Gauge-invariant ideal structure of C*-algebras associated with strong compactly aligned product systems

Thursday 19th September 16:00-17:00 Maths 311B

Product systems represent powerful contemporary tools in the study of mathematical

structures. A major success in the theory came from Katsura (2007), who provided a complete

description of the gauge-invariant ideals of many important C*-algebras arising from product

systems over Z+. This result recaptures existing results from the literature, illustrating the

versatility of product system theory. The question now becomes whether or not Katsura’s result

can be bolstered to product systems over semigroups other than Z+ and, if so, what applications

do we obtain? An answer has been elusive, owing to the more pathological nature of product

systems over general semigroups. However, recent strides by Dor-On and Kakariadis (2018)

supply a more tractable subclass of product systems that still includes the important cases of

C*-dynamics, row-finite higher-rank graphs and regular product systems. In this talk we will

build a parametrisation of the gauge-invariant ideals, starting from first principles and gradually

increasing in complexity. We will pay particular attention to the higher-rank subtleties that are

not witnessed in Katsura’s theorem, and comment on the applications. Time permitting, we

will also look at the connection with the recent work of Bilich (2023)

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