Injective Envelopes and Local Multiplier Algebras of C*-algebras

The local multiplier C*-algebra Mloc(A) of any C*-algebra A can be ∗-isomorphicly embedded into the injective envelope I(A) of A in such a way that the canonical embeddings of A into both these C*-algebras are identified. If A is commutative then Mloc(A) ≡ I(A). The injective envelopes of A and Mloc(A) always coincide, and every higher order local multiplier C*-algebra of A is contained in the regular monotone completion A ⊆ I(A) of A. The center of the local multiplier C*-algebra of A is the local multiplier C*-algebra of the center of A, and both they are ∗-isomorphic to the injective envelope of the center of A. A Wittstock type extension theorem for completely bounded bimodule maps on operator bimodules taking values in Mloc(A) is proven to hold if and only if Mloc(A) ≡ I(A). In general, a solution of the problem for which C*algebras A the C*-algebras Mloc(A) is injective is shown to be equivalent to the solution of I. Kaplansky’s 1951 problem whether all AW*-algebras are monotone complete. The injective envelope of a C*-algebra in the category of C*-algebras and completely positive linear maps is defined by an extrinsic algebraic characterization. M. Hamana showed its general existence and uniqueness up to ∗-isomorphism, cf. [15]. The main problem is to determine the injective envelope I(A) of a given C*-algebra A from the structure of A, i.e. intrinsicly. For commutative C*-algebras an intrinsic characterization was given by H. Gonshor in [13, 14]. He relied on I. M. Gel’fand’s theorem for commutative C*-algebras and on the topology of the locally compact Hausdorff space X corresponding to a commutative C*-algebra A = C0(X). Unfortunately, there seems to be no obvious way to extend his results to the non-commutative case. The injective envelope of noncommutative C*-algebras has been described only for some examples and special classes of C*-algebras yet. One of the goals of this short note is an intrinsic algebraic characterization of the injective envelopes of commutative C*-algebras identifying them with their local multiplier algebras. In the general non-commutative case we show the existence of a canonical injective ∗-homomorphism mapping the local multiplier algebra into the corresponding injective envelope (Th. 1). The question whether the image of this map coincides with the entire injective envelope has a negative answer, in general, and is connected with I. Kaplansky’s still unsolved problem on the monotone completeness of AW*-algebras in the case ofAW*-algebras. So this class of C*-algebras cannot be characterized in full at present. We indicate some examples (Cor. 7). However, the center of the local multiplier C*-algebra of A always coincides with the local multiplier C*-algebra of the center of A, and both they can be identified with the injective envelope of the center of A: Z(Mloc(A)) ≡ Mloc(Z(A)) ≡ I(Z(A)) (Th. 2). Moreover, the injective envelopes 1991 Mathematics Subject Classification. Primary 46L05; Secondary 46L08, 46L35, 46M10.