  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.
