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Journal of Operator Theory

Volume 87, Issue 2, Spring 2022  pp. 295-317.

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

Authors:  Alexandru Chirvasitu
Author institution:Department of Mathematics, University at Buffalo, Buffalo, NY 14260-2900, U.S.A.

Summary: We prove a number of results on the automorphisms 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 $\widehat{\mathcal{T}(D)}$.

Keywords: bounded symmetric domain, Toeplitz $C^*$-algebra, tube-type, Jordan triple system, tripotent, Shilov boundary, spectrum, solvable $C^*$-algebra

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