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**Locally pseudoconvex Gel’fand-Mazur algebras.**
*(Russian.
English summary)*
Zbl 0677.46030

Summary: The by now classical theorem of Gelfand-Mazur has been generalized to many different classes of topological algebras. The purpose of the present paper is to give some generalizations of this theorem for complex locally pseudoconvex algebras. (We recall that a complex locally pseudoconvex algebra is a locally pseudoconvex space over \({\mathbb{C}}\) which is at the same time an associative algebra with separately continuous multiplication.)

Let comm X denote the closed commutator ideal of a topological algebra X (i.e. the smallest closed two-sided ideal containing all commutators), let \({\mathfrak M}_{\ell}(X)\) (\({\mathfrak M}_ r(X))\) denote the set of all maximal regular left (respectively right) ideals of X and let \({\mathfrak M}_ C(X)\) denote the subset of all closed ideals of \({\mathfrak M}_{\ell}(X)\cap {\mathfrak M}_ r(X).\)

We say that a complex topological algebra X is a Gelfand-Mazur algebra if for every \(M\in {\mathfrak M}_ C(X)\), the quotient algebra X/M is topologically isomorphic to \({\mathbb{C}}\). Moreover, we say that a Gelfand- Mazur algebra X is a strictly Gelfand-Mazur algebra if the set \({\mathfrak M}_ C(X)\) is nonempty.

Theorem 1. Let X be a complex locally pseudoconvex algebra. then X is a Gelfand-Mazur algebra if any one of the following possibilities holds.

(a) For each \(x\in X\) there exists \(\lambda\in {\mathbb{C}}\setminus \{0\}\) such that the sequence \((x/\lambda)^ n\) converges to zero in X.

(b) X is a topological algebra with continuous quasi-inverse.

(c) X is a locally A-pseudoconvex algebra.

(d) X is a Fréchet algebra.

Theorem 2. Let X be either a complex locally multiplicatively convex algebra with unit or a Gelfand-Mazur Q-algebra with unit. Then X is a strictly Gelfand-Mazur algebra if and only if comm \(X\neq X\).

Let comm X denote the closed commutator ideal of a topological algebra X (i.e. the smallest closed two-sided ideal containing all commutators), let \({\mathfrak M}_{\ell}(X)\) (\({\mathfrak M}_ r(X))\) denote the set of all maximal regular left (respectively right) ideals of X and let \({\mathfrak M}_ C(X)\) denote the subset of all closed ideals of \({\mathfrak M}_{\ell}(X)\cap {\mathfrak M}_ r(X).\)

We say that a complex topological algebra X is a Gelfand-Mazur algebra if for every \(M\in {\mathfrak M}_ C(X)\), the quotient algebra X/M is topologically isomorphic to \({\mathbb{C}}\). Moreover, we say that a Gelfand- Mazur algebra X is a strictly Gelfand-Mazur algebra if the set \({\mathfrak M}_ C(X)\) is nonempty.

Theorem 1. Let X be a complex locally pseudoconvex algebra. then X is a Gelfand-Mazur algebra if any one of the following possibilities holds.

(a) For each \(x\in X\) there exists \(\lambda\in {\mathbb{C}}\setminus \{0\}\) such that the sequence \((x/\lambda)^ n\) converges to zero in X.

(b) X is a topological algebra with continuous quasi-inverse.

(c) X is a locally A-pseudoconvex algebra.

(d) X is a Fréchet algebra.

Theorem 2. Let X be either a complex locally multiplicatively convex algebra with unit or a Gelfand-Mazur Q-algebra with unit. Then X is a strictly Gelfand-Mazur algebra if and only if comm \(X\neq X\).

### MSC:

46H05 | General theory of topological algebras |