PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 5, Pages 1251–1255 S 0002-9939(03)07198-3 Article electronically published on October 9, 2003 CONTRACTIBLE FR ´ ECHET ALGEBRAS RACHID EL HARTI (Communicated by Joseph A. Ball) Abstract. A unital Fr´ echet algebra A is called contractible if there exists an element d ∈ A ˆ ⊗A such that π A (d)=1 and ad = da for all a ∈ A where π A : A ˆ ⊗A → A is the canonical Fr´ echet A-bimodule morphism. We give a sufficient condition for an infinite-dimensional contractible Fr´ echet algebra A to be a direct sum of a finite-dimensional semisimple algebra M and a con- tractible Fr´ echet algebra N without any nonzero finite-dimensional two-sided ideal (see Theorem 1). As a consequence, a commutative lmc Fr´ echet 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 Fr´ echet 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 con- tractible (separable) if, and only if, it is a semisimple algebra [7]. An infinite- dimensional contractible Banach algebra has yet to be found. In the Fr´ echet alge- bra case, there exist some infinite-dimensional contractible algebras. Hence, it is very important to study contractible Fr´ echet algebras, which are useful as a class of topological algebras. We begin by introducing the concept of Fr´ echet modules, and we recall some results on projective Fr´ echet modules. Later, we prove some results concerning a class of contractible Fr´ echet algebras. A Fr´ echet space is a complete metrizable locally convex complex space. An algebra A will be called a Fr´ echet algebra if A is a Fr´ echet space with jointly continuous multiplication. For any two spaces X and Y , we write X ˆ ⊗Y for the completed projective tensorial product [3]. Let A be a unital Fr´ echet algebra. Following ([4], [5]), we define a Fr´ echet left A-module X to be a Fr´ echet space that is also a unital left A-module such that the linear map A ˆ ⊗X → X , a ⊗ x → ax, is continuous. Right modules are defined analogously. A Fr´ echet A-bimodule is a Fr´ echet space with structural A-bimodule such that the linear map A ˆ ⊗X ˆ ⊗A → X , a ⊗x ⊗b → axb is continuous. A Fr´ echet A-bimodule X is said to be projective if the canonical morphism π X : A ˆ ⊗X ˆ ⊗A → X has a right inverse Fr´ echet A-bimodule morphism. A morphism τ in the category of Fr´ echet modules is called C-split if its kernel and its image both have a direct complement as Fr´ echet 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 1251 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use