SDU Operator Algebras Seminar

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.

Title: Quantum graphs and spin models

Abstract: Spin models for singly-generated Yang-Baxter planar algebras are known to be determined by certain highly-regular classical graphs such as the pentagon or the Higman-Sims graph, which are extremely rare. Examples of spin models include the Jones and Kauffman polynomials. We will discuss the notion of higher-regularity for quantum graphs and give new examples of non-classical graphs yielding spin models. Time allowing, we will discuss some applications to quantum groups, topology and quantum information. 

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Title: Selfless Inclusions of C*-Algebras and Quantum Groups

Abstract: Recently, strong asymptotic freeness, or selflessness, in C*-algebras has emerged as a powerful technique to prove important regularity properties including simplicity, unique trace, stable rank 1, and strict comparison. In particular, in Fall 2024, Amrutam, Gao, Kunnwalkam Elayavalli, and I showed that the reduced group C*-algebras of all hyperbolic groups with trivial finite radical are selfless, which resolved the open problem of strict comparison for the reduced group C*-algebra of the free group on two generators. Since then, our result has been expanded to include a much larger class of groups. Work has also begun on isolating selflessness for C*-algebras not arising from groups, including the result of Hayes, Kunnawalkam Elayavalli, and Robert on selflessness of the reduced free product of a large class of C*-algebras (see also Flores-Klisse-Ó Cobhthaigh-Pagliero). I will introduce the general notion of a selfless inclusion of C*-algebras, with which we will see the selflessness of the reduced unitary compact matrix quantum groups. This work is joint with Ben Hayes, Srivatsav Kunnawalkam Elayavalli, and Leonel Robert.

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