Let A be a unital \(C^*\) -algebra. An A-multiplier cover is a \(C^*\) -algebra E together with a faithful non-degenerate \(*\) -homomorphism from A to M(E). We preorder such covers by A-preserving unital completely positive maps between their multiplier algebras. We prove that Hamana’s injective envelope I(A) is a greatest cover in this preorder and that the maximal rigid covers are precisely those whose multiplier algebra is canonically \(*\) -isomorphic to I(A). Consequently, a maximal rigid cover is greatest, rather than merely maximal among rigid covers. For \(A=C(X)\) , we further classify these covers: after the canonical identification with C(G(X)), where G(X) is the Gleason cover, their underlying ideals are exactly the algebras \(C_0(U)\) for dense open \(C^*\) -embedded subsets U of G(X). Dense cozero subsets provide an important special case.
