Rigidity results for automorphisms of Hardy-Toeplitz $C^*$-algebras

TL;DR

This paper establishes rigidity results for automorphisms of Hardy-Toeplitz $C^*$-algebras associated with bounded symmetric domains, showing how their structure determines the domain and constrains automorphisms.

Contribution

It proves that the stable isomorphism class of Hardy-Toeplitz algebras uniquely determines the underlying domain and characterizes automorphisms in relation to the domain's boundary.

Findings

Stable isomorphism class determines the domain even if reducible.

Automorphisms induce permutations of boundary components in reducible domains.

Automorphisms trivial on character spaces are trivial on the entire spectrum.

Abstract

We prove a number of results on the automorphisms of and isomorphisms between Hardy-Toeplitz algebras $\mathcal{T}(D)$ associated to bounded symmetric domains $D$: that the stable isomorphism class of $\mathcal{T}(D)$ determines $D$ (even when it is reducible), that for reducible domains $D=D_1\times\cdots \times D_s$ the automorphisms of the Shilov boundary $\check{S}(D)$ induced by those of $\mathcal{T}(D)$ permute the Shilov boundaries $\check{S}(D_i)$, and that by contrast to arbitrary solvable algebras, automorphisms of $\mathcal{T}(D)$ that are trivial on their character spaces $\check{S}(D)$ are trivial on the entire spectrum $\widehat{\mathcal{T}(D)}$.