Journal of Operator Theory
Volume 48, Issue 3, Supplementary 2002 pp. 487-502.
On well-behaved unbounded representations of ∗-algebrasAuthors: 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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