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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.
Rota-Baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics. Rota-Baxter algebras provide a construction of pre-Lie algebras from associative algebras
The program of matrix product states on the infinite tensor product ${mathcal A}^{otimes mathbb Z}$, initiated by Fannes, Nachtergaele and Werner in their seminal paper Commun. Math. Phys. Vol. 144, 443-490 (1992), is re-assessed in a context where $
We generalize the notion of a Rota-Baxter operator on groups and the notion of a Rota-Baxter operator of weight 1 on Lie algebras and define and study the notion of a Rota-Baxter operator on a cocommutative Hopf algebra $H$. If $H=F[G]$ is the group
This paper first introduces the notion of a Rota-Baxter operator (of weight $1$) on a Lie group so that its differentiation gives a Rota-Baxter operator on the corresponding Lie algebra. Direct products of Lie groups, including the decompositions of
Inspired by results for graph $C^*$-algebras, we investigate connections between the ideal structure of an inverse semigroup $S$ and that of its tight $C^*$-algebra by relating ideals in $S$ to certain open invariant sets in the associated tight grou