In this paper we give some
characterizations of M. Hamana’s injective envelope I(A) of a C∗-algebra A in the
setting of operator spaces and completely bounded maps. These characterizations
lead to simplifications and generalizations of some known results concerning
completely bounded projections onto C∗-algebras. We prove that I(A) is rigid for
completely bounded A-module maps. This rigidity yields a natural representation of
many kinds of multipliers as multiplications by elements of I(A). In particular, we
prove that the(n times iterated) local multiplier algebra of A embeds into
I(A).