Journal of Operator Theory

Volume 64, Issue 1, Summer 2010 pp. 103-116.

Strict essential extensions of $C^*$-algebras and Hilbert $C^*$-modules

Authors: Michael Frank (1) and Alexander A. Pavlov (2)
Author institution: (1) Hochschule fuer Technik, Wirtschaft und Kultur (HTWK) Leipzig, Fachbereich IMN, PF 301166, D-04251 Leipzig, F.R.Germany
(2) Moscow State University, Department of Geography and Department of Mechanics and Mathematics, 119 992 Moscow, Russia

Summary: In the present paper we develop both ideas of~\cite{Bakic} and the categorical approach to multipliers from~\cite{Lance}, \cite{Pav1}, \cite{Pav2} for the introduction and study of left multipliers of Hilbert $C^*$-modules. Some properties and, in particular, the property of maximality among all strict essential extensions of a Hilbert $C^*$-module for left multipliers are proved. Also relations between left essential and left strict essential extensions in different contexts are obtained. Left essential and left strict essential extensions of matrix algebras are considered. In the final paragraph the topological approach to the left multiplier theory of Hilbert $C^*$-modules is worked out.

Keywords: Hilbert $C^*$-modules, $C^*$-algebras of operators, left multipliers, left strict essential extensions.

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