Injective envelopes and local multiplier algebras of C*-algebras

TL;DR

The paper investigates how local multiplier algebras $M_{loc}(A)$ sit inside the injective envelope $I(A)$ of a C*-algebra $A$, establishing a canonical embedding that coincides with $I(A)$ when $A$ is commutative. It derives center-identification results under density assumptions, shows that $I(A)$, $I(M(A))$, and $I(M_{loc}(A))$ coincide, and that higher-order local multipliers lie in the regular monotone completion $\overline{A}$ inside $I(A)$. It further links the possibility of $M_{loc}(A)$ being injective to Wittstock-type extension theorems for completely bounded bimodule maps, and outlines several open problems (including Kaplansky’s monotone completeness question) and concrete examples. These results advance intrinsic characterizations of injective envelopes and the structure of multiplier-type algebras, with implications for AW*-algebras and centers of C*-algebras.

Abstract

The local multiplier C*-algebra M_{loc}(A) of any C*-algebra A can *-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 M_{loc}(A) = I(A) . The injective envelopes of A and M_{loc}(A) always coincide, and every higher order local multiplier C*-algebra of A is contained in the regular monotone completion \bar{A} in I(A) of A . In case the set Z(A).A is dense in 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 M_{loc}(A) is proven to hold if and only if M_{loc}(A) = I(A). In general, a solution of the problem for which C*-algebras A the C*-algebras M_{loc}(A) is injective is shown to be equivalent to the solution of I. Kaplansky's 1951 problem whether all AW*-algebras are monotone complete.