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Analysis Seminar
March 22, 2012 at 2:30 PM to 3:30 PM
| Location: | 4325 Herzberg Laboratories |
| Cost: | Free |
Speaker: Adam Fuller (University of Nebraska in Lincoln)
Title: Semicrossed Products by Abelian Semigroups
Abstract: Nonself-adjoint crossed product algebras were first introduced by Arveson in the late 1960s and have, over the years, been studied on and off by operator algebraists. In recent years work by Davidson and Katsoulis (and others) have shown these algebras to be exceptionally interesting class of algebras. In particular, they are a fertile ground for dilation theory and a place where one can often calculate $C^*$-envelopes.
Let $\mathcal{S}$ be a semigroup and $\mathcal{A}$ be a unital operator algebra. If $\mathcal{S}$ acts on the algebra $\mathcal{A}$ by endomorphisms $\{\alpha_s\}_{s\in\mathcal{S}}$ then we can form semicrossed product algebras to encode both the algebra $\mathcal{A}$ and the endomorphisms $\{\alpha_s\}_{s\in\mathcal{S}}$ into a single algebra.
In this talk we will discuss two types of nonself-adjoint semicrossed product algebras: the universal algebra for \emph{isometric} covariant representations of the dynamical system $(\mathcal{A},\mathcal{S},\alpha)$ and the universal algebra for \emph{contractive} covariant representations of the dynamical system $(\mathcal{A},\mathcal{S},\alpha)$. In particular we will be looking at semicrossed products by semigroups of the form $\mathcal{S}=\sum_{i=1}^{\oplus k}S_i$ where each $S_i$ is a countable subsemigroup of the positive real line.
We restrict ourselves to the the semicrossed product algebras which relate to what we call Nica-covariant (a.k.a. doubly-commuting) representations of the dynamical system. This restriction is partly due to the impossibility of forming a dilation theory for more general representations. We will conclude with a discussion of the $C^*$-envelope of these algebras.