Abstract. We prove that the crossed product Banach algebra 1 (G, A; α) that is associated with a C * -dynamical system (A, G, α) is amenable if G is a discrete amenable group and A is a commutative or finite dimensional C * -algebra. Perspectives for further developments are indicated.Mathematics Subject Classification. Primary 47L65; Secondary 43A07, 46H25, 46L55.
Abstract. A unital Fréchet algebra A is called contractible if there exists an element d ∈ A⊗A such that π A (d) = 1 and ad = da for all a ∈ A where π A : A⊗A → A is the canonical Fréchet A-bimodule morphism. We give a sufficient condition for an infinite-dimensional contractible Fréchet algebra A to be a direct sum of a finite-dimensional semisimple algebra M and a contractible Fréchet algebra N without any nonzero finite-dimensional two-sided ideal (see Theorem 1). As a consequence, a commutative lmc Fréchet Q-algebra is contractible if, and only if, it is algebraically and topologically isomorphic to C n for some n ∈ N. On the other hand, we show that a Fréchet algebra, that is, a locally C * -algebra, is contractible if, and only if, it is topologically isomorphic to the topological Cartesian product of a certain countable family of full matrix algebras.It is well known that in the finite-dimensional case, a complex algebra is contractible (separable) if, and only if, it is a semisimple algebra [7]. An infinitedimensional contractible Banach algebra has yet to be found. In the Fréchet algebra case, there exist some infinite-dimensional contractible algebras. Hence, it is very important to study contractible Fréchet algebras, which are useful as a class of topological algebras. We begin by introducing the concept of Fréchet modules, and we recall some results on projective Fréchet modules. Later, we prove some results concerning a class of contractible Fréchet algebras.A Fréchet space is a complete metrizable locally convex complex space. An algebra A will be called a Fréchet algebra if A is a Fréchet space with jointly continuous multiplication. For any two spaces X and Y , we write X⊗Y for the completed projective tensorial product [3]. Let A be a unital Fréchet algebra.
We give sufficient conditions that allow contractible (resp., reflexive amenable) Banach algebras to be finite-dimensional and semisimple algebras. Moreover, we show that any contractible (resp., reflexive amenable) Banach algebra in which every maximal left ideal has a Banach space complement is indeed a direct sum of finitely many full matrix algebras. Finally, we characterize Hermitian *-algebras that are contractible
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.