Abstract view
Derivations on Toeplitz Algebras


Published:20130208
Printed: Jun 2014
Michael Didas,
Fachrichtung Mathematik, Universität des Saarlandes, Postfach 15 11 50, D66041 Saarbrücken, Germany
Jörg Eschmeier,
Fachrichtung Mathematik, Universität des Saarlandes, Postfach 15 11 50, D66041 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 noninner
derivations on $\mathcal{T}(H^\infty+C(\partial \mathbb{B}_n))$ over
the unit ball $\mathbb{B}_n$ in dimension $n\gt 1$.
MSC Classifications: 
47B47, 47B35, 47L80 show english descriptions
Commutators, derivations, elementary operators, etc. Toeplitz operators, Hankel operators, WienerHopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
47B47  Commutators, derivations, elementary operators, etc. 47B35  Toeplitz operators, Hankel operators, WienerHopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] 47L80  Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
