The structure of bounded bilinear forms on products of $C^ *$-algebras

Abstract:

Let ${A_1}$ and ${A_2}$ be ${C^ * }$-algebras and $B:{A_1} \times {A_2} \to {\mathbf {C}}$ be a bounded bilinear form. It is proved that there exist a Hilbert space $H$, two Jordan morphisms ${\mu _i}:{A_i} \to L(H),i = 1,2$, and two vectors ${\xi _1},{\xi _2} \in H$ such that \[ B(x,y) = ({\mu _1}(x){\xi _1}\left | {{\mu _2}({y^ * }} \right .){\xi _2}){\text { for all }}x \in {A_1},y \in {A_2}.\] The proof depends on the Grothendieck-Pisier-Haagerup inequality and Halmos’s unitary dilation theorem. An extemely elementary proof of the latter is given.