Multiplicative versions of Klyachko's theorem in finite factors

Let A and B be invertible positive elements in a II1-factor U, and let mu(s), ((.)) be the singular number on U. We prove that exp integral(k) log mu(s)(AB)ds <= exp integral(I) log mu(s)(A)ds (.) exp integral(J) log mu(s)(B)ds, where {I, J, K} is an analogue of Klyachko's list. In this paper, this family I I, J, K I must satisfy some hypotheses which are specific to operators A and B. But, we show that Our family of inequalities includes the weak Gelfand-Naimark inequality for all positive operators A and B. (c) 2007 Elsevier Inc. All rights reserved.