Well-behaved unbounded operator representations and unbounded \(C^*\)-seminorms
The theory of unbounded \(*\)-representations of \(*\)-algebras as presented by \textit{K. Schmüdgen} [``Unbounded operator algebras and representation theory'' (Oper. Theory Adv. Appl. 37, Birk\-häuser Verlag, Basel) (1989; Zbl 0697.47048)] has been complemented by the notion of well behaved \(*\)-representations by \textit{S. J. Bhatt}, \textit{A. Inoue} and \textit{H. Ogi} [J. Oper. Theory 45, No. 1, 53--80 (2001; Zbl 0994.47074)]. In the present paper, problems investigated by Bhatt, Inoue and Ogi for \(*\)-algebras are transferred into the framework of locally convex \(*\)-algebras \({\mathcal A}\). Conditions for the existence of well-behaved \(*\)-representations are given, and the existence of spectral well-behaved \(*\)-representations is characterized. (A \(*\)-representation \(\pi\) of \({\mathcal A}\) is said to be spectral if \(Sp_{{\mathcal A}_b}(x)\subset Sp_{C^*_u(\pi)}(\overline{\pi(x)})\) for each \(x\in{\mathcal A}_b\), where \({\mathcal A}_b\) is the \(*\)-subalgebra of \({\mathcal A}\) generated by the hermitian part of the Allan bounded part of \({\mathcal A}\), and where \(C^*_u(\pi)\) denotes the \(C^*\)-algebra generated by \(\overline{\pi({\mathcal A}_b)}\).) A large number of examples demonstrates the range of the results.
existence of spectral \(*\)-representations
scientific article; zbMATH DE number 2110890