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# Derivations on Toeplitz Algebras

Published:2013-02-08
Printed: Jun 2014
• Michael Didas,
Fachrichtung Mathematik, Universität des Saarlandes, Postfach 15 11 50, D-66041 Saarbrücken, Germany
• Jörg Eschmeier,
Fachrichtung Mathematik, Universität des Saarlandes, Postfach 15 11 50, D-66041 Saarbrücken, Germany
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## Abstract

Let $H^2(\Omega)$ be the Hardy space on a strictly pseudoconvex domain $\Omega \subset \mathbb{C}^n$, and let $A \subset L^\infty(\partial \Omega)$ denote the subalgebra of all $L^\infty$-functions $f$ with compact Hankel operator $H_f$. Given any closed subalgebra $B \subset A$ containing $C(\partial \Omega)$, we describe the first Hochschild cohomology group of the corresponding Toeplitz algebra $\mathcal(B) \subset B(H^2(\Omega))$. In particular, we show that every derivation on $\mathcal{T}(A)$ is inner. These results are new even for $n=1$, where it follows that every derivation on $\mathcal{T}(H^\infty+C)$ is inner, while there are non-inner derivations on $\mathcal{T}(H^\infty+C(\partial \mathbb{B}_n))$ over the unit ball $\mathbb{B}_n$ in dimension $n\gt 1$.
 Keywords: derivations, Toeplitz algebras, strictly pseudoconvex domains
 MSC Classifications: 47B47 - Commutators, derivations, elementary operators, etc. 47B35 - Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] 47L80 - Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)