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Characterizations of essential ideals as operator modules over C-algebras.pdf

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Characterizations of essential ideals as operator modules over C-algebras

a r X i v : m a t h / 0 1 0 2 1 3 4 v 1 [ m a t h .O A ] 1 6 F e b 2 0 0 1 CHARACTERIZATIONS OF ESSENTIAL IDEALS AS OPERATOR MODULES OVER C*-ALGEBRAS MASAYOSHI KANEDA* AND VERN IVAL PAULSEN* Abstract. 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 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. Date: April 25, 2000. AMS 2000 subject classifications: Primary 46H10, 46H25, 46L07; Secondary 46A22, 46L05, 46L08, 46M10, 47D15, 47L25. Key Words: essential ideals; left ideals; C*-module extensions. * Supported by a grant from the NSF. 1 2 MASAYOSHI KANEDA* AND VERN IVAL PAULSEN* 1. Introduction. Let A denote a C*-algebra. In the C*-algebra literature a two-sided ideal, J , of A is called essential if aJ = {0} implies a = 0. Given a category and an object X contained in an object Y, then Y is called an essential extension of X provided that every morphism of Y that restricts to be an isomorphism of X must necessarily be an isomorphism of Y. In this paper we study the relationships between these two different approaches to essentiality as we vary the category. Every left ideal can be regarded as an object in the category of operator left A-modules, with morphisms either the completely contractive or completely bounded left A-module maps. Thus, it is interesting to know the relationships b