# Scottish Operator Algebras Seminar

## University of Glasgow, Friday, 14 March, 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 a dinner. A schedule will be posted soon.

## Speakers

Joan Bosa (University of Glasgow) - The category Cu. Which maps are the correct ones? *-homomorphisms or cpc order zero maps?
Slides

In this talk we focus on the fact that the map induced by a cpc order zero $$\varphi : A \to B$$ in the category Cu does not preserve the compactly containment relation. In particular, these kinds of maps are not in the category Cu, so that, in general, they may not be used in the classification of C*-algebras via the Cuntz Semigroup. Nevertheless, there is a subclass of these maps which preserves the relation, and so they can be used in the above mentioned classification. Our main result characterizes these maps via the positive element induced by the description of cpc order zero maps shown in [Winter, W. and Zacharias, J. Completely positive maps of order zero. Münster J. Math., 2, 2009, 311-324.].

Liam Dickson (University of Glasgow) - The row length of unital C*-algebras

The property of finite length for an operator algebra was introduced by Pisier, it reflects the existence of factorisations of elements in matrix amplifications of the algebra into scalar matrices and diagonal matrices over the algebra with the length and norm of the factorisations controlled simultaneously. Pisier showed that this property is equivalent to the similarity property for C*-algebras (the property that every bounded homomorphism from a C*-algebra into the bounded operators on a Hilbert space is necessarily similar to a *-homomorphism). Finite row length is a weakening of the finite length condition where we seek factorisations, as above, over row amplifications of the algebra only. We show that all unital C*-algebras have finite row length, in fact, have row length at most 2. Finally, we will present an application of this result in the perturbation theory of C*-algebras.

Greg Maloney (University of Newcastle) - Calculating K-theory of substitution tiling C*-algebras using dual tilings
(joint work with Franz Gähler and John Hunton)
Slides

When studying aperiodic tilings, rather than looking at a single tiling one typically considers a topological space, the points of which are tilings. Under standard assumptions, this topological space is pathological, in the sense that the orbit of any point under the translation action is dense. One method of dealing with this pathology is to build a C*-algebra from the space; this C*-algebra is the groupoid C*-algebra of the translation groupoid of the topological space.

In the special case that the tilings arise from a substitution, Anderson and Putnam have described a method that, in low dimensions, allows us to calculate the K-theory of this algebra. But this calculation can be difficult: some interesting examples are impossible to calculate without the aid of a computer, and other interesting examples are impossible to calculate even with the aid of a computer. I will describe a modification to the Anderson-Putnam method that enables us to calculate the K-theory in cases that were previously intractable, and I will mention some results that have been found using this method.

Yasuhiko Sato (Kyoto University) - Murray-von Neumann equivalence for positive elements and order zero c.p. maps
(joint work with Stuart White and Wilhelm Winter)

A completely positive map is called order zero if it preserves orthogonality. Recent developments in the classification theorem of C*-algebras suggest that order zero c.p. maps are very compatible with projectionless C*-algebras. In this talk, we investigate the Murray-von Neumann equivalence for positive elements and see that plays a crucial role for the understanding of order zero c.p. maps and their conjugacy classes. As a consequence of this study, we obtain an affirmative answer to the Toms and Winter conjecture for C*-algebras with a unique tracial state.

Ján Špakula (University of Southampton) - Operator theory and coarse geometry
(Some) operator theorists study Fredholmness of certain operators on $$l^2(Z^n)$$ using the so-called operator spectrum. John Roe, in 2004, explained that the operators of interest are really just elements of the Translation C*-algebra (also called the uniform Roe algebra) of $$Z^n$$, the C*-algebra encoding the large scale (or coarse) structure of $$Z^n$$. In this talk, I will explain 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 big question on Limit Operators (on $$Z^n$$)").

Due to a cancelled flight, Ján Špakula was unable to attend. In his place, Yemon Choi (Lancaster University) graciously gave a talk on Amenable operator algebras.

## Schedule

The meeting will take place in room 504 of the Joseph Black Building (School of Chemistry); this is on the other side of University Avenue than the Maths Building. Tea and coffee will be in the Maths Lounge.

 11:30-12:00 Tea and coffee 12:00-12:45 Liam Dickson 12:45-13:45 Lunch 13:45-14:30 Yasuhiko Sato 14:40-15:25 Jan Spakula 15:25-15:55 Tea and coffee 15:55-16:40 Greg Maloney 16:50-17:35 Joan Bosa 18:15 Dinner

## Expected Participants

Rob Archbold (University of Aberdeen)
Joan Bosa (University of Glasgow)
Jorge Castillejos Lopez (University of Glasgow)
Yemon Choi (Lancaster University)
Liam Dickson (University of Glasgow)
Sam Evington (University of Glasgow)
Lefteris Kastis (Lancaster University)
Linda Mawhinney (Queen's University Belfast)
Greg Maloney (University of Newcastle)
Martin Mathieu (Queen's University Belfast)
Stephen Power (Lancaster University)
Yasuhiko Sato (Kyoto University)
Ján Špakula (University of Southampton)
Aaron Tikuisis (University of Aberdeen)
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.