Multiplicative versions of Klyachko’s theorem in finite factors Original Research Article

Let A and B be invertible positive elements in a II1-factor image, and let μs(·) be the singular number on image. We prove thatexp∫Klogμs(AB)dsless-than-or-equals, slantexp∫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.