Inmathematics, the multiplier algebra, denoted by M(A), of a C*-algebra A is a unital C*-algebra that is the largest unital C*-algebra that contains A as an ideal in a "non-degenerate" way. It is the noncommutative generalization of Stone–Čech compactification. Multiplier algebras were introduced by Busby (1968).
For example, if A is the C*-algebra of compact operators on a separable Hilbert space, M(A) is B(H), the C*-algebra of all bounded operatorsonH.
An ideal I in a C*-algebra B is said to be essentialifI ∩ J is non-trivial for every ideal J. An ideal I is essential if and only if I⊥, the "orthogonal complement" of I in the Hilbert C*-module B is {0}.
Let A be a C*-algebra. Its multiplier algebra M(A) is any C*-algebra satisfying the following universal property: for all C*-algebra D containing A as an ideal, there exists a unique *-homomorphism φ: D → M(A) such that φ extends the identity homomorphism on A and φ(A⊥) = {0}.
Uniqueness up to isomorphism is specified by the universal property. When A is unital, M(A) = A. It also follows from the definition that for any D containing A as an essential ideal, the multiplier algebra M(A) contains D as a C*-subalgebra.
The existence of M(A) can be shown in several ways.
Adouble centralizer of a C*-algebra A is a pair (L, R) of bounded linear maps on A such that aL(b) = R(a)b for all a and binA. This implies that ||L|| = ||R||. The set of double centralizers of A can be given a C*-algebra structure. This C*-algebra contains A as an essential ideal and can be identified as the multiplier algebra M(A). For instance, if A is the compact operators K(H) on a separable Hilbert space, then each x ∈ B(H) defines a double centralizer of A by simply multiplication from the left and right.
Alternatively, M(A) can be obtained via representations. The following fact will be needed:
Lemma.IfI is an ideal in a C*-algebra B, then any faithful nondegenerate representation πofI can be extended uniquelytoB.
Now take any faithful nondegenerate representation πofA on a Hilbert space H. The above lemma, together with the universal property of the multiplier algebra, yields that M(A) is isomorphic to the idealizerofπ(A) in B(H). It is immediate that M(K(H)) = B(H).
Lastly, let 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.IfI is an ideal in a C*-algebra B, then any faithful nondegenerate *-homomorphism πofI into B(E)can be extended uniquelytoB.
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.
Consider the topology on M(A) specified by the seminorms {la, ra}a ∈ A, where
The resulting topology is called the strict topologyonM(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.
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 spectrumofCb(X). Y is in fact homeomorphic to the Stone–Čech compactification βXofX.
The coronaorcorona algebraofA 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.
