Activity on Operator algebras in Oslo and Trondheim 2003-2004

(Part of a QSNM report)

*Sergey Neshveyev and Lars Tuset study the Dirac operator on the quantum sphere, and obtain a local index formula for it. Their considerations yield one of a few examples of a successful combination of the techniques of $q$-differential geometry and of Connes' non-commutative geometry. The non-classical behaviour of the geodesic flow that is observed, may be a key for the analysis of higher dimensional examples.

Reference: S. Neshveyev and L. Tuset, A local index formula for the

quantum sphere, to appear in Comm. Math. Phys.

(Team 5, C1)

*Sergey Neshveyev and Lars Tuset define the Martin boundary of a discrete quantum group in terms of certain quantum random walks. This is illustrated for the dual group of the quantum group $SU_{q}(2)$. Together with M. Izumi they compute the Poisson boundary of the dual of $SU_{q}(n)$, which is shown to be isomorphic to the quantum flag manifold. The proof relies on a connection between the Poisson integral and the Berezin transform.

References: 1) S. Neshveyev and L. Tuset, The Martin boundary of a

discrete quantum group, J.reine u. angew. Math. 568(2004), 23-70.

2) M. Izumi, S. Neshveyev and L. Tuset, Poisson boundary of

the dual of $SU_{q}(n)$, preprint.

3) S. Neshveyev and L. Tuset, Quantum random walks and their

boundaries, to appear in RIMS, Kyoto Symposium Proceedings.

(Team 5, B7 and C2)

*Sergey Neshveyev and M. Laca generalize and shed light on earlier results concerning KMS-states of the gauge actions on Cuntz algebras,

Cuntz-Krieger algebras and crossed products by endomorphisms. It is also shown how to derive in a unified way examples of KMS-states of

$C^{*}$-dynamical systems previously studied under various guises.

References: S. Neshveyev and M. Laca, KMS-states of quasi-free dynamics on

Pimsner algebras, J. Funct. Anal. 211(2004), 457-482.

(Team 5, B7)

*Christian Skau, Thierry Giordano and Ian Putnam have studied in depth how the property of being affable, i.e. having an AF-equivalence relation structure, is preserved under extensions and restrictions. This will be be crucial in order to show that a free and minimal continuous action of an amenable countable group on the Cantor set is orbit equivalent to a $\mathbb{Z}$-action. At this stage results are obtained for $\mathbb{Z}^{2}$-actions.

References: 1) T. Giordano, I. Putnam and C. Skau, Affable equivalence

relations and orbit structure of Cantor dynamical systems, Ergod. Th. & Dynam. Sys. 24(2004), 441-475.

2) T. Giordano, I. Putnam and C. Skau, Cantor minimal

$\mathbb{Z}^{2}$-systems, preprint.

(Team 5, B7, B8)

*Sergey Neshveyev and Erling Størmer have studied non-commutative ergodic theory. Specifically, they have studied how the position of the group von Neumann algebra $L(G)$ inside the crossed product $M\times G$, where $G$ acts on the von Neumann algebra $M$, gives information about the original action. If $G$ has property $T$ it was known that this gives a lot of information, but, rather surprisingly, it is shown that even beyond property $T$ the same is true.

Reference: S. Neshveyer and E. Størmer, Maximal abelian subalgebras of the hyperfinite factor, entropy and ergodic theory, Acta Math. Sinica, (English Series), 19(2003), 599-604.

(Team 5, B7)

*Ola Bratteli, P.E.T. Jørgensen and V. Ostrovsky have studied permutative

representations, which are related to wavelets. They computed the

associated modular group, and found the fixed point algebra. This turns

out to be an AF-algebra and they described its structure in detail, by

first computing its dimension group.

Reference: O. Bratteli, P.E.T. Jørgensen and V. Ostrovski, Representation theory and numerical AF-invariants. The representation and centralizers of certain states on $O_{d}$, Mem. Amer. Math. Soc. 168(2004), no. 797, 178pp.

(Team 5, B1)

*Erik Bedos and Lars Tuset, together with Gerard Murphy and Roberto Conti, have written a series of joint papers, where they study the concept of amenability (and coamenability) in the quantum group setting. A simple definiton of coamenability is given,which is shown to be equivalent to properties that have obvious classical analogues. It is shown that coamenability implies that the dual quantum group is amenable, and the converse implication holds if the quantum group is compact, generalizing an old result about Kac algebras.

References: 1) E. Bedos and L. Tuset, Amenability and coamenability for locally compact quantum groups, Int. J. Math. 14(2003), 865-884.

2) E. Bedos, R. Conti and L. Tuset, Amenability and

coamenability of algebraic quantum groups and their corepresentations, to appear in Canadian J. of Math.

3) E. Bedos, G. J. Murphy and Lars Tuset, Amenability and

co-amenability for algebraic quantum groups II, J. Funct. Anal. 201(2003), 303-340.

(Team 5, D2)

*Sergey Neshveyev and Lars Tuset give a complete procedure for computing the Hopf equivariant K-groups associated to actions of Hopf algebras or, more generally, quantum group actions on algebras. A relation is established between equivariant K-theory and ordinary K-theory of the associated $C^{*}$-crossed product, and one is thus able to give a characterization of equivariant vector bundles on quantum homogeneous spaces.

Reference: S. Neshveyev and L. Tuset, Hopf algebra equivariant cyclic

cohomology, K-theory and index formulas, to appear in K-theory.

(Team 5, D2)

*Lars Tuset, G.J. Murphy and J. Kustermans have in a series of papers

explored the possibility of connecting Connes' theory of non-commutative geometry with the theory of quantum groups and $q$-deformed homogeneous spaces. The Chern character for quantum groups are defined by introducing a twisted cyclic cohomology. A characteristic homomorphism is defined from Hopf cyclic cohomology (as defined by Connes and Moscovici) into the twisted cyclic cohomology above, which preservers the long exact IBS-sequence. Eigenvalues of the Hodge-Laplace operator associated to a compact quantum group is studied in detail. The commutators of the Dirac

operators are no longer bounded operators, but can be reduced to the

bounded case by a key formula.

References: 1) J. Kustermans, G. J. Murphy and L. Tuset, Differential

calculi over quantum groups and twisted cyclic cocycles, J. Geom. Phys. 044(2003), 570-594.

2) J. Kustermans, G. J. Murphy and L. Tuset, Quantum groups,

differential calculi and the eigenvalues of the Laplacian, to appear in

Trans. Amer. Math. Soc.

3) J. Kustermans, J. Rognes and Lars Tuset, The modular square for quantum groups, to appear in Banach Center Publ.

(Team 5, D2)

*Lars Tuset, M. Müger and J.E. Roberts study the question as to what

extent a compact quantum group is determined by its tensor category of finite-dimensional corepresentations. This tensor category turns out to have what is called duality, and Woronowicz showed that this special property is sufficient to reconstruct the quantum group, thereby generalizing the Tannaka-Krein theorem for compact groups.

References: 1) M. Müger, J. E. Roberts and L. Tuset, Representations of algebraic quantum groups and reconstruction theorems for tensor

categories, to appear in Algebra and Reprs.

2) M. Müger and L. Tuset, Representation of algebraic quantum

groups and embedding theorems, preprint.

(Team 5, D3)

*Eric Bedos and Roberto Conti have studied infinite tensor products of

discrete groups. Special attention is given to regular representations

twisted by 2-cocycles, and projective representations associated with

CCR-representations of bilinear maps.

Reference: E. Bedos and R. Conti, On infinite tensor products of

projective unitary representations, Rocky Mount. J. Math. 34(2004),

467-494.

(Team 5, B7)

*Toke Carlsen, Søren Eilers and Kengo Matsumoto have studied various

dynamical systems of subshift type - in particular, substitution and sofic shifts - and the $C^{*}$-algebras and dimension groups associated to these. They are able to compute in an algorithmic efficient way the dimension group invariant associated to these subshifts.

References: 1) T. Carlsen, $C^{*}$-algebras associated to sofic shifts, J. Operator Theory 49(2003), 203-212.

2) T. Carlsen and K. Matsumoto, Some remarks on the

$C^{*}$-algebras associated with subshifts, to appear in Math. Scand.

3) T. Carlsen and S. Eilers, Augmenting dimenson group

invariants for substitution dynamics, to appear in Erg. Th. & Dynam. Sys.

4) T. Carlsen and S. Eilers, A graph approach to computing

non-determinacy in substitutional dynamical systems, preprint.

(Team 5, B1, B7)

*Magnus Landstad, S. Kaliszewski and John Quigg show that the study of certain Hecke algebras can be simplified by a direct use of Morita-Rieffel equivalence. This gives a natural framework to study the Hecke algebras arising from number theory, first studied by Bost and Connes.

Reference: S. Kaliszewski, M. Landstad and J. Quigg, Hecke algebras and groups, preprint (math. QA/0311222)

(Team 5, B6)

*Magnus Landstad and A. van Daele use the fact that the existence of a compact open subgroup of a group is equivalent to the existence of a

non-trivial pair of a convolution operator and a multiplication operator

which commute, to deduce consequences for general quantum groups.

In particular,the relation between multiplier Hopf algebras and a notion

of compact open quantum subgroups is investigated.

Reference: M. Landstad and A. van Daele, Multiplier Hopf $^{*}$-algebras

and groups with compact open subgroups, preprint.

(Team 5, D5)

*Nadia Larsen study actions of injective, extendible endomorphisms of an Ore semigroup on a $C^{*}$-algebra. She shows that there is a partial action on the fixed-point algebra under the canonical coaction of the enveloping group of the semigroup. The resulting crossed products, though characterized by different properties, are the same.

Reference: N. Larsen, Crossed products by semigroups of endomorphisms and groups of particular automorphisms, Canadian Math. Bull. 46(2003), 98-112.

(Team 5, B7)

*Nadia Larsen and M. Laca consider group - subgroup pairs in which the

group is a semidirect product and the subgroup is contained in the normal part. Conditions are given for the pair to be a Hecke pair, and it is shown that the enveloping Hecke algebra and Hecke $C^{*}$-algebra are canonically isomorphic to semigroup crossed products.

Reference: M. Laca and N. Larsen, Hecke algebras of semidirect products, Proc. Amer. Math. Soc. 131(2003), 2189-2199.

(Team 5, B7)

*Anne Louise Svendsen in a series of papers has studied automorphisms of certain subfactors of a von Neumann algebra.

References: 1) A. L. Svendsen, Automorphisms of subfactors from commuting squares, Trans. Am. Math. Soc. 356(2004), 2515-2543.

2) A. L. Svendsen, Outer automorphisms of a series of

non-amenable subfactors, to appear in Proceedings of the 2003 Sinai

conference.

3) A. L. Svendsen, Endomorphisms and automorphisms from

subfactors illustrating non-commutative entropy, preprint.

(Team 5, B7)