nLab multiplier algebra

Idea

Given a nonunital C*-algebra $A$, the multiplier algebra $M(A)$ is the maximal unital extension of $A$ in which $A$ is an essential ideal (an ideal having nonzero intersection with any other proper ideal). In the commutative case, every unital commutative extension in which $A$ is an essential ideal corresponds to a compactification of the spectrum $Spec(A)$; $M(A)$ is in that case the $C^*$-algebra of functions on the Stone–Čech compactification $\beta(Spec(A))$.

Definition

The multiplier algebra $M(A)$ of a (not necessarily unital) $C^*$-algebra $A$ is the $C^*$-algebra satisfying the following universal property: for any $C^*$-algebra $B$ containing $A$ as an ideal, there exists a unique $*$-homomorphism $\phi:B\to M(A)$ such that $\phi$ extends the identity homomorphism on $A$ and $\phi(A^\perp)=\{0\}$.

The multiplier algebra $M(A)$ can be realized as the set of 2-sided multipliers in the enveloping von Neumann algebra of $A$. If $A\subset B$, $b\in B$ is a multiplier (operator) for $A$ if $\forall a\in A$, $b a\in A$ and $a b\in A$.

Of course, if $A$ is already unital, then $M(A) = A$.

Related concepts

There are also larger local multiplier algebra?s $M_{loc}(A)$.

Literature

• eom: Gert K. Pedersen, multiplier algebra

• wikipedia: multiplier algebra

• Ch.A. Akemann, G.K. Pedersen, J. Tomiyama, Multipliers of $C^*$-algebras, J. Funct. Anal. 13 (1973) 277–301 MR470685

• Paul Skoufranis, An introduction to multiplier algebras, (pdf)

• Corran Webster, On unbounded operators affiliated with $C^*$-algebras, J. Operator Theory 51 (2004) 237-244 dvi pdf ps

We show that the multipliers of Pedersen’s ideal of a $C^*$-algebra $A$ correspond to the densely defined operators on $A$ which are affiliated with $A$ in the sense defined by Woronowicz, and whose domains contain Pedersen’s ideal. We also extend the theory of q-continuity developed by Akemann to unbounded operators and show that these operators correspond to self-adjoint operators affiliated with $A$.

• S. L. Woronowicz, $C^*$-algebras generated by unbounded elements, pdf; Unbounded operators in the context of $C^*$-algebras, slides pdf

• Bruce Blackadar, K-Theory for Operator Algebras, Cambridge University Press 1998.

There is also a notion of multiplier Hopf algebra and even multiplier bimonoid and Hopf monoid:

• Gabriella Böhm, Stephen Lack, Multiplier Hopf monoids, arxiv/1511.03806; A category of multiplier bimonoids, arxiv/1509.07171

category: operator algebras, noncommutative geometry, analysis