Multiplier Algebra

Idea

“Minimal” compactification: Given a locally compact Hausdorff space $X$, the $C^\ast$-algebra $A=C_0(X)$ is commutative. If $A$ can be embedded as an ideal in a unital $C^\ast$-algebra $D$ as an ideal, this corresponds to the embedding of $X$ as an open set in a compact Hausdorff space $Y$. If $Y=X^\dagger$ is the on-point compactification, this embedding is minimal. The $C^\ast$-analog is to take $D=A^\ast$, the unitization.

“Maximal” compactification: Another way to compactify $X$ is to take the Stone-Cech compactification $\beta X$. The $C^\ast$-analog for this is to take the multiplier algebra $M(A)$ of $A$. Namely, for $X$ a locally compact Hausdorff space, we have $M(C_0(X))\cong C_b(X)$ and also $\operatorname{Spec}(C_b(X))\cong \beta X$.

Definition

Definition(Multiplier algebra) . Let $A$ be a $C^\ast$-algebra. The multiplier algebra $M(A)$ of $A$ is the unital $C^\ast$-algebra that
  • Contains $A$ as an essential ideal;
  • If $A$ is in an ideal of a $C^\ast$-algebra $D$, the identity map on $A$ extends uniquely to a $\ast$-homomorphism $D\to M(A)$ whose kernel is $A_D^\bot$.
Where $A^\bot$ are the annihilators of $A$ in the respective $C^\ast$-algebra and recall that $J$ is an essential ideal in $B$ iff $J^\bot=\{0\}$ or equivalently $J\cap I \neq \{0\}$ for every nonzero closed ideal $I$ of $B$.

It can be proved that $M(A)$ exists and is unique up to isomorphism over $A$.

Crossed product

For the crossed product algebra $A\rtimes_\alpha G$ of a $C^\ast$-algebra $A$ and a locally compact topological group $G$ together with a continuous (for the point-norm topology) action $\alpha:G\to \operatorname{Aut}(A)$, $A$ and the group of unitaries isomorphic to $G$ sits inside the multiplier algebra $M(A\rtimes_\alpha G)$.