E be a Hilbert C*-module and B(E) (resp. K(E)) be the adjointable (resp. compact) operators on E M(A) can be identified via a *-homomorphism of A into B(E). Something similar to the above lemma is true: Lemma. If I is an ideal in a C*-algebra B, then any faithful nondegenerate *-homomorphism π of I into B(E)can be extended uniquely to B. Consequently, if π is a faithful nondegenerate *-homomorphism of A into B(E), then M(A) is isomorphic to the idealizer of π(A). For instance, M(K(E)) = B(E) for any Hilbert module E. The C*-algebra A is isomorphic to the compact operators on the Hilbert module A. Therefore, M(A) is the adjointable operators on A. Strict topology Consider the topology on M(A) specified by the seminorms {la, ra}a ∈ A, where The resulting topology is called the strict topology on M(A). A is strictly dense in M(A) . When A is unital, M(A) = A, and the strict topology coincides with the norm topology. For B(H) = M(K(H)), the strict topology is the σ-strong* topology. It follows from above that B(H) is complete in the σ-strong* topology. Commutative case Let X be a locally compact Hausdorff space, A = C0(X), the commutative C*-algebra of continuous functions that vanish at infinity. Then M(A) is Cb(X), the continuous bounded functions on X. By the Gelfand–Naimark theorem, one has the isomorphism of C*-algebras where Y is the spectrum of Cb(X). Y is in fact homeomorphic to the Stone–Čech compactification βX of X. Corona algebra The corona or corona algebra of A is the quotient M(A)/A. For example, the corona algebra of the algebra of compact operators on a Hilbert space is the Calkin algebra. The corona algebra is a noncommutative analogue of the corona set of a topological space. References B. Blackadar, K-Theory for Operator Algebras, MSRI Publications, 1986.