Symmetric topological*-algebras

Abstract

V. Ptak's 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 Raikov's 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 Frechet 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.