The Effros-Ruan conjecture for bilinear forms on C^*-algebras

Abstract

In 1991 Effros and Ruan conjectured that a certain Grothendieck-type inequality for a bilinear form on C$^*$-algebras holds if (and only if) the bilinear form is jointly completely bounded. In 2002 Pisier and Shlyakhtenko proved that this inequality holds in the more general setting of operator spaces, provided that the operator spaces in question are exact. Moreover, they proved that the conjecture of Effros and Ruan holds for pairs of C$^*$-algebras, of which at least one is exact. In this paper we prove that the Effros-Ruan conjecture holds for general C$^*$-algebras, with constant one. More precisely, we show that for every jointly completely bounded (for short, j.c.b.) bilinear form on a pair of C$^*$-algebras $A$ and $B$, there exist states $f_1$, $f_2$ on $A$ and $g_1$, $g_2$ on $B$ such that for all $a\in A$ and $b\in B$, |u(a, b)| \leq ||u||_{jcb}(f_1(aa^*)^{1/2}g_1(b^*b)^{1/2} + f_2(a^*a)^{1/2}g_2(bb^*)^{1/2}) . While the approach by Pisier and Shlyakhtenko relies on free probability techniques, our proof uses more classical operator algebra theory, namely, Tomita-Takesaki theory and special properties of the Powers factors of type III$_λ$, $0< λ< 1$ .