Multiplier algebra


In mathematics, the multiplier algebra, denoted by M, 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.
For example, if A is the C*-algebra of compact operators on a separable Hilbert space, M is B, the C*-algebra of all bounded operators on H.

Definition

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.
Let A be a C*-algebra. Its multiplier algebra M 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 such that φ extends the identity homomorphism on A and φ =.
Uniqueness up to isomorphism is specified by the universal property. When A is unital, M = A. It also follows from the definition that for any D containing A as an essential ideal, the multiplier algebra M contains D as a C*-subalgebra.
The existence of M can be shown in several ways.
A double centralizer of a C*-algebra A is a pair of bounded linear maps on A such that aL = R'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. For instance, if A is the compact operators K on a separable Hilbert space, then each x ∈ B defines a double centralizer of A by simply multiplication from the left and right.
Alternatively, M can be obtained via representations. The following fact will be needed:
Lemma. 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 is isomorphic to the idealizer of π in B. It is immediate that M = B.
Lastly, let E be a Hilbert C*-module and B be the adjointable operators on E M can be identified via a *-homomorphism of A into B. 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 Bcan be extended uniquely to B.
Consequently, if π is a faithful nondegenerate *-homomorphism of A into B, then M is isomorphic to the idealizer of π. For instance, M = B for any Hilbert module E.
The C*-algebra A is isomorphic to the compact operators on the Hilbert module A. Therefore, M is the adjointable operators on A''.

Strict topology

Consider the topology on M specified by the seminorms a ∈ A, where
The resulting topology is called the strict topology on M. A is strictly dense in M.
When A is unital, M = A, and the strict topology coincides with the norm topology. For B = M, the strict topology is the σ-strong* topology. It follows from above that B is complete in the σ-strong* topology.

Commutative case

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