##
**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.

(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 |