Multiplicative versions of Klyachko’s theorem in finite factors

Let A and B be invertible positive elements in a II1-factor AA, and let μs(·) be the singular number on AA. We prove thatexp∫Klogμs(AB)ds⩽exp∫Ilogμs(A)ds·exp∫Jlogμs(B)ds,exp∫Klogμs(AB)ds⩽exp∫Ilogμs(A)ds·exp∫Jlogμs(B)ds,where {I, J, K} is an analogue of Klyachko’s list. In this paper, this family {I, J, K} 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.