# Q-MEASURES ON THE DUAL UNIT BALL OF A JB⁎-TRIPLE

Edwards, C. Martin;Oliveira, Lina

• Accepted : 2018.08.02
• Published : 2019.01.01
• 58 1

#### 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 ${\tilde{\mathcal{U}}}_c(A)$.

#### Keywords

$JB^*$-triple;$C^*$-algebra;Q-topology;Q-measure;compact support tripotent

#### References

1. C. A. Akemann, The general Stone-Weierstrass problem, J. Functional Analysis 4 (1969), 277-294. https://doi.org/10.1016/0022-1236(69)90015-9
2. C. A. Akemann and S. Eilers, Regularity of projections revisited, J. Operator Theory 48 (2002), no. 3, suppl., 515-534.
3. E. M. Alfsen, Compact Convex Sets and Boundary Integrals, Springer-Verlag, New York, 1971.
4. E. M. Alfsen and F. W. Shultz, State spaces of Jordan algebras, Acta Math. 140 (1978), no. 3-4, 155-190. https://doi.org/10.1007/BF02392307
5. E. M. Alfsen, F. W. Shultz, and E. Strmer, A Gelfand-Naimark theorem for Jordan algebras, Adv. Math. 28 (1978), 11-56. https://doi.org/10.1016/0001-8708(78)90044-0
6. L. Asimow and A. J. Ellis, Convexity Theory and its Applications in Functional Analysis, London Mathematical Society Monographs, 16, Academic Press, Inc., London, 1980.
7. T. J. Barton, T. Dang, and G. Horn, Normal representations of Banach Jordan triple systems, Proc. Amer. Math. Soc. 102 (1988), no. 3, 551-555. https://doi.org/10.1090/S0002-9939-1988-0928978-2
8. T. J. Barton and R. M. Timoney, Weak*-continuity of Jordan triple products and its applications, Math. Scand. 59 (1986), no. 2, 177-191. https://doi.org/10.7146/math.scand.a-12160
9. M. Battaglia, Order theoretic type decomposition of JBW*-triples, Quart. J. Math. Oxford Ser. (2) 42 (1991), no. 166, 129-147. https://doi.org/10.1093/qmath/42.1.129
10. F. F. Bonsall and J. Duncan, Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, London Mathematical Society Lecture Note Series, 2, Cambridge University Press, London, 1971.
11. H. G. Dales, A. T-M. Lau, and D. Strauss, Second duals of measure algebras, Dissertationes Mathematicae (Rozprawy Matematyczne) 481 (2012), 1-121.
12. S. Dineen, Complete holomorphic vector fields on the second dual of a Banach space, Math. Scand. 59 (1986), no. 1, 131-142. https://doi.org/10.7146/math.scand.a-12158
13. S. Dineen, The second dual of a JB* triple system, in Complex analysis, functional analysis and approximation theory (Campinas, 1984), 67-69, North-Holland Math. Stud., 125, Notas Mat., 110, North-Holland, Amsterdam, 1986.
14. C. M. Edwards, On Jordan W*-algebras, Bull. Sci. Math. (2) 104 (1980), no. 4, 393-403.
15. C. M. Edwards, F. J. Fernandez-Polo, C. S. Hoskin, and A. M. Peralta, On the facial structure of the unit ball in a JB*-triple, J. Reine Angew. Math. 641 (2010), 123-144.
16. C. M. Edwards and R. V. Hugli, Decoherence in pre-symmetric spaces, Rev. Mat. Complut. 21 (2008), no. 1, 219-249.
17. C. M. Edwards, R. V. Hugli, and G. T. Ruttimann, A geometric characterization of structural projections on a JBW*-triple, J. Funct. Anal. 202 (2003), no. 1, 174-194. https://doi.org/10.1016/S0022-1236(02)00064-2
18. C. M. Edwards and L. Oliveira, Local facial structure and norm-exposed faces of the unit ball in a JB*-triple, J. Math. Anal. Appl. 421 (2015), no. 2, 1315-1333. https://doi.org/10.1016/j.jmaa.2014.07.073
19. C. M. Edwards and G. T. Ruttimann, On the facial structure of the unit balls in a GL-space and its dual, Math. Proc. Cambridge Philos. Soc. 98 (1985), no. 2, 305-322. https://doi.org/10.1017/S0305004100063489
20. C. M. Edwards and G. T. Ruttimann, On the facial structure of the unit ball of a GM-space, Math. Z. 193 (1986), no. 4, 597-611. https://doi.org/10.1007/BF01160478
21. C. M. Edwards and G. T. Ruttimann, On the facial structure of the unit balls in a JBW*-triple and its predual, J. London Math. Soc. (2) 38 (1988), no. 2, 317-332.
22. C. M. Edwards and G. T. Ruttimann, Compact tripotents in bi-dual JB*-triples, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 1, 155-173. https://doi.org/10.1017/S0305004100074740
23. C. M. Edwards and G. T. Ruttimann, The lattice of compact tripotents in bi-dual JB*-triples, Atti Sem. Mat. Fis. Univ. Modena 45 (1997), no. 1, 155-168.
24. C. M. Edwards and G. T. Ruttimann, Exposed faces of the unit ball in a JBW*-triple, Math. Scand. 82 (1998), no. 2, 287-304. https://doi.org/10.7146/math.scand.a-13838
25. C. M. Edwards and G. T. Ruttimann, Orthogonal faces of the unit ball in a Banach space, Atti Sem. Mat. Fis. Univ. Modena 49 (2001), no. 2, 473-493.
26. E. G. Effros, Order ideals in a C*-algebra and its dual, Duke Math. J. 30 (1963), 391-411. https://doi.org/10.1215/S0012-7094-63-03042-4
27. F. J. Fernandez-Polo and A. M. Peralta, Compact tripotents and the Stone-Weierstrass theorem for C*-algebras and JB*-triples, J. Operator Theory 58 (2007), no. 1, 157-173.
28. F. J. Fernandez-Polo and A. M. Peralta, Non-commutative generalisations of Urysohn's lemma and hereditary inner ideals, J. Funct. Anal. 259 (2010), no. 2, 343-358. https://doi.org/10.1016/j.jfa.2010.04.003
29. F. J. Fernandez-Polo and A. M. Peralta, On the facial structure of the unit ball in the dual space of a JB*-triple, Math. Ann. 348 (2010), no. 4, 1019-1032. https://doi.org/10.1007/s00208-010-0511-9
30. Y. Friedman, Bounded symmetric domains and the JB*-triple structure in physics, in Jordan algebras (Oberwolfach, 1992), 61-82, de Gruyter, Berlin, 1994.
31. Y. Friedman, Physical Applications of Homogeneous Balls, Progress in Mathematical Physics, 40, Birkhauser Boston, Inc., Boston, MA, 2005.
32. Y. Friedman and Y. Gofman, Why does the geometric product simplify the equations of physics?, Internat. J. Theoret. Phys. 41 (2002), no. 10, 1841-1855. https://doi.org/10.1023/A:1021048722241
33. Y. Friedman and Y. Gofman, Relativistic linear spacetime transformations based on symmetry, Found. Phys. 32 (2002), no. 11, 1717-1736. https://doi.org/10.1023/A:1021450706566
34. Y. Friedman and B. Russo, Structure of the predual of a JBW*-triple, J. Reine Angew. Math. 356 (1985), 67-89.
35. H. Hanche-Olsen and E. Strmer, Jordan Operator Algebras, Monographs and Studies in Mathematics, 21, Pitman (Advanced Publishing Program), Boston, MA, 1984.
36. D. J. Hebert and H. E. Lacey, On supports of regular Borel measures, Pacific J. Math. 27 (1968), 101-118. https://doi.org/10.2140/pjm.1968.27.101
37. G. Horn, Characterization of the predual and ideal structure of a JBW*-triple, Math. Scand. 61 (1987), no. 1, 117-133. https://doi.org/10.7146/math.scand.a-12194
38. N. Jacobson, Structure and Representations of Jordan Algebras, Amer. Math. Soc. Publications 39, Providence, Rhode Island, 1968.
39. W. Kaup, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z. 183 (1983), no. 4, 503-529. https://doi.org/10.1007/BF01173928
40. W. Kaup, Contractive projections on Jordan C*-algebras and generalizations, Math. Scand. 54 (1984), no. 1, 95-100. https://doi.org/10.7146/math.scand.a-12043
41. O. Loos, Jordan pairs, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, Berlin, 1975.
42. K. McCrimmon, A Taste of Jordan Algebras, Universitext, Springer-Verlag, New York, 2004.
43. M. Neal, Inner ideals and facial structure of the quasi-state space of a JB-algebra, J. Funct. Anal. 173 (2000), no. 2, 284-307. https://doi.org/10.1006/jfan.2000.3577
44. E. Neher, Jordan Triple Systems by the Grid Approach, Lecture Notes in Mathematics 1280, Springer-Verlag, Berlin, 1987.
45. G. K. Pedersen, C*-Algebras and their Automorphism Groups, London Mathematical Society Monographs 14, Academic Press, London, 1979.
46. S. Sakai, C*-Algebras and W*-Algebras, Springer, Berlin,1971.
47. F. W. Shultz, On normed Jordan algebras which are Banach dual spaces, J. Funct. Anal. 31 (1979), no. 3, 360-376. https://doi.org/10.1016/0022-1236(79)90010-7
48. M. Takesaki, Faithful states of a C*-algebra, Pacific J. Math. 52 (1974), 605-610. https://doi.org/10.2140/pjm.1974.52.605
49. M. Tomita, Spectral theory of operator algebras. I, Math. J. Okayama Univ. 9 (1959/1960), 63-98.
50. H. Upmeier, Symmetric Banach manifolds and Jordan C*-algebras, North-Holland Mathematics Studies, 104, North-Holland Publishing Co., Amsterdam, 1985.
51. H. Upmeier, Jordan Algebras in Analysis, Operator Theory, and Quantum Mechanics, CBMS Regional Conference Series in Mathematics, 67, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1987.
52. J. D. M. Wright, Jordan C*-algebras, Michigan Math. J. 24 (1977), no. 3, 291-302. https://doi.org/10.1307/mmj/1029001946
53. M. A. Youngson, A Vidav theorem for Banach Jordan algebras, Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 2, 263-272. https://doi.org/10.1017/S0305004100055092

#### Acknowledgement

Supported by : FCT/Portugal