Multiplier algebra

An ideal I is essential if and only if I⊥, the "orthogonal complement" of I in the Hilbert C*-module B is {0}.

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.

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 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.

If I is an ideal in a C*-algebra B, then any faithful nondegenerate representation π of I can be extended uniquely to B.

Now take any faithful nondegenerate representation π of A 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 idealizer of π(A) in B(H).

Let X be a locally compact Hausdorff space, A = C0(X), the commutative C*-algebra of continuous functions that vanish at infinity.