Journal of Operator Theory

Volume 49, Issue 2, Spring 2003 pp. 245-262.

Characterizations of essential ideals as operator modules over $C^*$-algebras

Authors: Masayoshi Kaneda (1) and Vern Ival Paulsen (2)
Author institution: (1) Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston, TX 77204-3008, USA
(2) Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston, TX 77204-3008, USA and Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, Japan

Summary: In this paper we give characterizations of essential left ideals of a $C^*$-algebra $A$ in terms of their properties as operator $A$-modules. Conversely, we seek $C^*$-algebraic characterizations of those ideals $J$ in $A$ such that $A$ is an essential extension of $J$ in various categories of operator modules. In the case of two-sided ideals, we prove that all the above concepts coincide. We obtain results, analogous to \mbox{M. Hamana's} results, which characterize the injective envelope of a $C^*$-algebra as a maximal essential extension of the $C^*$-algebra, but with completely positive maps replaced by completely bounded module maps. By restricting to one-sided ideals, module actions reveal clear differences which do not show up in the two-sided case. Throughout this paper, module actions are crucial.

Keywords: Essential ideals, left ideals, $C^*$-module extensions

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