Extending representations of Banach algebras to their biduals

• Published in: Mathematische Zeitschrift Vol. 294; no. 3-4; pp. 1341 - 1354
• Main Authors: Gardella, Eusebio, Thiel, Hannes
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2020; Springer Nature B.V
• Subjects: Algebra; Banach spaces; Fourier analysis; Harmonic analysis; Mathematics; Mathematics and Statistics; Operators (mathematics); Representations; 46L05; Primary 46H15; Secondary 46E30; 47L10
• ISSN: 0025-5874; 1432-1823
• DOI: 10.1007/s00209-019-02315-8

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 → 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 ∈ [ 1 , ∞ ) \ { 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.