Injective envelopes of a Hilbert \(C^*\)-module (Q2764277)

As in homology, a Hilbert \(C^*\)-module \(H\) over a \(C^*\)-algebra \(A\) is called \textit{injective} if for any Hilbert \(C^*\)-modules \(X\subset Y\) over \(A\) every bounded \(A\)-module operator \(T:X\to H\) extends to an operator \(Y\to H\) with the same norm. A Hilbert \(C^*\)-module \(E\supset H\) over \(A\) is an \textit{injective envelope} of \(H\) if \(E\) is injective and \(id_E\) is the only contractive \(A\)-module operator that extends the inclusion \(H\subset E\). It is shown that every Hilbert \(C^*\)-module over a \(W^*\)-algebra has a unique injective envelope. For general \(C^*\)-algebras a Hilbert \(C^*\)-module may have no injective envelope, but it is unique if it exists.