The purpose of the present paper is to put into the context of the theory of tensor products of topological algebras [9], [10] some recent results concerning certain properties of permanence of a tensor product of Banach algebras [4], [5]. A brief report of the main results of this paper has been given in [11]. In particular, we are concerned with the question to what extent the tensor product (in an appropriate topology) of two semi-simple topological algebras is an algebra of the same kind; this seems to have a certain particular interest. The case of Banach algebras has been considered in [16], [4] and [5]. A partial answer to this question (independently of [4], [5]) has been given in [10]. In this paper starting from a result of [5] we complete our previous result in [10] and also strengthen it to include the corresponding results of [4], [5] (cf. Th. 2.1, 2.2 below). Similarly Th. 3.1 strengthens the corresponding result in [10] extending in this manner those of [16], [4]. Moreover Th. 4.1 extends a result of [4]. Finally the last section contains some applications to certain function algebras.