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**Topological algebras. Selected topics.**
*(English)*
Zbl 0597.46046

North-Holland Mathematics Studies, 124. Notas de Matemática, 109. Amsterdam etc.: North-Holland. XVII, 535 p. $ 64.75; Dfl. 175.00 (1986).

The general concept of a normed algebra was introduced in 1936 by M. Nagumo. The study of ideals in normed algebras was achieved by I. Gel’fand in a series of papers published in 1941. The topological imbedding of what we call today a \(C^*\)-algebra into the algebra of linear bounded operators on a Hilbert space was proved in 1943 by I. Gel’fand and M. A. Naĭmark. These are some of the main stages which led to the development of the contemporary Banach algebra theory.

Nevertheless, the existence of topological algebras that are not normed, which are of interest for both the theory and applications, has forced the mathematicians to investigate the possibility of adaptation of the framework of Banach algebra theory to more general situations. The present book is dedicated to such investigations. The first part of the book deals with the general theory of topological algebras, in particular of locally convex ones. Topological tensor products of topological algebras are presented in the second part of the book. The author tried, and we think that he succeeded, to state the theorems in their ”most useful generality”. The book also contains a large variety of examples. The occurence of monographs dedicated to this subject is a necessity and many specialists in functional analysis will welcome the present work.

Nevertheless, the existence of topological algebras that are not normed, which are of interest for both the theory and applications, has forced the mathematicians to investigate the possibility of adaptation of the framework of Banach algebra theory to more general situations. The present book is dedicated to such investigations. The first part of the book deals with the general theory of topological algebras, in particular of locally convex ones. Topological tensor products of topological algebras are presented in the second part of the book. The author tried, and we think that he succeeded, to state the theorems in their ”most useful generality”. The book also contains a large variety of examples. The occurence of monographs dedicated to this subject is a necessity and many specialists in functional analysis will welcome the present work.

Reviewer: F.-H.Vasilescu

### MSC:

46H05 | General theory of topological algebras |

46M05 | Tensor products in functional analysis |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |