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Journal of Operator Theory

Volume 48, Issue 3, Supplementary 2002  pp. 487-502.

On well-behaved unbounded representations of $*$-algebras

Authors Konrad Schmudgen
Author institution: Fakultat fur Mathematik und Informatik, Universitaet Leipzig, Augustusplatz 10, 04109 Leipzig, Germany

Summary:  A general approach to well-behaved unbounded $\ast$-repre\-sen\-tations of a $\ast$-algebra $\X$ is proposed. Let $\B$ be a normed $\ast$-algebra equipped with a left action $\ang$ of $\X$ on $\B$ such that $(x\ang a)^+ b=a^+(x^+\ang b)$ for $a,b\in\B$ and $x\in\X$. Then the pair $(\X,\B)$ is called a {\it compatible pair}. For any continuous non-degenerate $\ast$-representation $\rho$ of $\B$ there exists a closed $\ast$-representation $\rho^\prime$ of $\X$ such that $\rho^\prime(x)\rho(b)=\rho(x\ang b)$, where $x\in\X$ and $b\in\B$. The $\ast$-representations $\rho^\prime$ are called the {\it well-behaved $\ast$-representations} associated with the compatible pair $(\X,\B)$. A number of examples illustrating this concept ar e developed in detail.

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