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 $\ast$-representations of a $\ast$-algebra $\mathcal X$ is proposed. Let $\mathcal B$ be a normed $\ast$-algebra equipped with a left action $\triangleright$ of $\mathcal X$ on $\mathcal B$ such that $(x\triangleright a)^+ b=a^+(x^+\triangleright b)$ for $a,b\in\mathcal B$ and $x\in\mathcal X$. Then the pair $(\mathcal X,\mathcal B)$ is called a compatible pair. For any continuous non-degenerate $\ast$-representation $\rho$ of $\mathcal B$ there exists a closed $\ast$-representation $\rho^\prime$ of $\mathcal X$ such that $\rho^\prime(x)\rho(b)=\rho(x\triangleright b)$, where $x\in\mathcal X$ and $b\in\mathcal B$. The $\ast$-representations $\rho^\prime$ are called the well-behaved $\ast$-representations associated with the compatible pair $(\mathcal X,\mathcal B)$. A number of examples illustrating this concept are developed in detail.

Keywords: Unbounded representations, quantum groups

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