## Locally convex algebras.(English)Zbl 1010.46046

The paper under review (the author’s thesis) consists of four Chapters (from 0 to 3). The notations are given in Chapter 0 and the main notions, illustrative examples and auxiliary results in Chapter 1. All topological algebras considered here have a jointly continuous multiplication. Examples of topological algebras $$A$$, which are not idempotent topological algebras, or whose topological dual space $$A'=\{\theta_A\}$$ (here $$\theta _A$$ denotes the zero element in $$A$$) are presented. It is shown that
(a) every jointly continuous bilinear map from $$X\times Y$$ into $$Z$$ has a jointly continuous bilinear extension to $$\tilde{X}\times \tilde{Y}$$ in the case when $$X,\;Y,\;Z,\;\tilde{X}$$ and $$\tilde{Y}$$ are Hausdorff linear spaces (by this result it is easy to show that the completion of any topological Hausdorff algebra with a jointly continuous multiplication is a complete Hausdorff algebra with a jointly continuous multiplication);
(b) every locally $$m$$-convex (complete locally $$m$$-convex) Hausdorff algebra is topologically isomorphic to the dense subalgebra of the projective limit of Banach algebras;
(c) the quasi–inversion in every locally $$m$$-convex Hausdorff algebra is continuous;
(d) the spectrum of every element of any complex locally $$m$$-convex Hausdorff algebra is not empty and
(e) the multiplication in $$A[X]$$ (the algebra of polynomials with coefficients in a locally convex algebra $$A$$ endowed with the direct sum topology) is jointly continuous, if the multiplication in $$A$$ is jointly continuous.
Examples of such topological linear spaces $$X$$, which have (or have not) the so-called ”countable neighbourhood property” (i.e., for any sequence $$(U_n)$$ of neighbourhoods of zero in $$X$$ there is a sequence $$(\varrho_n)$$ of positive numbers such that the intersection of all sets $$\varrho_nU_n$$ is a neighbourhood of zero in $$X$$) are introduced and topological algebras $$(A,\tau)$$, which are a topological semidirect product of his two-sided ideal $$C$$ and subalgebra $$B$$ (then $$A=C+B$$, $$C\cap B=\{\theta _A\}$$ and the map $$(c,b)\mapsto c+b$$ is a homeomorphism from $$(C,\tau\cap C)\times (B,\tau\cap B)$$ onto $$(A,\tau)$$) are described. It is shown, that every locally convex algebra, which is a topological semidirect product of a locally $$m$$-convex two-sided ideal and a locally $$m$$-convex subalgebra, is a locally $$m$$-convex algebra.
Properties connected with (continuous) linear multiplicative functionals on a locally convex algebra $$A$$ are considered in Chapter 2. It is shown that every locally complete locally $$m$$-convex Hausdorff algebra with a fundamental sequence of bounded sets is functionally bounded (i.e., every linear multiplicative functional $$\chi$$ on $$A$$ has the property: $$\chi (B)$$ is bounded on all bounded subsets $$B\subset A$$) and that every locally $$m$$-convex Fréchet algebra $$A$$ is functionally continuous (i.e., every linear multiplicative functional $$\chi$$ on $$A$$ is continuous) if and only if all locally complete and locally $$m$$-convex Hausdorff algebras are functionally bounded. Moreover, it is shown that every nontrivial linear multiplicative functional $$\Phi$$ on $$C(X,A)$$ (the algebra of all continuous $$A$$-valued functions on $$X$$ with point-wise algebraic operations) defines a point $$x\in X$$ and a nontrivial linear multiplicative functional $$\phi$$ on $$A$$ such that $$\Phi(f)=\phi(f(x))$$ for each $$f\in C(X,A)$$ in the following cases: (a) $$X$$ is a realcompact space and $$A$$ is a metrizable algebra with unity and (b) $$X$$ is a compact space and $$A$$ is a locally convex Hausdorff algebra, which satisfies the strict Mackey condition (i.e., for each bounded subset $$B\subset A$$ there is an absolutely convex bounded subset $$D\subset A$$ such that $$B\subset D$$ and the Minkowski funktional $$p_D$$ of $$D$$ induces the original topology on $$B$$). Similar descriptions of nontrivial linear multiplicative functionals on several other algebras of vector-valued functions are given, too.
Locally $$m$$-convex inductive topologies on countable inductive limits of locally $$m$$-convex algebras are considered in Chapter 3. It is shown that the finest locally convex topology on a countable inductive limit $$A$$ of seminormed algebras $$A_n$$ (or commutative locally $$m$$-convex algebras $$A_n$$, which satisfy the countable neighbourhood condition) is locally $$m$$-convex and every countable inductive limit of locally convex algebras $$A_n$$, which satisfy the countable neighbourhood condition, is a locally convex algebra, if every $$A_n$$ is continuously embedded into $$A_{n+1}$$ and all the inclusions $$A_n\to A$$ are continuous. Conditions which yield that the inductive limit of Moscatelli type of locally convex (locally $$m$$-convex) algebras is again a locally convex locally $$m$$-convex) algebra, are considered separately.
Several results presented in this paper are known, but reproved here. Some of these hold also in the locally pseudoconvex or locally $$m$$-pseudoconvex case.
Reviewer: Mati Abel (Tartu)

### MSC:

 46H05 General theory of topological algebras 46A04 Locally convex Fréchet spaces and (DF)-spaces 46E25 Rings and algebras of continuous, differentiable or analytic functions 46H20 Structure, classification of topological algebras

### Keywords:

locally convex algebras