# Scottish Operator Algebras Seminar

## University of Aberdeen, Friday, 28 November, 2014

This operator algebras meeting is one in a series of meetings between Aberdeen and Glasgow. Participants from other universities are of course welcome to attend, and may contact Aaron Tikuisis for more details. The talks will go from about 12 noon until 5:30 pm, and followed by dinner.

We will be having a joint dinner with the Scottish topology seminar at Nazma. Please email Aaron if you plan to come to dinner.

## Speakers

Slides from the talks are available by clicking on the talk title.

Rob Archbold (University of Aberdeen) - Some representation theory for compact transformation groups
Motivated by earlier work of Echterhoff, we discuss the representation theory of the crossed-product C*-algebra associated to a compact group G acting on a locally compact space X, in the case where the stability subgroups vary discontinuously.
Our main result applies when G has a principal stability subgroup or X is locally of finite G-orbit type. Then the upper multiplicity of the representation of the crossed product induced from an irreducible representation V of a stability subgroup is obtained by restricting V to a certain closed subgroup of the stability subgroup and taking the maximum of the multiplicities of the irreducible summands occurring in the restriction of V.
As a corollary we obtain that when the trivial subgroup is a principal stability subgroup, the crossed product is a Fell algebra if and only if every stability subgroup is abelian. A second corollary is that the C*-algebra of the motion group $$\mathbb{R} ^n\rtimes {\rm SO}(n)$$ is a Fell algebra. This uses the classical branching theorem for the special orthogonal group $${\rm SO}(n)$$ with respect to $${\rm SO}(n-1)$$.
Since proper transformation groups are locally induced from the actions of compact subgroups, some of the results can be extended to locally proper (i.e. Cartan) transformation groups.
This is joint work with Astrid an Huef (University of Otago). Click here for the article. (You may also contact Rob Archbold if you are unable to access the article in this link.)

Olivier Gabriel (University of Glasgow) - Actions of compact quantum groups and inclusions of C*-algebras
Any finite dimensional representation of a compact quantum group $$(H, \Delta)$$ induces an action on the Cuntz algebra $$\mathcal O_d$$. We prove that the fixed point algebra is Kirchberg and in the bootstrap class, hence classified by its K-theory. Moreover, one can prove that this K-theory only depends on $$H$$ through its representation category and its fusion rules. In particular, for actions of $$SU_q(2)$$, the fixed points are the Cuntz algebra $$\mathcal O_\infty$$.
Can we recover the compact quantum group from the inclusion $$\mathcal O_\infty \to \mathcal O_2$$ only?

Evgenios Kakariadis (Newcastle University) - Semigroup actions on operator algebras
The study of operator algebras arising from dynamical systems is almost as old as the study of operator algebras themselves. Recently the research has turned the focus to actions of semigroups on an arbitrary operator algebra by endomorphisms and their dilation to group actions. In this talk we will present recent findings for the semigroup $$\mathbb{Z}_+^n$$. These include the Cuntz-Nica-Pimsner algebra that generalises the Cuntz-Pimsner algebra of the one variable case and the usual C*-crossed product for group actions. In particular we will give a dilation technique from n commuting endomorphisms to n commuting automophisms on a larger C*-algebra such that the corresponding (minimal) Cuntz-Nica-Pimsner algebras are strong Morita equivalent. Hence we can reduce problems on semigroup actions to problems on group actions. Further consequences of our analysis include the association of the ideal structure/nuclearity/exactness of the Nica-Pimsner algebras with minimality-freeness/nuclearity/exactness of the C*-dynamics.
The talk is based on a joint work with Ken Davidson and Adam Fuller.

David McConnell (University of Glasgow) - Tensor products of C*-bundles
It is often useful to decompose a C*-algebra A into the algebra of sections of a bundle of C*-algebras over a suitable base space. Of particular interest are a certain class of C*-bundles known as $$C_0$$(X)-algebras.
Here we study the tensor product of a $$C_0$$(X)-algebra A and a $$C_0$$(Y)-algebra B, and analyse the structure of their tensor product A$$\otimes$$B as a $$C_0$$(X$$\times$$Y)-algebra (generally with respect to the minimal C*-tensor norm). When A and B define continuous C*-bundles over X and Y, we investigate the question of continuity of the bundle arising from the $$C_0$$(X$$\times$$Y)-algebra A$$\otimes$$B, with reference to exactness of the C*-algebras A and B under consideration. We apply these results in particular to study the stability of the class of quasi-standard C*-algebras (a particularly well-behaved class of $$C_0$$(X)-algebras) under tensor products.

Ján Špakula (University of Southampton) - Operator theory and coarse geometry
(Some) operator theorists study Fredholmness of certain operators on $$l^2(\mathbb{Z}^n)$$ using the so-called operator spectrum. John Roe, in 2004, explained that their setup is in fact coarse-geometric in nature. For instance, the operators of interest are really just elements of the Translation C*-algebra (also called the uniform Roe algebra) of $$\mathbb{Z}^n$$, the C*-algebra encoding the large scale (or coarse) structure of $$\mathbb{Z}^n$$.
In this talk, I will explain how to generalise the limit operator theory framework not only to other discrete groups, but to general discrete metric spaces. Furthermore, I will show how to further exploit the inherent connections to coarse geometry to generalise a recent result of Lindner and Siedel, which significantly simplifies the Fredholmness criterion (they refer to the problem they solve as "The core issue on Limit Operators (on $$\mathbb{Z}^n$$)").
This is a joint work with Rufus Willett.

## Schedule

The meeting will take place in Seminar Room 156 of the Fraser Noble Building. Tea and coffee will be in the Maths Lounge, outside of the seminar room.

 12:00-12:45 Olivier Gabriel 12:45-13:45 Lunch 13:45-14:30 Evgenios Kakariadis 14:40-15:25 David McConnell 15:25-15:45 Tea and coffee 15:45-16:30 Ján Špakula 16:40-17:25 Rob Archbold 18:30 Joint dinner with Scottish Topology Seminar

## Participants

Rob Archbold (University of Aberdeen)
Jorge Castillejos Lopez (University of Glasgow)
Sam Evington (University of Glasgow)
Olivier Gabriel (University of Glasgow)
Ilja Gogic (Trinity College Dublin)
Greg Maloney (Newcastle University)
Martin Mathieu (Queen's University Belfast)
David McConnell (University of Glasgow)
Andrew Monk (University of Glasgow)
Mark Paulin (University of Aberdeen)
Luis Santiago (University of Aberdeen)
Ján Špakula (University of Southampton)
Aaron Tikuisis (University of Aberdeen)
Richard Timoney (Trinity College Dublin)
Gabriele Tornetta (University of Glasgow)
Christian Voigt (University of Glasgow)
Simon Wassermann (University of Glasgow)
Stuart White (University of Glasgow)
Joachim Zacharias (University of Glasgow)

The meeting is organised by Aaron Tikuisis and Stuart White.

## Support

Funding for this meeting is provided by the Glasgow Mathematical Journal trust.