A geometric interpretation of Segal’s inequality $\Vert e^ {X+Y}\Vert \leq \Vert e^ {X/2}e^ Ye^ {X/2}\Vert$

Abstract:

It is shown that the exponential mapping of the manifold of positive elements of a ${C^*}$-algebra (provided with its natural connection) increases distances (when measured in the natural Finsler structure). The proof relies on Segal’s inequality $||{e^{X + Y}}|| \leq ||{e^{X/2}}{e^Y}{e^{X/2}}||$, valid for all symmetric $X,Y$ in any ${C^*}$-algebra. In turn, this geometric inequality implies Segal’s’ inequality.