Grothendieck's theorem for noncommutative C1-algebras,with an Appendix on Grothendieck's constants

We study a conjecture of Grothendieck on bilinear forms on a C¹-algebra $\text{Ol}$. We prove that every “approximable” operator from $\text{Ol}$ into $\text{Ol}$¹ 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 C¹-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.