# Scottish Operator Algebras Seminar

## University of Aberdeen, 22-23 June, 2015

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. Please email Aaron if you plan to come to dinner.

The Scottish Operator Algebras Seminar will be immediately followed by the North British Functional Analysis Seminar, with two lectures by Ilijas Farah (York University, Toronto).

## Speakers

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

Ilijas Farah (York University, Toronto) - Ultrapowers and relative commutants of operator algebras (NBFAS)
Let U be maximal filter on the power-set of some set (i.e., an ultrafilter). Some categories can be equipped with a functor associated to U, the so-called ultrapower. Although this functor has applications in a wide variety of subjects (ranging from logic and combinatorics to functional analysis), it can be uniquely characterized by two of its abstract properties. In the case of operator algebras, one has an even more important related construction, the relative commutant. In the last few minutes of my talk I will present some recent results on the structure of relative commutants.

Naimark’s problem (NBFAS)
Representation theory of a separable C*-algebra is either very simple (for type I algebras) or extremely complicated. In particular, algebras of compact operators on separable Hilbert spaces are the only separable C*-algebras with unique irreducible representation up to the spatial isomorphism. This result does not carry over to the nonseparable C*-algebras. In a remarkable 2004 result Akemann and Weaver used Jensen’s diamond principle to construct a simple nonseparable C*-algebra with a unique irreducible representation. Diamond principle is a strengthening of the Continuum Hypothesis and it is not known whether such algebra can be constructed without using additional set-theoretic axioms. I will talk about some recent progress on this (still open) problem.

Xin Li (Queen Mary University London) - Topological dynamics and C*-algebras

We discuss interactions between dynamical systems and operator algebras. The key notions are continuous orbit equivalence and Cartan subalgebras in C*-algebras

Richard Timoney (Trinity College Dublin) - Fixed points of ternary involutions and applications
Ternary rings of operators (TROs) T play a central role in operator space theory and are usually treated as off-diagonal corners of their linking C*-algebra. However if T is already ternary isomorphic to a C*-algebra, the canonical linking algebra is arguably bigger than necessary. In the case of weak*-closed TROs U, we have a natural decomposition of U as a maximal square ideal U$$_s$$ (which is ternary isomorphic to a W*-algebra) and two other ideals U$$_r$$ and U$$_l$$ which are right and left ideals of W*-algebras. We apply this to analyse ternary involutions (ternary anti-isomorphisms which their own inverse, like a transpose) and their fixed points. Further applications are to surjective linear isometries between TROs. This is related to the universal TRO of a JC*-triple and the concepts of a universal TRO and of universal reversibility for JC*-triples. (Joint work with L. J. Bunce.)

Gabriele Tornetta (University of Glasgow) - A bivariant extension of open projection Motivated by a bivariant theory for the Cuntz semigroup and the open projection picture of its ordinary theory I will introduce a bivariant extension of open projections and some notions of comparisons of positive elements by using completely positive and orthogonality preserving maps between C*-algebras.

Stefaan Vaes (Katholieke Universiteit Leuven) - Representation theory and (co)homology for subfactors, $$\lambda$$-lattices and C*-tensor categories

I will first give an introduction to Jones' theory of subfactors and their invariants: the standard invariant, which is a $$\lambda$$-lattice of multimatrix algebras, and the associated C*-tensor category of bimodules. I then present a joint work with Sorin Popa in which we develop a representation theory for such $$\lambda$$-lattices and for rigid C*-tensor categories. This includes a definition of their universal C*-algebra and a systematic account of approximation and rigidity properties for subfactors and tensor categories, like (weak) amenability, the Haagerup property and property (T). I will also explain the relations between our representation theory and the recent approaches using unitary half braidings (Neshveyev-Yamashita), and representations of Ocneanu's tube algebra (Ghosh-Jones). Finally I will present a joint work with Sorin Popa and Dimitri Shlyakhtenko in which we define (co)homology for $$\lambda$$-lattices and for rigid C*-tensor categories and use this to define their $$L^2$$-Betti numbers.

Simon Wassermann (University of Glasgow) - Tensor primeness for certain large C*-algebras

Stuart White (University of Glasgow) - C*-rigidity and Bieberbach groups
This is a report on joint work in progress with Søren Knudby and Hannes Thiel.

## 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.

Monday, 22 June
 12:00-13:00 Simon Wassermann 13:00-14:00 Lunch 14:00-15:00 Richard Timoney 15:00-15:30 Tea and coffee 15:30-16:20 Gabriele Tornetta 16:30-17:30 Stefaan Vaes 18:45 Dinner: Goulash restaurant

Tuesday, 23 June
 9:00-10:00 Xin Li 10:00-10:30 Tea and coffee 10:30-11:30 Stefaan Vaes 11:40-12:30 Stuart White 12:30-14:30 Lunch 14:30-15:30 Ilijas Farah (NBFAS) 15:30-16:00 Tea and coffee 16:00-17:00 Ilijas Farah (NBFAS) 18:30 Dinner: Nargile (NBFAS)

## Participants

Rob Archbold (University of Aberdeen)
Ilijas Farah (York University, Toronto)
Olivier Gabriel (University of Glasgow)
Jason Hancox (Lancaster University)
Xin Li (QMU London)
Martin Mathieu (Queen's University Belfast)
David McConnell (University of Glasgow)
Andrew Monk (University of Glasgow)
David O'Sullivan (University of Sheffield)
Mark Paulin (University of Aberdeen)
Aaron Tikuisis (University of Aberdeen)
Richard Timoney (Trinity College Dublin)
Gabriele Tornetta (University of Glasgow)
Stefaan Vaes (KU Leuven)
Christian Voigt (University of Glasgow)
Simon Wassermann (University of Glasgow)
Jared White (Lancaster University)
Stuart White (University of Glasgow)
Joachim Zacharias (University of Glasgow)

The meeting is organised by Aaron Tikuisis and Stuart White.

## Accommodation

A list of hotels in Aberdeen can be found here. University accommodation is also available; rates are as follows:
Single room, En-suite facilities £45.5
Twin room, En-suite facilities £66.5
Single room, Shared toilet and showers facilities £39
Twin room, Shared toilet and showers facilities £60.

All rooms include breakfast and have shared kitchen facilities.

To book a room, contact the Accommodation Team at Crombie Hall Reception.
Tel: 01224 272218
Email: accommodation@abdn.ac.uk
Visit: www.abdn.ac.uk/summer_accommodation

## Support

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