Positivity pp 27-71 | Cite as

Positivity in Operator Algebras and Operator Spaces

  • David P. Blecher
Part of the Trends in Mathematics book series (TM)


This article is aimed at a general reader familiar with the basics of functional analysis. It begins with a quick summary of the most basic ‘facts of life’ of positivity for Hilbert space operators, or for algebras of operators on a Hilbert space. It being impossible to adequately survey the fundamental role of positivity in the field of operator algebras, since this is so extensive and ubiquitous, in the present article we review selectively some of the general principles in the subject, and give some examples of how positivity plays a central role in the field, even in settings where positivity is not at first in evidence. The topics become more progressively more specialized towards our own current interests, ending with some very recent work of ours and of others.


Operator Space Operator Algebra Boundary Representation Approximate Identity Complete Positivity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. Agler, An abstract approach to model theory, pp. 1–23 in Surveys of some recent. results in operator theory, Vol. II, Longman Sci. Tech., Harlow 1988.MathSciNetGoogle Scholar
  2. [2]
    C.A. Akemann, The general Stone-Weierstrass problem, J. Funct. Anal. 4 (1969), 277–294.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    C.A. Akemann, Left ideal structure of C*-algebras, J. Funct. Anal. 6 (1970), 305–317.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    C.A. Akemann, A Gelfand representation theory for C*-algebras, Pacific J. Math. 39 (1971), 1–11.MATHMathSciNetGoogle Scholar
  5. [5]
    C.A. Akemann and G.K. Pedersen, Facial structure in operator algebra theory, Proc. London Math. Soc. 64 (1992), 418–448.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    E.M. Alfsen, Compact convex sets and boundary integrals, Springer-Verlag, New York-Heidelberg, 1971.MATHGoogle Scholar
  7. [7]
    E.M. Alfsen and F.W. Schultz, Geometry of state spaces of operator algebras, Birkhäuser Boston, Inc., Boston, MA, 2003.MATHGoogle Scholar
  8. [8]
    E.M. Alfsen and F.W. Shultz, State spaces of operator algebras. Basic theory, orientations, and C*-products, Birkhäuser Boston, Inc., Boston, MA, 2001.Google Scholar
  9. [9]
    W.B. Arveson, Subalgebras of C*—algebras, Acta Math. 123 (1969), 141–224.MATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    W.B. Arveson, Subalgebras of C*-algebras II, Acta Math. 128 (1972), 271–308.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    W.B. Arveson, Noncommutative dynamics and E-semigroups, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.MATHGoogle Scholar
  12. [12]
    W.B. Arveson, Notes on extensions of C*-algebras, Duke Math. J. 44 (1977), 329–355.MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    W.B. Arveson, The noncommutative Choquet boundary, Preprint 2007, math.OA/0701329, to appear in Journal of Amer. Math Soc. Google Scholar
  14. [14]
    M. Battaglia, Order theoretic type decomposition of JBW*-triples, Quart. J. Math. Oxford, 42 (1991), 129–147.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    J. Bendat and S. Sherman, Monotone and convex operator functions, Trans. Amer. Math. Soc. 79 (1955), 58–71.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    S.K. Berberian, Baer *-rings, Springer-Verlag, New York-Berlin, 1972.Google Scholar
  17. [17]
    B. Blackadar, K-theory for operator algebras, Second edition, Math. Sci. Res. Inst. Pub, 5, Cambridge University Press, Cambridge, 1998.MATHGoogle Scholar
  18. [18]
    B. Blackadar, Operator algebras. Theory of C*-algebras and von Neumann algebras, Encyclopaedia of Mathematical Sciences, 122, Springer-Verlag, Berlin, 2006.Google Scholar
  19. [19]
    B. Blackadar and E. Kirchberg, Generalized inductive limits of finite-dimensional C*-algebras, Math. Ann. 307 (1997), 343–380.MATHMathSciNetCrossRefGoogle Scholar
  20. [20]
    D.P. Blecher, A generalization of Hilbert modules, J. Funct. Anal. 136 (1996), 365–421.MATHMathSciNetCrossRefGoogle Scholar
  21. [21]
    D.P. Blecher, The Shilov boundary of an operator space and the characterization theorems, J. Funct. Anal. 182 (2001), 280–343.MATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    D.P. Blecher, Multipliers and dual operator algebras, J. Funct. Anal. 183 (2001), 498–525.MATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    D.P. Blecher, Multipliers, C*-modules, and algebraic structure in spaces of Hilbert space operators, pp. 85–128 in Operator algebras, quantization, and noncommutative. geometry, Contemp. Math., 365, Amer. Math. Soc., Providence, RI, 2004.Google Scholar
  24. [24]
    D.P. Blecher, E.G. Effros, and V. Zarikian, One-sided M-ideals and multipliers in operator spaces, I, Pacific J. Math. 206 (2002), 287–319.MATHMathSciNetGoogle Scholar
  25. [25]
    D.P. Blecher, D.M. Hay, and M. Neal, Hereditary subalgebras of operator algebras, to appear J. Operator Theory, math.OA/0512417Google Scholar
  26. [26]
    D.P. Blecher, K. Kirkpatrick, M. Neal, and W. Werner, Ordered involutive operator spaces, to appear in Positivity.Google Scholar
  27. [27]
    D.P. Blecher and L.E. Labuschagne, Von Neumann algebraic H p theory, To appear in Proceedings of fifth conference on function spaces, Contemp. Math., math.OA/0611879Google Scholar
  28. [28]
    D.P. Blecher and C. Le Merdy, Operator algebras and their modules — an operator. space approach, Oxford Univ. Press, Oxford (2004).MATHGoogle Scholar
  29. [29]
    D.P. Blecher and B. Magajna, Duality and operator algebras: automatic weak* continuity and applications, J. Funct. Anal. 224 (2005), 386–407.MATHMathSciNetCrossRefGoogle Scholar
  30. [30]
    D.P. Blecher and M. Neal, Open partial isometries and positivity in operator spaces, Preprint 2006, math.OA/0606661Google Scholar
  31. [31]
    D.P. Blecher and V.I. Paulsen, Multipliers of operator spaces, and the injective envelope, Pacific J. Math. 200 (2001), 1–17.MATHMathSciNetCrossRefGoogle Scholar
  32. [32]
    D.P. Blecher, Z.-J. Ruan, and A.M. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89 (1990), 188–201.MATHMathSciNetCrossRefGoogle Scholar
  33. [33]
    D.P. Blecher and W. Werner, Ordered C*-modules, Proc. London Math. Soc. 92 (2006), 682–712.MATHMathSciNetCrossRefGoogle Scholar
  34. [34]
    D.P. Blecher and V. Zarikian, The calculus of one-sided M-ideals and multipliers. in operator spaces, Mem. Amer. Math. Soc. 842 (2006).Google Scholar
  35. [35]
    L.G. Brown, Semicontinuity and multipliers of C*-algebras, Canad. J. Math. 40 (1988), 865–988.MATHMathSciNetGoogle Scholar
  36. [36]
    L.G. Brown, R.G. Douglas, P.A. Fillmore, Extensions of C*-algebras and K-homology, Ann. of Math. 105 (1977), 265–324.MathSciNetCrossRefGoogle Scholar
  37. [37]
    N.P. Brown, On quasidiagonal C*-algebras, Operator algebras and applications, 19–64, Adv. Stud. Pure Math. 38 Math. Soc. Japan, Tokyo, 2004.Google Scholar
  38. [38]
    M.-D. Choi, A Schwarz inequality for positive linear maps on C*-algebras, Illinois. J. Math. 18 (1974), 565–574.MATHMathSciNetGoogle Scholar
  39. [39]
    M.-D. Choi, Completely positive linear maps on complex matrices, Linear Alg. and. Applns, 10 (1975), 285–290.MATHCrossRefGoogle Scholar
  40. [40]
    M.-D. Choi and E. G. Effros, The completely positive lifting problem for C*-algebras, Ann. of Math. 104 (1976), 585–609.MathSciNetCrossRefGoogle Scholar
  41. [41]
    M.-D. Choi and E. G. Effros, Injectivity and operator spaces, J. Funct. Anal. 24 (1977), 156–209.MATHMathSciNetCrossRefGoogle Scholar
  42. [42]
    F. Combes, Sur le faces d’une C*-algébre, Bull. Sci. Math. 93 (1969), 57–100.MathSciNetGoogle Scholar
  43. [43]
    A. Connes, A view of mathematics, Preprint (2005), available from
  44. [44]
    J.B. Conway, A Course in Operator Theory, Graduate Studies in Mathematics, 21, Amer. Math. Soc. Providence, RI, 2000.MATHGoogle Scholar
  45. [45]
    J. Dixmier, C*-algebras, North-Holland Publ. Co., Amsterdam, 1977.MATHGoogle Scholar
  46. [46]
    M. Dritschel and S. McCullouch, Boundary representations for families of representations of operator algebras and spaces, J. Operator Theory, 53 (2005), 159–167.MATHMathSciNetGoogle Scholar
  47. [47]
    C.M. Edwards and G.T. Rüttimann, On the facial structure of the unit balls in a. JBW*-triple and its predual, J. London Math. Soc. 38 (1988), 317–332.MATHMathSciNetCrossRefGoogle Scholar
  48. [48]
    C.M. Edwards and G.T. Rüttimann, Inner ideals in C*-algebras, Math. Ann. 290 (1991), 621–628.MATHMathSciNetCrossRefGoogle Scholar
  49. [49]
    E.G. Effros, Order ideals in a C*-algebra and its dual, Duke Math. J. 30 (1963), 391–411.MATHMathSciNetCrossRefGoogle Scholar
  50. [50]
    E.G. Effros, Aspects of noncommutative order, pp. 1–40 in C*-algebras and applications. to physics, Lecture Notes in Math., 650, Springer, Berlin, 1978.CrossRefGoogle Scholar
  51. [51]
    E.G. Effros and Z.-J. Ruan, Operator Spaces, London Mathematical Society Monographs, New Series, 23, The Clarendon Press, Oxford University Press, New York, 2000.MATHGoogle Scholar
  52. [52]
    E.G. Effros and S. Winkler, Matrix convexity: operator analogues of the bipolar and Hahn-Banach theorems, J. Funct. Anal. 144 (1997), 117–152.MATHMathSciNetCrossRefGoogle Scholar
  53. [53]
    D.R. Farenick, Extremal matrix states on operator systems, J. London Math. Soc. 61 (2000), 885–892.MATHMathSciNetCrossRefGoogle Scholar
  54. [54]
    D.R. Farenick, Pure matrix states on operator systems, Linear Algebra Appl. 393 (2004), 149–173.MATHMathSciNetCrossRefGoogle Scholar
  55. [55]
    D.R. Farenick and P.B. Morentz, C*-extreme points in the generalized state spaces of a C*-algebra, Trans. Amer. Math. Soc. 349 (1997), 1725–1748.MATHMathSciNetCrossRefGoogle Scholar
  56. [56]
    T.W. Gamelin, Uniform Algebras, Second edition, Chelsea, New York, 1984.Google Scholar
  57. [57]
    R. Giles and H. Kummer, A non-commutative generalization of topology, Indiana. Univ. Math. J., 21 (1971/72), 91–102MathSciNetCrossRefGoogle Scholar
  58. [58]
    U. Haagerup, Decompositions of completely bounded maps on operator algebras, Unpublished manuscript (1980).Google Scholar
  59. [59]
    D. Hadwin, Completely positive maps and approximate equivalence, Indiana Univ. Math J. 36 (1987), 211–228.MATHMathSciNetCrossRefGoogle Scholar
  60. [60]
    M. Hamana, Injective envelopes of operator systems, Publ. R.I.M.S. Kyoto Univ. 15 (1979), 773–785.MATHMathSciNetGoogle Scholar
  61. [61]
    M. Hamana, Regular embeddings of C*-algebras in monotone complete C*-algebras, J. Math. Soc. Japan 33 (1981), 159–183.MathSciNetCrossRefGoogle Scholar
  62. [62]
    M. Hamana, Modules over monotone complete C*-algebras, Internat. J. Math. 3 (1992), 185–204.MATHMathSciNetCrossRefGoogle Scholar
  63. [63]
    M. Hamana, Triple envelopes and Silov boundaries of operator spaces, Math. J. Toyama University 22 (1999), 77–93.MATHMathSciNetGoogle Scholar
  64. [64]
    F. Hansen, An operator inequality, Math. Ann. 246 (1979/80), 249–250.MathSciNetCrossRefGoogle Scholar
  65. [65]
    P. Harmand, D. Werner, and W. Werner, M-ideals in Banach spaces and Banach. algebras, Lecture Notes in Math., 1547, Springer-Verlag, Berlin-New York, 1993.Google Scholar
  66. [66]
    D.M. Hay, Closed projections and peak interpolation for operator algebras, to appear J. Int. Eq. Oper. Th., math.OA/0512353Google Scholar
  67. [67]
    M. Henle, A Lebesgue decomposition theorem for C*-algebras, Canad. Math. Bull. 15 (1972), 87–91.MATHMathSciNetGoogle Scholar
  68. [68]
    F. Hiai and H. Kosaki, Means of Hilbert space oeprators. Lectures Notes in Mathematics, 1820. Springer-Verlag, Berlin, 2003.Google Scholar
  69. [69]
    M. Junge, Minimal sets of complete contractions on operator systems (tentative title), Draft (2005) and revision (March 2007).Google Scholar
  70. [70]
    M. Junge and G. Pisier, Bilinear forms on exact operator spaces and B(H)⊗B(H), Geom. Funct. Anal. 5 (1995), 329–363.MATHMathSciNetCrossRefGoogle Scholar
  71. [71]
    R.V. Kadison, A representation theory for commutative topological algebras, Mem. Amer. Math. Soc. 7 (1951).Google Scholar
  72. [72]
    R.V. Kadison, A generalized Schwarz inequality, Ann. of Math. 56 (1952), 494–503.MathSciNetCrossRefGoogle Scholar
  73. [73]
    R.V. Kadison, Operator algebras with a faithful weakly-closed representation, Ann. of Math. 64 (1956), 175–181.MathSciNetCrossRefGoogle Scholar
  74. [74]
    R.V. Kadison and J.R. Ringrose, Fundamentals of the theory of operator algebras, Graduate Studies in Mathematics, 15, Amer. Math. Soc. Providence, RI, 1997.Google Scholar
  75. [75]
    M. Kaneda and V.I. Paulsen, Quasi-multipliers of operator spaces, J. Funct. Anal. 217 (2004), 347–365.MATHMathSciNetCrossRefGoogle Scholar
  76. [76]
    A.K. Karn and R. Vasudevan, Characterization of matricially Riesz normed spaces, Yokohama Math. J. 47 (2000), 143–153.MATHMathSciNetGoogle Scholar
  77. [77]
    A.K. Karn, Adjoining an order unit to a matrix ordered space, Positivity 9 (2005), 207–223.MATHMathSciNetCrossRefGoogle Scholar
  78. [78]
    A.K. Karn, Corrigedem to the paper “Adjoining an order unit to a matrix ordered space”, Draft, 2006.Google Scholar
  79. [79]
    E. Kirchberg, On nonsemisplit extensions, tensor products and exactness of group C*-algebras, Invent. Math. 112 (1993), 449–489.MATHMathSciNetCrossRefGoogle Scholar
  80. [80]
    E. Kirchberg, On subalgebras of the CAR-algebra, J. Funct. Anal. 129 (1995), 35–63.MATHMathSciNetCrossRefGoogle Scholar
  81. [81]
    E. Kirchberg, On restricted peturbations in inverse images and a description of normalizer algebras in C*-algebras, J. Funct. Anal. 129 (1995), 1–34.MATHMathSciNetCrossRefGoogle Scholar
  82. [82]
    E. Kirchberg and N.C. Phillips, Embedding of exact C*-algebras in the Cuntz algebra O 2, J. Reine Angew. Math. 525 (2000), 17–53.MATHMathSciNetGoogle Scholar
  83. [83]
    E. Kirchberg and S. Wassermann, C*-algebras generated by operator systems, J. Funct. Anal. 155 (1998), 324–351.MATHMathSciNetCrossRefGoogle Scholar
  84. [84]
    E.C. Lance, Hilbert C*-modules — A toolkit for operator algebraists, London Math. Soc. Lecture Notes, 210, Cambridge University Press, Cambridge, 1995.Google Scholar
  85. [85]
    A.T.-M. Lau and R.J. Loy, Contractive projections on Banach algebras, Preprint 2006.Google Scholar
  86. [86]
    C. Le Merdy, An operator space characterization of dual operator algebras, Amer. J. Math. 121 (1999), 55–63.MATHMathSciNetCrossRefGoogle Scholar
  87. [87]
    R.I. Loebl and V.I. Paulsen, Some remarks on C*-convexity, Linear Algebra Appl. 35 (1981), 63–78.MATHMathSciNetCrossRefGoogle Scholar
  88. [88]
    K. Löwner, Über monotone matrixfunktionen, Math. Z. 38 (1934), 177–216.MATHMathSciNetCrossRefGoogle Scholar
  89. [89]
    B. Magajna, C*-convex sets and completely bounded bimodule homomorphisms, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 375–387.MATHMathSciNetCrossRefGoogle Scholar
  90. [90]
    R. Meyer, Adjoining a unit to an operator algebra, J. Operator Theory 46 (2001), 281–288.MATHMathSciNetGoogle Scholar
  91. [91]
    P.B. Morentz, The structure of C*-convex sets, Canad. J. Math. 46 (1994), 1007–1026.MathSciNetGoogle Scholar
  92. [92]
    P.S. Muhly and B. Solel, An algebraic characterization of boundary representations, pp. 189–196 in Nonselfadjoint operator algebras, operator theory, and related topics, Oper. Th. Adv. Appl., 104, Birkhäuser, Basel, 1998.Google Scholar
  93. [93]
    M. Nielsen and I. Chuang, Quantum computation and quantum information, Cambridge University Press, 2000.Google Scholar
  94. [94]
    N. Ozawa, About the QWEP conjecture, Internat. J. Math. 15 (2004), 501–530.MATHMathSciNetCrossRefGoogle Scholar
  95. [95]
    V.I. Paulsen, Every completely polynomially bounded operator is similar to a contraction, J. Funct. Anal. 55 (1984), 1–17.MATHMathSciNetCrossRefGoogle Scholar
  96. [96]
    V.I. Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Math., 78, Cambridge University Press, Cambridge, 2002.MATHGoogle Scholar
  97. [97]
    G.K. Pedersen, C*-algebras and their automorphism groups, Academic Press, London (1979).MATHGoogle Scholar
  98. [98]
    G.K. Pedersen, Analysis now, Graduate Texts in Mathematics, 118, Springer-Verlag, New York, 1989.MATHGoogle Scholar
  99. [99]
    R.R. Phelps, Lectures on Choquet’s theorem, 2nd Edition, Lecture Notes in Mathematics, Vol. 1757, Springer-Verlag, Berlin, 2001.MATHGoogle Scholar
  100. [100]
    G. Pisier, Similarity problems and completely bounded maps, Second, expanded edition, Lecture Notes in Math., 1618, Springer-Verlag, Berlin, 2001.MATHGoogle Scholar
  101. [101]
    G. Pisier, Introduction to operator space theory, London Math. Soc. Lecture Note Series, 294, Cambridge University Press, Cambridge, 2003.MATHGoogle Scholar
  102. [102]
    Z.-J. Ruan, Injectivity of operator spaces, Trans. Amer. Math. Soc. 315 (1989), 89–104.MATHMathSciNetCrossRefGoogle Scholar
  103. [103]
    B. Russo, Structure of JB*-triples, Jordan algebras (Oberwolfach, 1992), 209–280, de Gruyter, Berlin, 1994.Google Scholar
  104. [104]
    W.J. Schreiner, Matrix regular operator spaces, J. Funct. Anal. 152 (1998), 136–175.MATHMathSciNetCrossRefGoogle Scholar
  105. [105]
    D. Sherman, On the dimension theory of von Neumann algebras, to appear, Math. Scand., math.OA/0503747Google Scholar
  106. [106]
    R.R. Smith and J.D. Ward, Matrix ranges for Hilbert space operators, Amer. J. Math. 102 (1980), 1031–1081.MATHMathSciNetCrossRefGoogle Scholar
  107. [107]
    W.F. Stinespring, Positive functions on C*-algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216.MATHMathSciNetCrossRefGoogle Scholar
  108. [108]
    E. Størmer, Positive linear maps of operator algebras, Acta Math. 110 (1963), 233–278.MATHMathSciNetCrossRefGoogle Scholar
  109. [109]
    E. Størmer, Positive linear maps of C*-algebras, pp. 85–106 in Foundations of quantum mechanics and ordered linear spaces, Lecture Notes in Phys., Vol. 29, Springer, Berlin, 1974.Google Scholar
  110. [110]
    E. Størmer, Multiplicative properties of positive maps, Math. Scand. 100 (2007), 184–192.MathSciNetGoogle Scholar
  111. [111]
    E.L. Stout, The theory of uniform algebras, Bogden and Quigley, 1971.Google Scholar
  112. [112]
    S. Stratila, Modular theory in operator algebras, Editura Acedemiei and Abacus Press, 1981.Google Scholar
  113. [113]
    S. Stratila and L. Zsidó, Operator algebras, the general Banach algebra background (Tentative title), Theta Series in Advanced Mathematics, Bucharest, to appear.Google Scholar
  114. [114]
    M. Takesaki, Theory of Operator Algebras I, Springer, New York, 1979.MATHGoogle Scholar
  115. [115]
    M.A. Youngson, Completely contractive projections on C*-algebras, Quart. J. Math. Oxford 34 (1983), 507–511.MATHMathSciNetCrossRefGoogle Scholar
  116. [116]
    C. Webster and S. Winkler, The Krein-Milman theorem in operator convexity, Trans. Amer. Math. Soc. 351 (1999), 307–322.MATHMathSciNetCrossRefGoogle Scholar
  117. [117]
    W. Werner, Subspaces of L(H) that are *-invariant, J. Funct. Anal. 193 (2002), 207–223.MATHMathSciNetCrossRefGoogle Scholar
  118. [118]
    W. Werner, Multipliers on matrix ordered operator spaces and some K-groups, J. Funct. Anal. 206 (2004), 356–378.MATHMathSciNetCrossRefGoogle Scholar
  119. [119]
    S. Winkler, Matrix convexity, Ph.D. thesis, U.C.L.A., 1996.Google Scholar
  120. [120]
    G. Wittstock, Extensions of completely bounded C*-module homomorphisms, pp. 238–250 in Operator algebras and group representations, Vol. II (Neptun, 1980), Monogr. Stud. Math., 18, Pitman, Boston, MA, 1984.Google Scholar
  121. [121]
    G. Wittstock, On matrix order and convexity, pp. 175–188 in Functional analysis: surveys and recent results, North-Holland Math. Stud., 90, North-Holland, Amsterdam, 1984.Google Scholar
  122. [122]
    S.L. Woronowicz, Nonextendible positive maps, Commun. Math. Phys. 51 (1976), 243–282.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag AG2007 2007

Authors and Affiliations

  • David P. Blecher
    • 1
  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA

Personalised recommendations