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


Author: Kari Ylinen
Journal: Proc. Amer. Math. Soc. 102 (1988), 599-602
MSC: Primary 46L05
DOI: https://doi.org/10.1090/S0002-9939-1988-0928987-3
MathSciNet review: 928987
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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

$\displaystyle B(x,y) = ({\mu _1}(x){\xi _1}\left\vert {{\mu _2}({y^ * }} \right.){\xi _2}){\text{ for all }}x \in {A_1},y \in {A_2}.$

The proof depends on the Grothendieck-Pisier-Haagerup inequality and Halmos's unitary dilation theorem. An extemely elementary proof of the latter is given.

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DOI: https://doi.org/10.1090/S0002-9939-1988-0928987-3
Article copyright: © Copyright 1988 American Mathematical Society

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