Characterizations of Essential Ideals as Operator Modules over C*-algebras
نویسنده
چکیده
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.
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