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Contractible Fréchet algebras

Author: Rachid El Harti
Journal: Proc. Amer. Math. Soc. 132 (2004), 1251-1255
MSC (2000): Primary 13E40, 46H05, 46J05, 46K05
Published electronically: October 9, 2003
MathSciNet review: 2053328
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Abstract: A unital Fréchet algebra $A$ is called contractible if there exists an element $d \in A \hat{\otimes} A$ such that $\pi_A (d) = 1$ and $ad = da $ for all $ a\in A$ where $\pi_A: A \hat{\otimes} A \to A$ 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 ${\mathbb {C}}\sp n$ for some $n \in \mathbb {N}$. On the other hand, we show that a Fréchet algebra, that is, a locally $C\sp*$-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.

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Additional Information

Rachid El Harti
Affiliation: University Hassan I, Department of Mathematics, FST of Settat, BP 577, Settat, Morocco

Received by editor(s): March 14, 2002
Received by editor(s) in revised form: January 8, 2003
Published electronically: October 9, 2003
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2003 American Mathematical Society

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