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Nonseparable approximate equivalence


Author: Donald W. Hadwin
Journal: Trans. Amer. Math. Soc. 266 (1981), 203-231
MSC: Primary 46L05
DOI: https://doi.org/10.1090/S0002-9947-1981-0613792-6
MathSciNet review: 613792
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Abstract: This paper extends Voiculescu's theorem on approximate equivalence to the case of nonseparable representations of nonseparable $ {C^ \ast }$-algebras. The main result states that two representations $ f$ and $ g$ are approximately equivalent if and only if $ {\text{rank}}f(x) = {\text{rank}}g(x)$ for every $ x$. For representations of separable $ {C^ \ast }$-algebras a multiplicity theory is developed that characterizes approximate equivalence. Thus for a separable $ {C^ \ast }$-algebra, the space of representations modulo approximate equivalence can be identified with a class of cardinal-valued functions on the primitive ideal space of the algebra. Nonseparable extensions of Voiculescu's reflexivity theorem for subalgebras of the Calkin algebra are also obtained.


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  • [AFV 1] C. Apostol, C. Foias and D. Voiculescu, Strongly reductive operators are normal, Acta Sci. Math. (Szeged) 38 (1976), 261-263. MR 0433241 (55:6219)
  • [AFV 2] -, On strongly reductive algebras, Rev. Roumaine Math. Pures Appl. 21 (1976), 633-642. MR 0417804 (54:5852)
  • [AF] C. Apostol and C. K. Fong, Invariant subspaces for algebras generated by strongly reductive operators, Duke Math. J. 42 (1975), 495-498. MR 0372647 (51:8854)
  • [Ar 1] W. Arveson, Notes on extensions of $ {C^ \ast }$-algebras, Duke Math. J. 44 (1977), 329-355. MR 0438137 (55:11056)
  • [Ar 2] -, An invitation to $ {C^ \ast }$-algebras, Graduate Texts in Math., vol. 39, Springer-Verlag, New York, 1976.
  • [B] I. D. Berg, An extension of the Weyl-von Neumann theorem to normal operators, Trans. Amer. Math. Soc. 160 (1971), 365-371. MR 0283610 (44:840)
  • [BDF] L. G. Brown, R. G. Douglas and P. A. Fillmore, Unitary equivalence modulo the compact operators and extensions of $ {C^ \ast }$-algebras, Lecture Notes in Math., vol. 345, Springer-Verlag, Berlin and New York, 1973. MR 0380478 (52:1378)
  • [BuDe] J. W. Bunce and J. A. Deddens, Subspace approximants and GCR operators, Indiana Univ. Math. J. 24 (1974), 341-349. MR 0350473 (50:2965)
  • [CG] S. L. Campbell and R. Gellar, On asymptotic properties of several classes of operators, Proc. Amer. Math. Soc. 66 (1977), 79 - 84. MR 0461187 (57:1172)
  • [Di 1] J. Dixmier, Les $ {C^ \ast }$-algèbres et leurs représentations, Gauthier-Villars, Paris, 1964. MR 0171173 (30:1404)
  • [Di 2] -, Les algèbres d'opérateurs dans l'espace Hilbertien, Gauthier-Villars, Paris, 1957.
  • [EEL] G. Edgar, J. Ernest and S. G. Lee, Weighing operator spectra, Indiana Univ. Math. J. 21 (1971), 61-80. MR 0417836 (54:5884)
  • [GP] R. Gellar and L. Page, Limits of unitarily equivalent normal operators, Duke Math. J. 41 (1974), 319-322. MR 0338817 (49:3581)
  • [H 1] D. W. Hadwin, An operator-valued spectrum, Indiana Univ. Math. J. 26 (1977), 329-340. MR 0428089 (55:1118)
  • [H 2] -, Continuous functions of operators: a functional calculus, Indiana Univ. Math. J. 27 (1978), 113-125. MR 0467367 (57:7226)
  • [H 3] -, Closures of unitary equivalence classes and completely positive maps, Notices Amer. Math. Soc. 23 (1976), A-587.
  • [H 4] -, Approximating direct integrals of operators by direct sums, Michigan Math. J. 25 (1978), 123-127. MR 0500245 (58:17920)
  • [H 5] -, An asymptotic double commutant theorem for $ {C^ \ast }$-algebras, Trans. Amer. Math. Soc. 244 (1978), 273-297. MR 506620 (81b:47027)
  • [H 6] -, Non-separable approximate equivalence, Notices Amer. Math. Soc. 25 (1978), A-8.
  • [PRH 1] P. R. Halmos, Irreducible operators, Michigan Math. J. 15 (1968), 215-223. MR 0231233 (37:6788)
  • [PRH 2] -, Some unsolved problems of unknown depth about operators on Hilbert space, Proc. Roy. Soc. Edinburgh Sect. A 76A (1976), 67-76. MR 0451002 (56:9292)
  • [Ha] K. J. Harrison, Strongly reductive operators, Acta Sci. Math. (Szeged) 37 (1975), 205-212. MR 0388129 (52:8966)
  • [PS] C. Pearcy and N. Salinas, Finite-dimensional representations of $ {C^ \ast }$-algebras and the reducing material spectra of an operator, Rev. Roumaine Math. Pures Appl. 20 (1975), 567-598. MR 0390786 (52:11609)
  • [Si] W. Sikonia, The von Neumann converse of Weyl's theorem, Indiana Univ. Math. J. 21 (1971/72), 121-124. MR 0285928 (44:3145)
  • [Th] F. J. Thayer, Quasidiagonal $ {C^ \ast }$-algebras, J. Functional Analysis 25 (1977), 50-57. MR 0448098 (56:6408)
  • [V] D. Voiculescu, A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1976), 97-113. MR 0415338 (54:3427)

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DOI: https://doi.org/10.1090/S0002-9947-1981-0613792-6
Article copyright: © Copyright 1981 American Mathematical Society

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