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• W2592709341 abstract "We show that a representation of a Banach algebra $A$ on a Banach space $X$ can be extended to a canonical representation of $A^{**}$ on $X$ if and only if certain orbit maps $Ato X$ are weakly compact. When this is the case, we show that the essential space of the representation is complemented if $A$ has a bounded left approximate identity. This provides a tool to disregard the difference between degenerate and nondegenerate representations. Our results have interesting consequences both in C*-algebras and in abstract harmonic analysis. For example, a C*-algebra $A$ has an isometric representation on an $L^p$-space, for $pin[1,infty)setminus{2}$, if and only if $A$ is commutative. Moreover, the $L^p$-operator algebra of a locally compact group is universal with respect to arbitrary representations on $L^p$-spaces." @default.
• W2592709341 created "2017-03-16" @default.
• W2592709341 creator A5016962095 @default.
• W2592709341 creator A5034930549 @default.
• W2592709341 date "2017-03-02" @default.
• W2592709341 modified "2023-09-27" @default.
• W2592709341 title "Extending representations of Banach algebras to their biduals" @default.
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• W2592709341 hasPublicationYear "2017" @default.
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