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


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