SDU Operator Algebras Seminar

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Title: Universal C*-Algebras from Graph Products

Abstract: In recent years, graph products of operator algebras have attracted growing interest, particularly in relation to free probability, Popa’s deformation/rigidity theory, and approximation properties. Furthermore, von Neumann algebras arising from group-theoretic graph products have been studied intensively. In contrast, comparatively little is known about their C*-algebraic counterparts beyond the free product case.

In this talk, I present a framework for analyzing structural properties of graph product C*-algebras by introducing a natural class of C*-algebras generated by reduced graph products and families of projections associated with words in right-angled Coxeter groups. These algebras possess a rich and tractable combinatorial structure, which enables the deduction of a variety of properties. Among other things, I will discuss universal properties, approximation properties, and analyze the ideal structure. I will then explain how to leverage this framework to derive new insights into the structure of graph product C*-algebras – many of which are novel even in the case of free products.

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Title: Applications of Operator System Methods to Questions about Exactness, LLP, and the Existence of Amenable Traces

Abstract: Given a C*-algebra generated by a finite set of unitaries or isometries many properties of the C*-algebra are determined by properties of the operator system, which can then be reduced to the study of numerical properties of that operator system.  For example, let A be generated by a set of n unitaries and S denotes the operator system that they span. We prove that A is exact if and only if the supremum of the cb-norms of unital k-positive maps with range S tends to 1 as k tends to infinity. Using this fact, we give a new proof of Wasserman's result that the C*-algebra of the free groups on n generators is not exact. In a similar vein we show that if A is generated by n isometries, then A has an amenable trace if and only if the free joint numerical range, which is another operator system parameter, of the set of isometries is n. This leads to some new proofs and extensions of results of Bekka about the existence of amenable traces on reduced group C*-algebras. There are analogous results concerning lifting properties.

This talk is based on ongoing joint work with K. Davidson, M. Rahaman, and E. Samei.

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