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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.

References [Enhancements On Off] (What's this?)

  • [1] G. Corach, H. Porta, and L. Recht, The geometry of the space of self-adjoint invertible elements of a $ {C^*}$-algebra, preprint form in Trabajos de Matemática no. 149, Instituto Argentino de Matemática, December 1989.
  • [2] M. Kobayashi and K. Nomizu, Foundations of differential geometry, Interscience, New York, London, and Sydney, 1969.
  • [3] M. Reed and B. Simon, Methods of modern mathematical physics, Academic Press, New York, San Francisco, and London, 1975. MR 0493420 (58:12429b)
  • [4] I. Segal, Notes toward the construction of nonlinear relativistic quantum fields. III, Bull. Amer. Math. Soc. 75 (1969), 1390-1395. MR 0251992 (40:5217)

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