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Q-measures on the dual unit ball of a JB${}^*$-triple

C. Martin Edwards|Lina Oliveira
대한수학회 (The Korean Mathematical Society)|2019.01|대한수학회지 (Journal of the KMS)|제56권 제1호|pp. 197-224|DOI 10.4134/JKMS.j180138

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참고문헌

키워드

JB$^*$-tripleC$^*$-algebraQ-topologyQ-measurecompact support tripotent
주제분류수학

초록

Let $A$ be a JB$^*$-triple with Banach dual space $A^*$ and bi-dual the JBW$^*$-triple $A^{**}$. Elements $x$ of $A^*$ of norm one may be regarded as normalised `Q-measures' defined on the complete ortho-lattice $\tilde{\mathcal U}(A^{**})$ of tripotents in $A^{**}$. A Q-measure $x$ possesses a support $e(x)$ in $\tilde{\mathcal U}(A^{**})$ and a compact support $e_c(x)$ in the complete atomic lattice $\tilde{\mathcal U}_c(A)$ of elements of $\tilde{\mathcal U}(A^{**})$ compact relative to $A$. Necessary and sufficient conditions for an element $v$ of $\tilde{\mathcal U}_c(A)$ to be a compact support tripotent $e_c(x)$ are given, one of which is related to the Q-covering numbers of $v$ by families of elements of $\tilde{\mathcal U}_c(A)$.

Abstract

Let $A$ be a JB$^*$-triple with Banach dual space $A^*$ and bi-dual the JBW$^*$-triple $A^{**}$. Elements $x$ of $A^*$ of norm one may be regarded as normalised `Q-measures' defined on the complete ortho-lattice $\tilde{\mathcal U}(A^{**})$ of tripotents in $A^{**}$. A Q-measure $x$ possesses a support $e(x)$ in $\tilde{\mathcal U}(A^{**})$ and a compact support $e_c(x)$ in the complete atomic lattice $\tilde{\mathcal U}_c(A)$ of elements of $\tilde{\mathcal U}(A^{**})$ compact relative to $A$. Necessary and sufficient conditions for an element $v$ of $\tilde{\mathcal U}_c(A)$ to be a compact support tripotent $e_c(x)$ are given, one of which is related to the Q-covering numbers of $v$ by families of elements of $\tilde{\mathcal U}_c(A)$.

참고문헌 (53)

  1. C. A. Akemann, "The general Stone-Weierstrass problem", J. Functional Analysis, 4, pp.277~294 (1969)
  2. C. A. Akemann, "Regularity of projections revisited", J. Operator Theory, 48(3, suppl), pp.515~534 (2002)
  3. E. M. Alfsen, "Compact Convex Sets and Boundary Integrals" (1971), Springer-Verlag
  4. E. M. Alfsen, "State spaces of Jordan algebras", Acta Math, 3(4), pp.155~190 (1978)
  5. E. M. Alfsen, "A Gelfand-Naimark theorem for Jordan algebras", Adv. Math, 28, pp.11~56 (1978)
  6. L. Asimow, "Convexity Theory and its Applications in Functional Anal-ysis" (1980), Academic Press, Inc
  7. T. J. Barton, "Normal representations of Banach Jordan triple systems", Proc. Amer. Math. Soc, 102(3), pp.551~555 (1988)
  8. T. J. Barton, "Weak*-continuity of Jordan triple products and its applications", Math. Scand, 59(2), pp.177~191 (1986)
  9. M. Battaglia, "Order theoretic type decomposition of JBW*-triples", Quart. J. Math. Oxford Ser, 42(166), pp.129~147 (1991)
  10. F. F. Bonsall, "Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras" (1971), Cambridge University Press
  11. H. G. Dales, "Second duals of measure algebras", Dissertationes Mathematicae (Rozprawy Matematyczne), 481, pp.1~121 (2012)
  12. S. Dineen, "Complete holomorphic vector fields on the second dual of a Banach space", Math. Scand, 59(1), pp.131~142 (1986)
  13. S. Dineen, "Complex analysis, functional analysis and approximation theory", pp.67~69 (1986), North-Holland
  14. C. M. Edwards, "On Jordan W*-algebras", Bull. Sci. Math, 104(4), pp.393~403 (1980)
  15. C. M. Edwards, "On the facial structure of the unit ball in a JB*-triple", J. Reine Angew. Math, 641, pp.123~144 (2010)
  16. C. M. Edwards, "Decoherence in pre-symmetric spaces", Rev. Mat. Complut, 21(1), pp.219~249 (2008)
  17. C. M. Edwards, "A geometric characterization of structural projections on a JBW*-triple", J. Funct. Anal, 202(1), pp.174~194 (2003)
  18. C. M. Edwards, "Local facial structure and norm-exposed faces of the unit ball in a JB*-triple", J. Math. Anal. Appl, 421(2), pp.1315~1333 (2015)
  19. C. M. Edwards, "On the facial structure of the unit balls in a GL-space and its dual", Math. Proc. Cambridge Philos. Soc, 98(2), pp.305~322 (1985)
  20. C. M. Edwards, "On the facial structure of the unit ball of a GM-space", Math. Z, 193(4), pp.597~611 (1986)
  21. C. M. Edwards, "On the facial structure of the unit balls in a JBW*-triple and its predual", J. London Math. Soc, 38(2), pp.317~332 (1988)
  22. C. M. Edwards, "Compact tripotents in bi-dual JB*-triples", Math. Proc. Cambridge Philos. Soc, 120(1), pp.155~173 (1996)
  23. C. M. Edwards, "The lattice of compact tripotents in bi-dual JB*-triples", Atti Sem. Mat. Fis. Univ. Modena, 45(1), pp.155~168 (1997)
  24. C. M. Edwards, "Exposed faces of the unit ball in a JBW*-triple", Math. Scand, 82(2), pp.287~304 (1998)
  25. C. M. Edwards, "Orthogonal faces of the unit ball in a Banach space", Atti Sem. Mat. Fis. Univ. Modena, 49(2), pp.473~493 (2001)
  26. E. G. Effros, "Order ideals in a C*-algebra and its dual", Duke Math. J, 30, pp.391~411 (1963)
  27. F. J. Fernandez-Polo, "Compact tripotents and the Stone-Weierstrass theorem for C*-algebras and JB*-triples", J. Operator Theory, 58(1), pp.157~173 (2007)
  28. F. J. Fernandez-Polo, "Non-commutative generalisations of Urysohn's lemma and hereditary inner ideals", J. Funct. Anal, 259(2), pp.343~358 (2010)
  29. F. J. Fernandez-Polo, "On the facial structure of the unit ball in the dual space of a JB*-triple", Math. Ann, 348(4), pp.1019~1032 (2010)
  30. Y. Friedman, "Jordan algebras", pp.61~82 (1994), de Gruyter
  31. Y. Friedman, "Physical Applications of Homogeneous Balls" (2005), Birkhauser Boston, Inc
  32. Y. Friedman, "Why does the geometric product simplify the equations of physics?", Internat. J. Theoret. Phys, 41(10), pp.1841~1855 (2002)
  33. Y. Friedman, "Relativistic linear spacetime transformations based on symmetry", Found. Phys, 32(11), pp.1717~1736 (2002)
  34. Y. Friedman, "Structure of the predual of a JBW*-triple", J. Reine Angew. Math, 356, pp.67~89 (1985)
  35. H. Hanche-Olsen, "Jordan Operator Algebras" (1984), Pitman (Advanced Publishing Program)
  36. D. J. Hebert, "On supports of regular Borel measures", Pacific J. Math, 27, pp.101~118 (1968)
  37. G. Horn, "Characterization of the predual and ideal structure of a JBW*-triple", Math. Scand, 61(1), pp.117~133 (1987)
  38. N. Jacobson, "Structure and Representations of Jordan Algebras" (1968), Amer. Math. Soc. Publications
  39. W. Kaup, "A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces", Math. Z, 183(4), pp.503~529 (1983)
  40. W. Kaup, "Contractive projections on Jordan C*-algebras and generalizations", Math. Scand, 54(1), pp.95~100 (1984)
  41. O. Loos, "Jordan pairs" (1975), Springer-Verlag
  42. K. McCrimmon, "A Taste of Jordan Algebras" (2004), Springer-Verlag
  43. M. Neal, "Inner ideals and facial structure of the quasi-state space of a JB-algebra", J. Funct. Anal, 173(2), pp.284~307 (2000)
  44. E. Neher, "Jordan Triple Systems by the Grid Approach" (1987), Springer-Verlag
  45. G. K. Pedersen, "C*-Algebras and their Automorphism Groups" (1979), Academic Press
  46. S. Sakai, "C*-Algebras and W*-Algebras" (1971), Springer
  47. F. W. Shultz, "On normed Jordan algebras which are Banach dual spaces", J. Funct. Anal, 31(3), pp.360~376 (1979)
  48. M. Takesaki, "Faithful states of a C*-algebra", Pacific J. Math, 52, pp.605~610 (1974)
  49. M. Tomita, "Spectral theory of operator algebras. I", Math. J. Okayama Univ, 9, pp.63~98 (1959)
  50. H. Upmeier, "Symmetric Banach manifolds and Jordan C*-algebras" (1985), North-Holland Publishing Co
  51. H. Upmeier, "Jordan Algebras in Analysis, Operator Theory, and Quantum Mechanics" (1987), Conference Board of the Mathematical Sciences
  52. J. D. M. Wright, "Jordan C*-algebras", Michigan Math. J, 24(3), pp.291~302 (1977)
  53. M. A. Youngson, "A Vidav theorem for Banach Jordan algebras", Math. Proc. Cambridge Philos. Soc, 84(2), pp.263~272 (1978)

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