## Contractible Fréchet algebras

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- by Rachid El Harti
- Proc. Amer. Math. Soc.
**132**(2004), 1251-1255 - DOI: https://doi.org/10.1090/S0002-9939-03-07198-3
- Published electronically: October 9, 2003
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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 ${\mathbb {C}}^ n$ for some $n \in \mathbb {N}$. 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.## References

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## Bibliographic Information

**Rachid El Harti**- Affiliation: University Hassan I, Department of Mathematics, FST of Settat, BP 577, Settat, Morocco
- Email: elharti@uh1.ac.ma
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**132**(2004), 1251-1255 - MSC (2000): Primary 13E40, 46H05, 46J05, 46K05
- DOI: https://doi.org/10.1090/S0002-9939-03-07198-3
- MathSciNet review: 2053328