Inventiones mathematicae

, Volume 174, Issue 1, pp 139–163

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


DOI: 10.1007/s00222-008-0137-7

Cite this article as:
Haagerup, U. & Musat, M. Invent. math. (2008) 174: 139. doi:10.1007/s00222-008-0137-7


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 f1, f2 on A and g1, g2 on B such that for all aA and bB,
$$|u(a, b)|\leq\|u\|_{\mathrm{jcb}}\big(f_1(aa^*)^{1/2}g_1(b^*b)^{1/2}+f_2(a^*a)^{1/2}g_2(bb^*)^{1/2}\big).$$
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 IIIλ, 0<λ<1.

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdense MDenmark
  2. 2.Department of Mathematical SciencesUniversity of MemphisMemphisUSA

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