$*$-Bimodules

TL;DR

This paper introduces the concept of $*$-bimodules for unital $*$-algebras, providing an algebraic framework, defining Hilbert space representations, and establishing a GNS-like theorem for these structures.

Contribution

It develops an algebraic model for $*$-bimodules, explores their Hilbert space representations, and proves a GNS-like representation theorem.

Findings

Provided an algebraic model for $*$-bimodules.

Defined and studied Hilbert space representations.

Established a GNS-like representation theorem.

Abstract

A $*$-bimodule for a unital $*$-algebra $A$ is an $A$-bimodule $X$ which is a vector space with involution $x\mapsto x^+$ satisfying $(a\cdot x\cdot b)^+=b^+\cdot x^+\cdot b^+$ for $x\in X$ and $a,b\in A$. An algebraic model for $*$-bimodules is given. Hilbert space representations of $*$-bimodules are defined and studied. A GNS-like representation theorem is obtained.