Double centralizer of a C*-algebra

Let $A$ be a $C^*$ -algebra. A double centralizer of $A$ is a pair $(L,R)$ of bounded linear maps $A\to A$ satisfying

L(ab)=L(a)b,\qquad R(ab)=aR(b),\qquad aL(b)=R(a)b

for all $a,b\in A$ . Thus $L$ is a left multiplier, $R$ is a right multiplier, and the last identity couples them. For $x\in A$ , multiplication gives the double centralizer $L_x(a)=xa$ , $R_x(a)=ax$ , but nonunital algebras generally have double centralizers not arising from elements of $A$ .

C^*

-algebra operations

Double centralizers form a unital $C^*$ -algebra. Products reverse the order on the right:

(L_1,R_1)(L_2,R_2) =(L_1\mathbin{\circ}L_2,R_2\mathbin{\circ}R_1).

The adjoint is

(L,R)^*=(L^\sharp,R^\sharp),\qquad L^\sharp(a)=R(a^*)^*,\quad R^\sharp(a)=L(a^*)^*,

and $\lVert(L,R)\rVert=\lVert L\rVert=\lVert R\rVert$ . The identity pair is $(\operatorname{id}_A,\operatorname{id}_A)$ . These formulas are forced by viewing a pair as left and right multiplication by one generalized element.

Intrinsic multiplier algebra

The algebra of all double centralizers is canonically the multiplier algebra $M(A)$ . The map

A\longrightarrow M(A),\qquad x\longmapsto(L_x,R_x),

is an injective $*$ -homomorphism whose image is an essential ideal. If $A$ is unital, every double centralizer is multiplication by the single element $L(1)=R(1)$ , so $M(A)=A$ . For nonunital $A$ , this construction adjoins multipliers without choosing a representation on a Hilbert space.

Concrete model and distinction

For a nondegenerate concrete representation $A\subseteq B(\mathcal H)$ on a Hilbert space, $M(A)$ can be realized as the operators $T\in B(\mathcal H)$ satisfying $TA\subseteq A$ and $AT\subseteq A$ . In particular, $M(K(\mathcal H))=B(\mathcal H)$ . A left multiplier alone is not a double centralizer until a compatible right multiplier is supplied; nor is an arbitrary bounded endomorphism of the Banach space $A$ a multiplier. Busby's construction makes this distinction precise.

References

Robert C. Busby, “Double Centralizers and Extensions of $C^*$ -Algebras,” Transactions of the American Mathematical Society 132 (1968), 79–99. AMS DOI record. Relevant: the double-centralizer construction and its $C^*$ -algebra structure.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §3.12 on multiplier algebras and double centralizers.