Dense subalgebras in noncommutative Jordan topological algebras

Abstract

Wilansky conjectured in [12] that normed dense Q-algebras are full subalgebras of Banach algebras. Beddaa and  Oudadess proved in [2] that Wilansky’s conjecture was true. They showed that k-normed Q-algebras are full subalgebras of k-Banach algebras for each k∈(0,1]. Moreover, J. Pérez, L. Rico and A. Rodríguez showed in [8], Theorem 4, that this was also true in the case of noncommutative Jordan-Banach algebras. In the present paper this problem has been studied in a more general case. It is proved that all dense Q-subalgebras of topological algebras and of noncommutative Jordan topological algebras with continuous multiplication are full subalgebras. Some equivalent conditions that a dense subalgebra would be a Q-algebra (in subspace topology) in Q-algebras and in nonassociative Jordan Q-algebras with continuous multiplication are given.