Contractible Fréchet Algebras
نویسندگان
چکیده
A unital Fréchet algebra A is called contractible if there exists an element d ∈ A⊗̂A such that πA(d) = 1 and ad = da for all a ∈ A where πA : 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 Cn 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. Following ([4], [5]), we define a Fréchet left A-module X to be a Fréchet space that is also a unital left A-module such that the linear map A⊗̂X → X , a ⊗ x → ax, is continuous. Right modules are defined analogously. A Fréchet A-bimodule is a Fréchet space with structural A-bimodule such that the linear map A⊗̂X⊗̂A→ X , a⊗x⊗b→ axb is continuous. A Fréchet A-bimodule X is said to be projective if the canonical morphism πX : A⊗̂X⊗̂A → X has a right inverse Fréchet A-bimodule morphism. A morphism τ in the category of Fréchet modules is called C-split if its kernel and its image both have a direct complement as Fréchet subspaces. Received by the editors March 14, 2002 and, in revised form, January 8, 2003. 2000 Mathematics Subject Classification. Primary 13E40, 46H05, 46J05, 46K05. c ©2003 American Mathematical Society
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