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
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.)
