  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.
