OPERATOR ALGEBRAS OF SUBPRODUCT SYSTEMS BY EXAMPLE

Speaker(s)

ADAM DOR-ON

Affiliation

Københavns Universitet

Date

March 3, 2021, 5:15 p.m.

Information about the event

Seminar 2021-03-03 17:15:00

Seminar

North Atlantic Noncommutative Geometry Seminar

In this talk, we will discuss subproduct systems as introduced by Shalit and Solel in 2009 following a definition given by Bhat and Mukherjee. Subproduct systems were originally defined for the purpose of classifying CP-semigroups, but they also give rise to natural (non-self-adjoint) tensor, Toeplitz and Cuntz type operator algebras. The latter C*-algebra was defined and studied in greater generality by Viselter in 2011 as a generalization of the well-known Cuntz-Pimsner algebra. It is unknown whether C*-algebras arising from subproduct systems admit a universal property in general (and this is a deep question), but they are concretely defined as bounded operators on analogues of the Fock space. Still, they share many of the familiar properties of Cuntz-Pimsner algebras, and give rise to new and interesting examples of operator algebras in several scenarios. We will showcase some examples coming from ideals of polynomial relations (Davidson, Kennedy, Shalit, Ramsey, etc.), some more recent examples coming from SU(2)-symmetries (Arici and Kaad), and focus on examples coming from stochastic matrices and random walks (Markiewicz, Linden, myself, etc.). It turns out that a new notion of ratio limit boundary arises from the lens of Cuntz algebras of random walks, spurring further research in probability.