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Derivations on Toeplitz Algebras
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- Journal:
- Canadian Mathematical Bulletin / Volume 57 / Issue 2 / 14 June 2014
- Published online by Cambridge University Press:
- 20 November 2018, pp. 270-276
- Print publication:
- 14 June 2014
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- Article
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Let
${{H}^{2}}\left( \Omega \right)$ be the Hardy space on a strictly pseudoconvex domain 
$\Omega \,\subset \,{{\mathbb{C}}^{n}}$, and let 
$A\,\subset \,{{L}^{\infty }}\left( \partial \Omega \right)$ denote the subalgebra of all 
${{L}^{\infty }}$-functions 
$f$ with compact Hankel operator 
${{H}_{f}}$. Given any closed subalgebra 
$B\,\subset \,A$ containing 
$C\left( \partial \Omega \right)$, we describe the first Hochschild cohomology group of the corresponding Toeplitz algebra 
$\mathcal{T}\left( B \right)\,\subset \,B\left( {{H}^{2}}\left( \Omega \right) \right)$. In particular, we show that every derivation on 
$\mathcal{T}\left( A \right)$ is inner. These results are new even for 
$n\,=\,1$, where it follows that every derivation on 
$\mathcal{T}\left( {{H}^{\infty }}\,+\,C \right)$ is inner, while there are non-inner derivations on 
$\mathcal{T}\left( {{H}^{\infty }}\,+\,C\left( \partial {{\mathbb{B}}_{n}} \right) \right)$ over the unit ball 
${{\mathbb{B}}_{n}}$ in dimension 
$n\,>\,1$.
