# Toeplitz algebras and their boundary quotients

### Marcelo Laca (University of Victoria)

Thursday 18th November, 2021 16:00-17:00 Maths 311B

#### Abstract

We define a universal Toeplitz C*-algebra $\mathcal T_u(P)$, given by a presentation based on  Xin Li's constructible ideals.  $\mathcal T_u(P)$ coincides with Li's semigroup C*-algebra  when $P$ satisfies the independence condition but  can be used in general to study the reduced Toeplitz C*-algebra $\mathcal T_\lambda(P)$ generated by the left regular representation of $P$. After briefly mentioning conditions for  faithful representations and a uniqueness theorem for $\mathcal T_\lambda(P)$, I will concentrate on the associated boundary quotient. This is initially defined by a presentation distilled from Sehnem's covariance algebra for the product system over $P$ with one dimensional fibres.  I will give conditions for the reduced boundary quotient $\partial\mathcal T_\lambda(P)$ to be purely infinite simple, and then discuss some new applications, mainly to monoids arising from non-maximal orders in number fields but also to right LCM monoids with nontrivial units.
This is joint work with Camila F. Sehnem.