Suppose that \(A\) is a unital \(C^*\)-algebra. The injective envelope \(I(A)\) of \(A\) was introduced by M. Hamana [J. Math. Soc. Japan 31, 181–197 (1979; Zbl 0395.46042)]. The present paper is essentially devoted to the study of the injective envelope \(I(A)\) as the “minimal” injective left operator \(A\)-module containing \(A\).
The authors discuss the rigidity as follows: If \(E\) is a subspace of \(I(A)\) with \(A\subseteq E\) and \(AE\subseteq E\) and \(\varphi:E\to I(A)\) is a completely bounded left \(A\)-module map such that \(\varphi(a)=a\) for all \(a\in A\), then \(\varphi(e)=e\) for all \(e\in E\). They conclude that \(A\) is an injective \(C^*\)-algebra if and only if \(A\) is an \(A\)-\({\mathbb C}\)-injective module.
A nice natural representation of many kinds of multipliers as multiplications by elements of \(I(A)\) is given. In particular, it is shown that \(I(A)\) contains the local multiplier algebra of \(A\).
Most results can be carried over to the case of non-unital \(C^*\)-algebra by adjoining a unity [see D. Blecher and V. I. Paulsen, Pac. J. Math. 200, No. 1, 1–17 (2001; Zbl 1053.46518)].