 1=2 ;

we get that
X

ı D ŒKC W K 1=2
D

dim.1gKC H 1eK / .g/

1=2

g2G=KC

X

#fe 2 E j s.e/ D g vC ; t.e/ D v g ŒStab.g vC / W KC 1=2 ŒKC W K 1=2

g 2 G=KC

D

X

#fe 2 E j s.e/ D w; t.e/ D v g ŒStab w W Stab v 1=2 :

w2VC

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

X

#fe 2 E j s.e/ D vC ; t.e/ D wg ŒStab w W Stab vC 1=2 :

w2V

To conclude the proof of the proposition, it remains to show that C 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 \ K appear in
`2 .paths starting at vC and ending at v / :


C -Tensor Categories and Subfactors for Totally Disconnected Groups

13

Since the graph is connected, the action of KC \ K on this set of paths is faithful
and the result follows. The other cases are proved in the same way.
t
u
Remark 2.6 Note that the subfactors N
M in Proposition 2.5 are 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 ŒKC W KC \ K  ŒK W
KC \ K .
We finally note that the rigid C -tensor categories C1;f .K < G/ and C3;f .K < G/
also arise in a different way as categories of bimodules over a II1 factor in the
case where K < G is the Schlichting completion of a Hecke pair
<
,
cf. [7, Section 4].
Recall that a Hecke pair consists of a countable group together with a subgroup
<
that is almost normal, meaning that g g 1 \ has finite index in for
all g 2 . The left translation action of on = gives a homomorphism of
to the group of permutations of = . The closure G of . / for the topology
of pointwise convergence is a totally disconnected group and the stabilizer K of
the point e 2 = is a compact open subgroup of G with the property that D
1
.K/. One calls .G; K/ the Schlichting completion of the Hecke pair . ; /. Note
that there is a natural identification of G=K and = .
Proposition 2.7 Let
<

be a Hecke pair with Schlichting completion K <
G. Choose an action
Õ˛ P of
by outer automorphisms of a II1 factor P.
Define N D P Ì and M D P Ì . Note that N
M is an irreducible, quasiregular inclusion of II1 factors. Denote by C the tensor category of finite index
N-N-bimodules generated by the finite index N-subbimodules of L2 .M/.
Then, C and the earlier defined C1;f .K < G/ and C3;f .K < G/ are naturally
equivalent rigid C -tensor categories.
Proof Define
C4 W

C5 W

the category of - -`1 . /-modules, i.e. Hilbert spaces H equipped with
two commuting unitary representations of and a representation of `1 . /
that are covariant with respect to the left and right translation actions Õ
;
the category of -`1 . = /-modules, i.e. Hilbert spaces equipped with
a unitary representation of
and a representation of `1 . = / that are
covariant with respect to the left translation action Õ = ;

with morphisms again given by bounded operators that intertwine the given
structure.
To define the tensor product of two objects in C4 , it is useful to view H 2 C4 as a
family of Hilbert spaces .Hg /g2 together with unitary operators .k/ W Hg ! Hkg
and .k/ W Hg ! Hgk 1 for all k 2 , satisfying the obvious relations. The tensor

14

Y. Arano and S. Vaes

product of two

- -`1 . /-modules H and K is then defined as

ˇ
n
ˇ
.H ˝ K /g D . h /h2 ˇ

h
hk

2 Hh ˝ Kh
1

X

D.

1g

;

.k/ ˝
o
k h k2 < 1

H

K

.k//. h / for all h 2 ; k 2

;

h2 =

with .k/ W .H ˝ K /g ! .H ˝ K /kg given by . .k/ /h D . H .k/ ˝ 1/ k 1 h
and .k/ W .H ˝ K /g ! .H ˝ K /gk 1 given by . .k/ /h D .1 ˝ K .k// .h/
for all k 2 , h 2 . Of course, choosing a section i W = ! , we have
M

.H ˝ K /g Š

.Hi.h/ ˝ Ki.h/

1g

/;

h2 =

but this isomorphism depends on the choice of the section.
As in Proposition 2.2, C4 and C5 are equivalent C -categories, where the
equivalence and its inverse are defined as follows.
• C4 ! C5 W H 7! K , with
ˇ

˚
Kg D . h /h2g ˇ

h

2 Hh ;

hk

1

D .k/

h

for all h 2 g ; k 2

«

and with the natural -`1 . = /-module structure. Note that Kg Š Hg , but
again, this isomorphism depends on a choice of section = ! .
• C5 ! C4 W K 7! H , with Hg D Kg and the obvious - -`1 . /-module
structure.
We say that an object H 2 C5 has finite rank if H is a finite dimensional
Hilbert space. This is equivalent to requiring that all Hilbert spaces Hg are finite
dimensional and that there are only finitely many double cosets g for which Hg
is nonzero. Similarly, we say that an object H 2 C4 has finite rank if all Hilbert
spaces Hg are finite dimensional and if there are only finitely many double cosets
g for which Hg is nonzero. Note here that an algebraic variant of the category
of finite rank objects in C4 was already introduced in [29].

In this way, we have defined the rigid C -tensor category C4;f . < / consisting
of the finite rank objects in C4 . Note that, in a different context, this rigid C -tensor
category C4;f . < / already appeared in [7, Section 4].
Denote by W ! G the canonical homomorphism. Identifying G=K and =
and using the homomorphism W ! K, every K-L1 .G=K/-module H also is a
-`1 . = /-module. This defines a functor C2 .K < G/ ! C5 . < / that is fully
faithful because . / is dense in K. Note however that this fully faithful functor
need not be an equivalence of categories: an object H 2 C5 . < / is isomorphic
with an object in the range of this functor if and only if the representation of on