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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: G. Corach, H. Porta and L. Recht
Journal: Proc. Amer. Math. Soc. 115 (1992), 229-231
MSC: Primary 46L99; Secondary 58B20
MathSciNet review: 1075945
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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 $ \vert\vert{e^{X + Y}}\vert\vert \leq \vert\vert{e^{X/2}}{e^Y}{e^{X/2}}\vert\vert$, valid for all symmetric $ X,Y$ in any $ {C^*}$-algebra. In turn, this geometric inequality implies Segal's' inequality.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1075945-8
PII: S 0002-9939(1992)1075945-8
Article copyright: © Copyright 1992 American Mathematical Society