A Cohomological Property of $pi$-invariant Elements

Canadian mathematical bulletin

M. Sangani Monfared²  M. Filali¹ 

[1] Department of Mathematical Sciences, University of Oulu, Oulu 90014, Finland;Department of Mathematics and Statistics, University of Windsor, Windsor, ON N9B 3P4

关键词: Banach algebras;    $pi$-invariance;    derivations;    representations;   

DOI  :  10.4153/CMB-2011-184-7

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

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$1le j le frak m$, $C_j=sum_{i=1}^{frak m}|pi_{ij}|_{A^*}lt infty$ and $sup_j C_jlt infty$. Under theseconditions, we call an element $overlinePhi in l^infty (frak m , A^{**})$ left $pi$-invariant if $acdot overlinePhi ={}^tpi (a) overlinePhi$ for all$ain A$. In this paper we prove a link between the existence of left $pi$-invariant elements and the vanishing of certainHochschild cohomology groups of $A$. Our results extend an earlierresult by Lau on $F$-algebras and recent results of Kaniuth-Lau-Pymand the second named author in the special case that $pi colon Alongrightarrow mathbf C$ is a non-zero character on $A$.

【 授权许可】

Unknown