Let A be a topological *-algebra, that is, a topological algebra on which an involution a [right arrow] [a.sup.*] has been given.

Now we consider the case when A is a topological *-algebra.

Let A be a sequentially Mackey complete topological *-algebra. If a [member of] A and S(a) is bounded in A, then there exists an element b [member of] A such that b [omicron] b = a and [[beta].sub.A](b) [less than or equal to] 1.

Let A be a unital sequentially Mackey complete topological *-algebra. If a [member of] A and S([e.sub.A] - a) is bounded in A, then there exists an element b [member of] A such that [b.sup.2] = a and [[beta].sub.A] ([e.sub.A] - b) [less than or equal to] 1.

Let A be a unital sequentially Mackey complete topological *-algebra. If a [member of] A is self-adjoint and S([e.sub.A] - a) is bounded in A, then there exists a self-adjoint element b [member of] A such that [b.sup.2] = a and [[beta].sub.A]([e.sub.A] - b) [less than or equal to] 1.

Let A be a unital sequentially Mackey complete topological *-algebra with continuous involution, for which [[beta].sub.A](a) = [[rho].sub.A](a) for each a [member of] A.

Among the topics are functional algebras of operators generated by a self-adjoint operator in Pontryagin space ?1, Wedderburn structure theorems for two-sided locally m-convex H*-algebras, main embedding theorems for symmetric spaces of measurable functions, discrete non-closed subsets in maximally non-discrete topological groups, faithfully representable topological

*-algebras: some spectral properties, and dense ideals in topological algebras: some results and open problems.