Let $H$ be a nonzero complex [Hilbert space](/page/Hilbert%20Space), and let $1_H:H\to H$ denote the identity operator. Let $B(H)$ denote the unital C*-algebra of bounded linear operators $H\to H$, and let $K(H)\subset B(H)$ denote the C*-algebra of compact operators on $H$, equivalently the operator-norm closure in $B(H)$ of the linear span of the rank-one operators on $H$. Let $M(K(H))$ denote the multiplier algebra of $K(H)$ realized as double centralizers: pairs $(L,R)$ of bounded linear maps $K(H)\to K(H)$ satisfying, for all $A,B\in K(H)$,

\begin{align*} L(AB)=L(A)B,\qquad R(AB)=AR(B),\qquad AL(B)=R(A)B. \end{align*}

The product and involution on double centralizers are given by

\begin{align*} (L,R)(L',R')=(L\circ L',R'\circ R) \end{align*}

and by $(L,R)^*=(L^\#,R^\#)$, where for every $S\in K(H)$,

\begin{align*} L^\#(S)=R(S^*)^*,\qquad R^\#(S)=L(S^*)^*. \end{align*}

Then there is a canonical unital $*$-isomorphism

\begin{align*} B(H) \cong M(K(H)). \end{align*}

Under this isomorphism, an operator $T\in B(H)$ corresponds to the double centralizer $(L_T,R_T)$ on $K(H)$ defined, for every $S\in K(H)$, by

\begin{align*} L_T(S)=TS,\qquad R_T(S)=ST. \end{align*}