Trace inequalities for spaces in spectral duality

O. Tikhonov

Studia Mathematica (1993)

  • Volume: 104, Issue: 1, page 99-110
  • ISSN: 0039-3223

Abstract

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Let (A,e) and (V,K) be an order-unit space and a base-norm space in spectral duality, as in noncommutative spectral theory of Alfsen and Shultz. Let t be a norm lower semicontinuous trace on A, and let φ be a nonnegative convex function on ℝ. It is shown that the mapping a → t(φ(a)) is convex on A. Moreover, the mapping is shown to be nondecreasing if so is φ. Some other similar statements concerning traces and real-valued functions are also obtained.

How to cite

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Tikhonov, O.. "Trace inequalities for spaces in spectral duality." Studia Mathematica 104.1 (1993): 99-110. <http://eudml.org/doc/215962>.

@article{Tikhonov1993,
abstract = {Let (A,e) and (V,K) be an order-unit space and a base-norm space in spectral duality, as in noncommutative spectral theory of Alfsen and Shultz. Let t be a norm lower semicontinuous trace on A, and let φ be a nonnegative convex function on ℝ. It is shown that the mapping a → t(φ(a)) is convex on A. Moreover, the mapping is shown to be nondecreasing if so is φ. Some other similar statements concerning traces and real-valued functions are also obtained.},
author = {Tikhonov, O.},
journal = {Studia Mathematica},
keywords = {order-unit space; base-norm space; spectral duality; noncommutative spectral theory of Alfsen and Shultz; norm lower semicontinuous trace},
language = {eng},
number = {1},
pages = {99-110},
title = {Trace inequalities for spaces in spectral duality},
url = {http://eudml.org/doc/215962},
volume = {104},
year = {1993},
}

TY - JOUR
AU - Tikhonov, O.
TI - Trace inequalities for spaces in spectral duality
JO - Studia Mathematica
PY - 1993
VL - 104
IS - 1
SP - 99
EP - 110
AB - Let (A,e) and (V,K) be an order-unit space and a base-norm space in spectral duality, as in noncommutative spectral theory of Alfsen and Shultz. Let t be a norm lower semicontinuous trace on A, and let φ be a nonnegative convex function on ℝ. It is shown that the mapping a → t(φ(a)) is convex on A. Moreover, the mapping is shown to be nondecreasing if so is φ. Some other similar statements concerning traces and real-valued functions are also obtained.
LA - eng
KW - order-unit space; base-norm space; spectral duality; noncommutative spectral theory of Alfsen and Shultz; norm lower semicontinuous trace
UR - http://eudml.org/doc/215962
ER -

References

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  1. [1] S. A. Ajupov, Extension of traces and type criterions for Jordan algebras of self-adjoint operators, Math. Z. 181 (1982), 253-268. 
  2. [2] E. M. Alfsen, Compact Convex Sets and Boundary Integrals, Ergeb. Math. Grenzgeb. 57, Springer, Berlin 1971. Zbl0209.42601
  3. [3] E. M. Alfsen and F. W. Shultz, Non-commutative spectral theory for affine function spaces on convex sets, Mem. Amer. Math. Soc. 172 (1976). Zbl0337.46013
  4. [4] M. A. Berdikulov, Traces on Jordan algebras, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 1986 (3), 11-15 (in Russian). Zbl0621.46055
  5. [5] F. A. Berezin, Convex operator functions, Mat. Sb. 88 (1972), 268-276 (in Russian). 
  6. [6] L. G. Brown and H. Kosaki, Jensen's inequality in semi-finite von Neumann algebras, J. Operator Theory 23 (1990), 3-19. Zbl0718.46026
  7. [7] T. Fack and H. Kosaki, Generalized s-numbers of τ-measurable operators, Pacific J. Math. 123 (1986), 269-300. Zbl0617.46063
  8. [8] A. Lieberman, Entropy of states of a gage space, Acta Sci. Math. (Szeged) 40 (1978), 99-105. Zbl0359.46039
  9. [9] D. Petz, Spectral scale of self-adjoint operators and trace inequalities, J. Math. Anal. Appl. 109 (1985), 74-82. Zbl0655.47032
  10. [10] D. Petz, Jensen's inequality for positive contractions on operator algebras, Proc. Amer. Math. Soc. 99 (1987), 273-277. Zbl0622.46044
  11. [11] O. E. Tikhonov, Inequalities for a trace on a von Neumann algebra, VINITI, Moscow 1982, No. 5602-82 (in Russian). 
  12. [12] O. E. Tikhonov, Convex functions and inequalities for traces, in: Konstr. Teor. Funktsiĭ i Funktsional. Anal. 6, Kazan Univ. 1987, 77-82 (in Russian). Zbl0719.46035
  13. [13] O. E. Tikhonov, Inequalities for spaces in spectral duality, connected with convex functions and traces, VINITI, Moscow 1987, No. 3591-B87 (in Russian). 
  14. [14] O. E. Tikhonov, On integration theory for spaces in spectral duality, in: Proc. 1st Winter School on Measure Theory (Liptovský Ján 1988), Slovak Acad. Sci., Bratislava 1988, 157-160. 
  15. [15] H. Upmeier, Automorphism groups of Jordan C*-algebras, Math. Z. 176 (1981), 21-34. Zbl0438.46050

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