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Nuclear faces of state spaces of $ C\sp{\ast} $-algebras


Author: C. J. K. Batty
Journal: Proc. Amer. Math. Soc. 86 (1982), 275-278
MSC: Primary 46L05
DOI: https://doi.org/10.1090/S0002-9939-1982-0667288-2
MathSciNet review: 667288
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Abstract: Let $ {B_1}$ and $ {B_2}$ be maximal abelian subalgebras of $ {C^* }$-algebras $ {A_1}$ and $ {A_2}$, and suppose that for each pure state $ {\psi _1}$ of $ {B_1}$, the von Neumann algebra $ {p_{{\psi _1}}}A_1^{**}{p_{{\psi _1}}}$ is injective, where $ {p_{{\psi _1}}}$ is the common support in $ A_1^{**}$ of all the states of $ {A_1}$ which extend $ {\psi _1}$. Then $ {B_1} \otimes {B_2}$ is a maximal abelian subalgebra of any $ {C^*}$-tensor product $ {A_1}{ \otimes _\beta }{A_2}$.


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  • [1] J. Anderson, Extensions, restrictions, and representations of states in $ {C^* }$-algebras, Trans. Amer. Math. Soc. 249 (1979), 303-329. MR 525675 (80k:46069)
  • [2] -, Extreme points in sets of positive linear maps on $ \mathcal{B}(\mathcal{H})$, J. Funct. Anal. 31 (1979), 195-217. MR 525951 (80k:46070)
  • [3] R. J. Archbold, On the centre of a tensor product of $ {C^* }$-algebras, J. London Math. Soc. (2) 10 (1975), 257-262. MR 0402512 (53:6331)
  • [4] -, Extensions of states of $ {C^* }$-algebras, J. London Math. Soc. (2) 21 (1980), 351-354. MR 575394 (83e:46053)
  • [5] R. J. Archbold, J. W. Bunce and K. Gregson, Extensions of states of $ {C^*}$-algebras. II, Proc. Roy. Soc. Edinburgh (to appear).
  • [6] C. J. K. Batty, Simplexes of states of $ {C^*}$-algebras, J. Operator Theory 4 (1980), 3-23. MR 587365 (82b:46067)
  • [7] -, Abelian faces of state spaces of $ {C^*}$-algebras, Comm. Math. Phys. 75 (1980), 43-50. MR 581568 (81k:46068)
  • [8] -, Simplexes of extensions of states of $ {C^*}$-algebras, Trans. Amer. Math. Soc. 272 (1982), 237-246. MR 656488 (83h:46071)
  • [9] M. D. Choi and E. G. Effros, Separable nuclear $ {C^*}$-algebras and injectivity, Duke Math. J. 43 (1976), 309-322. MR 0405117 (53:8912)
  • [10] -, Nuclear $ {C^*}$-algebras and injectivity: the general case, Indiana Univ. Math. J. 26 (1977), 443-446. MR 0430794 (55:3799)
  • [11] A. Connes, Classification of injective factors, Ann. of Math. (2) 109 (1976), 73-115. MR 0454659 (56:12908)
  • [12] -, On the equivalence between injectivity and semi discreteness for operator algebras, Colloques Internat. CNRS 274 (1979), 107-112.
  • [13] E. G. Effros and E. C. Lance, Tensor products of operator algebras, Adv. in Math. 25 (1977), 1-34. MR 0448092 (56:6402)
  • [14] R. V. Kadison and I. M. Singer, Extensions of pure states, Amer. J. Math. 81 (1959), 383-400. MR 0123922 (23:A1243)
  • [15] G. K. Pedersen, $ {C^*}$-algebras and their automorphism groups, Academic Press, London and New York, 1979. MR 548006 (81e:46037)
  • [16] M. Takesaki, Theory of operator algebras. I, Springer-Verlag, Berlin and New York, 1979. MR 548728 (81e:46038)
  • [17] J. Tomiyama, Tensor products and projections of norm one in von Neumann algebras, Seminar notes, Copenhagen Univ., 1971.
  • [18] A. S. Wassermann, The slice map problem for $ {C^*}$-algebras, Proc. London Math. Soc. (3) 32 (1976), 537-559. MR 0410402 (53:14152)
  • [19] -, Injective $ {W^*}$-algebras, Math. Proc. Cambridge Philos. Soc. 82 (1977), 39-47. MR 0448108 (56:6418)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0667288-2
Keywords: $ {C^* }$-tensor product, maximal abelian subalgebra, injective, nuclear
Article copyright: © Copyright 1982 American Mathematical Society

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