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

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.