Numerical Range Inclusion, Dilation, and Operator Systems

Abstract

Researchers have identified complex matrices $A$ such that a bounded linear operator $B$ acting on a Hilbert space will admit a dilation of the form $A \otimes I$ whenever the numerical range inclusion relation $W(B) \subseteq W(A)$ holds. Such an operator $A$ and the identity matrix will span a maximal operator system, i.e., every unital positive map from ${\rm span} \{I, A, A^*\}$ to ${\cal B}({\cal H})$, the algebra of bounded linear operators acting on a Hilbert space ${\cal H}$, is completely positive. In this paper, we identify $m$-tuple of matrices ${\bf A} = (A_1, \dots, A_m)$ such that any $m$-tuple of operators ${\bf B} = (B_1, \dots, B_m)$ satisfying the joint numerical range inclusion $W({\bf B}) \subseteq {\rm conv} W({\bf A})$ will have a joint dilation of the form $(A_1\otimes I, \dots, A_m\otimes I)$. Consequently, every unital positive map from ${\rm span} \{I, A_1, A_1^*, \dots, A_m, A_m^*\}$ to ${\cal B}({\cal H})$ is completely positive. New results and techniques are obtained relating to the study of numerical range inclusion, dilation, and maximal operator systems.