Injective Envelopes of Separable C * -algebras

Characterisations of those separable C∗-algebras that have type I injective envelopes or W∗-algebra injective envelopes are presented. An operator system I is injective if for every inclusion E ⊂ F of operator systems each completely positive linear map ω : E → I has a completely positive extension to F . An injective envelope of an operator system E is an injective operator system I such that E ⊆ I and I is minimal among all injective operator systems that contain E. That is, if E ⊆ I0 ⊆ I, with I0 injective, then I0 = I. Hamana [11] proved that every operator system E has an injective envelope and that all injective envelopes of E are completely isometric. Because every injective operator system is completely order isomorphic to an injective C-algebra [5], and because two Calgebras are ∗-isomorphic if and only if they are completely order isomorphic [4], one can unambiguously refer to “the” injective envelope of E, which is an injective C-algebra I(E) that contains E as an operator system. If E is a C-algebra A, then A is contained in I(A) as a C-subalgebra. The purpose of the present paper is to study how properties of a C-algebra A determine properties of its injective envelope, especially in the case of separable C-algebras A. The injective envelope I(A) of any C-algebra A is a monotone complete Calgebra. Thus, I(A) is a direct sum of AW-algebras of types I, II, and III. Herein we show that if A is separable, then I(A) has no direct summand that is finite and of type II. Further, we show that a separable C-algebra A has a type I injective envelope if and only if A has a liminal essential ideal. We also characterise those separable C-algebras A for which I(A) is a W-algebra. There are a number of other useful enveloping structures that contain a given C-algebra A as a C-subalgebra. Of these, the local multiplier algebra Mloc(A) [1, 10, 21] and the regular monotone completion A [25, 17, 12] have important roles in arriving at our our results. These structures, together with the injective envelope, are discussed in the following preliminary section. This research is supported in part by the Natural Sciences and Engineering Research Council of Canada. 2000 Mathematics Subject Classification: Primary 46L05; Secondary 46L07 .