Maria Fragoulopoulou

Symmetric topological*-algebras

V. Ptáks inequality is valid for every hermitian completeQ locallym-convex (:l.m.c.) algebra. Every algebra of the last kind is, in particular, symmetric. Besides, a (Hausdorff) locallyC*-algebra (being always symmetric) with the propertyQ is, within a topological algebraic isomorphism, aC*-algebra. Furthermore, a type of Raikovs criterion for symmetry is also valid for non-normed topological*-algebras. Concerning topological tensor products, one gets that symmetry of theπ-completed tensor product of two unital Fréchet l.m.c.*-algebrasE, F (π denotes the projective tensorial topology) is always passed toE, F, while the converse occurs when moreover either ofE, F is commutative.

Representations of tensor product l.m.c.*-algebras

If E is a l.m.c.*-algebra with a b.a.i., ℰ(E), ℛ(E) denote the enveloping algebra and the space of representations of E respectively, while ℬ(E) stands for the non-zero extreme points of the continuous positive linear forms on E. Thus, for suitable l.m.c.*-algebras E, F and an admissible topology on E ⊗ F, ℰ(E F) is given by the completedυ-tensor product of ⊗(E), ⊗(F) (whereυ is the projective tensorial l.m.c.C*-topology), while ℛ(E F) by the cartesian product of ℛ(E), ℛ(F). An analogous decomposition of ℬ(E F) is not valid in general.

Kadison's transitivity for locally C∗-algebras

Abstract The classical transitivity theorem of R. Kadison for C∗-algebras is here extended to the case of a locally C∗-algebra E. As a consequence, within the same context, various standard facts referred to the space of representations of E are obtained, broadening thus naturally out an earlier framework considered by this author, the relevant results being namely obtained hitherto only for bQ locally m-convex ∗-algebras.

∗-Semisimplicity of tensor product LMC ∗-algebras

Abstract Each (Hausdorff) lmc C∗-algebra is ∗-semisimple. The ∗-semisimplicity of two suitable lmc∗-algebras is passed on to their completed E -tensor product iff E is faithful. A sort of strong converse is also valid. In the commutative case, ∗-semisimplicity implies semisimplicity, whereas the converse occurs for suitable lmc∗-algebras.

Structure of contractible locally *-algebras

A locally C∗-algebra is contractible iff it is topologically isomorphic to the topological cartesian product of a certain family of full matrix algebras.

Abstract Bochner-Weil-Raikov theorem in topological algebras

Each continuous positive linear form on a commutative locally m -convex *-algebra E with a bounded approximate identity, accepts an integral representation on the hermitian spectrum (hermitian characters) of E . An alternative form of the latter is also obtained. The presented results constitute an abstract form of the Bochner-Weil-Raikov theorem within the frame of topological *-algebras.