On the A. M. Bikchentaev conjecture

In 1998 A. M. Bikchentaev conjectured that for positive $\tau$-measurable operators $a$ and $b$ affiliated with a semifinite von Neumann algebra, the operator $b^{1/2}ab^{1/2}$ is submajorized by the operator $ab$ in the sense of Hardy–Littlewood. We prove this conjecture in its full generality and obtain a number of consequences for operator ideals, Golden–Thompson inequalities, and singular traces.