Crossed products of Banach algebras. I

Abstract

We construct a crossed product Banach algebra from a Banach algebra dynamical system (A, G, α) and a given uniformly bounded class R of continuous covariant Banach space representations of that system. If A has a bounded left approximate identity, and R consists of non-degenerate continuous covariant representations only, then the non-degenerate bounded representations of the crossed product are in bijection with the non-degenerate R-continuous covariant representations of the system. This bijection, which is the main result of the paper, is also established for involutive Banach algebra dynamical systems and then yields the well-known representation theoretical correspondence for the crossed product C * -algebra as commonly associated with a C * -algebra dynamical system as a special case. Taking the algebra A to be the base field, the crossed product construction provides, for a given non-empty class of Banach spaces, a Banach algebra with a relatively simple structure and with the property that its non-degenerate contractive representations in the spaces from that class are in bijection with the isometric strongly continuous representations of G in those spaces. This generalizes the notion of a group C * -algebra, and may likewise be used to translate issues concerning group representations in a class of Banach spaces to the context of a Banach algebra, simpler than L 1 (G), where more functional analytic structure is present.

Amenable Crossed Product Banach Algebras Associated with a Class of $$\varvec{{\mathrm C}^*}$$ C ∗ -Dynamical Systems

“…Before doing so, we ought to note, in view of the considerations in the Introduction, that, according to [8, Theorem 5.13], ℓ 1 (G, A; α) is, in fact, a crossed product Banach algebra as in [4,Definition 3.2]. Furthermore, G is obviously a locally compact Hausdorff topological group, and, according to [16, p. 352], the C * -algebras under consideration are amenable.…”

“…Moreover, U is amenable (in our sense that there exists a left invariant mean on the space of bounded right uniformly continuous complex functions on U ) precisely when A is amenable. Furthermore, we see from [4,Proposition 6.4] that there are natural homomorphisms from G and A into the left centralizer algebra M l (L 1 (G, A; α)) of L 1 (G, A; α). The existence of these homomorphisms and their properties imply that M l (L 1 (G, A; α)) contains a group that is a homomorphic image of U ⋊ α G. With these ingredients, one could now attempt to incorporate modifications of the ideas in the proof of Theorem 2.4 into Johnson's proof of the amenability of L 1 (G) for general amenable G-which proceeds via the left centralizer algebra of L 1 (G))-and tackle the case of general G, A, and α. Needless to say, if such an approach is feasible, this will be considerably more technically demanding than the proof of Theorem 2.4.…”

“…According to [8,Theorem 5.13], L 1 (G) is an example of a crossed product Banach algebra that can be associated with a general Banach algebra dynamical system (A, G, α) as in [4, Definition 3.2]. More precisely: it is a member of a family of crossed product Banach algebras that can be associated with the C * -algebra dynamical system (C, G, triv), where the group acts as the identity on the C * -algebra C. Therefore, if G is amenable, the amenable Banach algebra L 1 (G) is a crossed product of the amenable group G and the amenable C * -algebra C. Extrapolating this quite a bit, could it perhaps be the case that the 'only if'-part of Johnson's theorem is a reflection of an underlying general principle, stating that, under appropriate additional conditions, crossed products of amenable locally compact Hausdorff topological groups and amenable C * -algebras, associated with C * -dynamical systems as in [4,Definition 3.2], are always amenable Banach algebras?…”

“…As we have formulated this result, we have used that amenability as a Banach algebra and nuclearity as a C * -algebra are equivalent for C * -algebras; see [2] and [7]. This formulation is deliberate, because [4,Remark 9.4] shows that A ⋊ α G is, in fact, a crossed product Banach algebra as in [4,Definition 3.2]. Therefore, a principle as alluded to above could conceivably explain, from a result in Banach algebra theory, why these C * -algebras A⋊ α G are amenable Banach algebras (and therefore nuclear C * -algebras).…”

“…Let us note, however, that not all Banach algebras that are a crossed product of an amenable C * -algebra and an amenable locally compact Hausdorff topological group as in [4,Definition 3.2] are amenable. This already fails for the C * -algebra C and the group Z.…”