Multiplier module - Knowlpedia

Let $E$ be a right Hilbert $C^*$ -module over a $C^*$ -algebra $A$ . Its multiplier module is

M(E)=\mathcal L_A(A,E),

the space of adjointable $A$ -linear maps from the standard Hilbert $A$ -module $A$ to $E$ . Since $\mathcal L_A(A)\cong M(A)$ , the multiplier algebra, $M(E)$ is a Hilbert $M(A)$ -module with

(T\cdot m)(a)=T(m(a)),\qquad \langle S,T\rangle_{M(A)}=S^*T.

The canonical map $E\to M(E)$ sends $x$ to the multiplier $a\mapsto xa$ .

Embedding and strict completion

The canonical map is an isometric $A$ -module embedding and identifies $E$ with $M(E)A$ . Its image is strictly dense in $M(E)$ : if $(e_\lambda)$ is an approximate identity for $A$ , then $T\cdot e_\lambda$ comes from $E$ and converges strictly to $T$ . The strict topology is generated by the seminorms

T\longmapsto\lVert T(a)\rVert,\qquad T\longmapsto\lVert T^*(x)\rVert

for $a\in A$ and $x\in E$ .

Examples and structure

For the standard module $E=A$ , one recovers $M(E)=M(A)$ . If $A$ is unital, evaluation at $1_A$ identifies $M(E)$ with $E$ , so the construction adds no new vectors. For a full $E$ , the multiplier module is full over $M(A)$ , and adjointable operators on $E$ extend uniquely to strictly continuous adjointable operators on $M(E)$ .

Conventions and scope

The construction is most useful for nonunital coefficient algebras and should not be confused with the multiplier algebra of the $C^*$ -algebra $\mathcal K_A(E)$ , although the two are closely related. Some sources reserve “multiplier module” for full Hilbert modules or describe it as a maximal essential extension; fullness is not needed for the displayed definition.

References

E. Christopher Lance, Hilbert $C^*$ -Modules: A Toolkit for Operator Algebraists, London Mathematical Society Lecture Note Series 210, Cambridge University Press, 1995. DOI record. Relevant: Chapter 2, “Multipliers and morphisms.”
Damir Bakić and Boris Guljaš, “Extensions of Hilbert $C^*$ -Modules,” Houston Journal of Mathematics 30 (2004), 537–558. Journal volume record. Relevant: the multiplier module as a maximal essential extension.