Injective envelopes and local multiplier algebras 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.

10 pages, LaTeX2e, one statement and its proof corrected

References in corpus (1)

• On Hahn-Banach type theorems for Hilbert C*-modules

Cited by in corpus (8)

• Maximal C*-algebras of quotients and injective envelopes of C*-algebras
• Characterizations of essential ideals as operator modules over C*-algebras
• On Hahn-Banach type theorems for Hilbert C*-modules
• Quasimultipliers of Operator Spaces
• When is the second local multiplier algebra of a C*-algebra equal to the first?
• Injective Envelopes and Local Multiplier Algebras of Some Spatial Continuous Trace C*-Algebras
• Injective Envelopes of Separable C*-algebras
• The second local multiplier algebra of a separable C*-algebra