Rota-Baxter $C^{\ast }$-algebras

Abstract

This paper introduces the notion of Rota-Baxter $C^{\ast }$-algebras. Here a Rota-Baxter $C^{\ast }$-algebra is a $C^{\ast }$-algebra with a Rota-Baxter operator. Symmetric Rota-Baxter operators, as special cases of Rota-Baxter operators on $C^{\ast }$-algebra, are defined and studied. A theorem of Rota-Baxter operators on concrete $C^{\ast }$-algebras is given, deriving the relationship between two kinds of Rota-Baxter algebras. As a corollary, some connection between $\ast$-representations and Rota-Baxter operators is given. The notion of representations of Rota-Baxter $C^{\ast }$-algebras are constructed, and a theorem of representations of direct sums of Rota-Baxter representations is derived. Finally using Rota-Baxter operators, the notion of quasidiagonal operators on $C^{\ast }$-algebra is reconstructed.

20 pages