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 \(A\rightarrow X\) 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 \(p\in [1,\infty ){\setminus }\{2\}\), 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.
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Acknowledgements
The authors would like to thank Philip G. Spain for valuable electronic correspondence. The authors would like to thank the staff and organizers, and Søren Eilers in particular, for the hospitality during their visits to the Research Program Classification of operator algebras, complexity, rigidity and dynamics, held at the Institut Mittag-Leffler, between January and April of 2016.
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Part of this research was conducted while the authors were taking part in the Research Program Classification of operator algebras, complexity, rigidity and dynamics, held at the Institut Mittag-Leffler, between January and April of 2016. The first named author was partially supported by a Postdoctoral Research Fellowship from the Humboldt Foundation. The authors were partially supported by the Deutsche Forschungsgemeinschaft (SFB 878 Groups, Geometry & Actions).
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Gardella, E., Thiel, H. Extending representations of Banach algebras to their biduals. Math. Z. 294, 1341–1354 (2020). https://doi.org/10.1007/s00209-019-02315-8
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DOI: https://doi.org/10.1007/s00209-019-02315-8