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Test problems for operator algebras


Author: Edward A. Azoff
Journal: Trans. Amer. Math. Soc. 347 (1995), 2989-3001
MSC: Primary 46L10; Secondary 16E50, 47D25
DOI: https://doi.org/10.1090/S0002-9947-1995-1321565-4
MathSciNet review: 1321565
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Abstract: Kaplansky's test problems, originally formulated for abelian groups, concern the relationship between isomorphism and direct sums. They provide a "reality check" for purported structure theories.

The present paper answers Kaplansky's problems in operator algebraic contexts including unitary equivalence of von Neumann algebras and equivalence of representations of (non self-adjoint) matrix algebras. In particular, it is shown that matrix algebras admitting similar ampliations are themselves similar.


References [Enhancements On Off] (What's this?)

  • [1] W.B. Arveson, An invitation to $ {C^ * }$-algebras, Springer-Verlag, New York, 1976.
  • [2] E.A. Azoff, On finite rank operators and preannihilators, Mem. Amer. Math. Soc. 357 (1986). MR 858467 (88a:47041)
  • [3] H. Bercovici, Three test problems for quasisimilarity, Canad. J. Math. 39 (1987), 880-892. MR 915020 (89a:47026)
  • [4] I. Bucur and A Deleanu, Introduction to the theory of categories and functors, Wiley, New York, 1968. MR 0236236 (38:4534)
  • [5] K.R. Davidson and D.A. Herrero, The Jordan form of a bitriangular operator, J. Funct. Anal. 94 (1990), 27-73. MR 1077544 (92a:47022)
  • [6] J. Ernest, Charting the operator terrain, Mem. Amer. Math. Soc. 171 (1976). MR 0463941 (57:3879)
  • [7] R.V. Kadison and I.M. Singer, Three test problems in operator theory, Pacific J. Math. 7 (1957), 1101-1106. MR 0092123 (19:1066e)
  • [8] I. Kaplansky, Infinite abelian groups, 2nd ed., The University of Michigan Press, Ann Arbor, 1965. MR 0065561 (16:444g)
  • [9] -, Fields and rings, 2nd ed., The University of Chicago Press, Chicago, 1972. MR 0349646 (50:2139)
  • [10] C. M. Pearcy, A complete set of unitary invariants for operators generating finite $ {W^*}$-algebras of type I, Pacific J. Math. 12 (1962), 1405-1416. MR 0149326 (26:6816)
  • [11] W. Specht, Zur Theorie der Matrizen II, Jber. Deutsch. Math. Verein. 50 (1940), 19-23. MR 0002830 (2:118g)
  • [12] M. Takesaki, Theory of operator algebras I, Springer-Verlag, New York, 1979. MR 548728 (81e:46038)
  • [13] D.M. Topping, Lectures on von Neumann algebras, Van Nostrand Reinhold, New York, 1971.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1321565-4
Keywords: Ampliation, direct sum, unitary equivalence, similarity
Article copyright: © Copyright 1995 American Mathematical Society

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