Corona algebra

In mathematics, the multiplier algebra, denoted by M(A), of a C*-algebra A is a unital C*-algebra which 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 operators on H.

An ideal I in a C*-algebra B is said to be essential if I ∩ J is non-trivial for all 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 the 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.

A double 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 b in A. 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.

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