Abstract
We study a conjecture of Grothendieck on bilinear forms on a C ∗-algebra Ol . We prove that every “approximable” operator from Ol into Ol ∗ factors through a Hilbert space, and we describe the factorization. In the commutative case, this is known as Grothendieck's theorem. These results enable us to prove a conjecture of Ringrose on operators on a C ∗-algebra. In the Appendix, we present a new proof of Grothendieck's inequality which gives an improved upper bound for the so-called Grothendieck constant k G .
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