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

TL;DR

This paper develops the theory of Rota-Baxter operators within $C^{}$-algebras, exploring their properties, representations, and connections to quasidiagonal operators, thereby extending algebraic structures in functional analysis.

Contribution

It introduces Rota-Baxter $C^{}$-algebras, studies symmetric Rota-Baxter operators, and establishes their relationships with representations and quasidiagonal operators.

Findings

Established a theorem relating Rota-Baxter operators on concrete $C^{}$-algebras.

Connected $st$-representations with Rota-Baxter operators.

Reconstructed the notion of quasidiagonal operators using Rota-Baxter operators.

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.