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# A Cohomological Property of $\pi$-invariant Elements

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Published:2012-02-03

• M. Filali,
Department of Mathematical Sciences, University of Oulu, Oulu 90014, Finland
• M. Sangani Monfared,
Department of Mathematics and Statistics, University of Windsor, Windsor, ON N9B 3P4
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## Abstract

Let $A$ be a Banach algebra and $\pi \colon A \longrightarrow \mathscr L(H)$ be a continuous representation of $A$ on a separable Hilbert space $H$ with $\dim H =\frak m$. Let $\pi_{ij}$ be the coordinate functions of $\pi$ with respect to an orthonormal basis and suppose that for each $1\le j \le \frak m$, $C_j=\sum_{i=1}^{\frak m} \|\pi_{ij}\|_{A^*}\lt \infty$ and $\sup_j C_j\lt \infty$. Under these conditions, we call an element $\overline\Phi \in l^\infty (\frak m , A^{**})$ left $\pi$-invariant if $a\cdot \overline\Phi ={}^t\pi (a) \overline\Phi$ for all $a\in A$. In this paper we prove a link between the existence of left $\pi$-invariant elements and the vanishing of certain Hochschild cohomology groups of $A$. Our results extend an earlier result by Lau on $F$-algebras and recent results of Kaniuth-Lau-Pym and the second named author in the special case that $\pi \colon A \longrightarrow \mathbf C$ is a non-zero character on $A$.

Keywords:  Banach algebras, $\pi$-invariance, derivations, representations

MSC Classifications:  46H15 - Representations of topological algebras 46H25 - Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 13N15 - Derivations
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