Some classes of locally linearly compact generalized bounded Algebras
by
Valeriu Bordan
Tiraspol State University headquarter at Chisinau (Str. Gh. Iablocichin 5, Chisinau, Republic of Moldova)

I introduce and study in this paper locally linearly compact algebras having neighborhoods of zero with special properties. The class of IN-algebras contain the class of all bounded locally linearly compact algebras. We give here necessary and sufficient conditions under which the semidirect product A x A* is IN-algebra.

I consider only topological algebras over discrete field. All topological algebras are assumed to be Hausdorff and associative.

If A is a locally linearly compact topological vector F-space, then denote by A* the dual space of A. By A x A* is denoted semidirect product of A and A*.

Definition
1.       A topological algebra A is called a LIN-algebra (RIN-algebra), if it contains an open linearly compact vector subspace V, such that A·V is included in V (V·A is included in V).

Definition
2.       A Topological algebra A is called an IN-algebra, if it contains an open two-sided ideal V which is a linearly compact space.

Theorem
For a locally linearly compact algebra A the following conditions are equivalent:

1) A x A* is an IN-algebra;

2) A x A* is a LIN-algebra;

3) A x A* is a RIN-algebras;

4) A is an IN-algebra and there exist an open linearly compact subspace U of A such that U² ={0}.

Date received: February 27, 2003