Extending representations of Banach algebras to their biduals

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 → 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.

Preduals and complementation of spaces of bounded linear operators

“…The results in this paper are the foundations for two other works of the authors; see [GT17] and [GT16].…”

Similarity of operators on $l^{p}$

“…So ∞ r=1 ψ t k (r) (a)P r − ψ 1 (a) has finite rank for every k ≥ 1. Hence to prove (iv), it suffices to show that The isometric version of this problem was solved by Gardella and Thiel [27]. A separable C * -algebra A is isometrically isomorphic to a subalgebra of B(l p ) if and only if A is commutative.…”

“…In 2012 and 2013, Phillips [51], [52], [53] introduced L p versions of some C * -algebra notions including UHF algebras and Cuntz algebras. See [25], [26], [32] [24], [27], [54] for some results in this direction. Most of the results about algebras of operators on L p that appear in the literature are in the "isometric theory," i.e., the "isomorphisms" (in the sense of category theory) between algebras of operators on L p are the isometric isomorphisms.…”

L^p operator algebras with approximate identities I

“…Subsequently others have followed him into this inquiry (for example, Gardella, Thiel, Lupini, and Viola; see e.g. [24,25,27,23,48]). However, as he has frequently stated, these investigations have been very largely focused on examples; one still lacks an abstract general theory in this setting.…”