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Maximal tracial algebras

  • Don Hadwin
  • Hassan YousefiEmail author
Original Paper

Abstract

We introduce and study the notion of maximal tracial algebras. We prove several results in a general setting based on dual pairs and multiplier pairs. In a special case that X is a Banach space we determine the abelian subalgebras of \({\mathcal {B}}\left( X\right) \) that are maximal tracial for rank-one tensors. In another special case that \(\mathcal {H\ }\)is a Hilbert space we show that a unital weak-operator closed subalgebra \( {\mathcal {A}}\) of \({\mathcal {B}}\left( {\mathcal {H}}\right) \) is abelian and transitive if and only if it is maximal \(e\otimes e\)-tracial for every unit vector e in \({\mathcal {H}}\). We also make slight connections between our ideas and the Kadison Similarity Problem and also the Connes’ Embedding Problem.

Keywords

Tracial algebra Dual pair Multiplier pair Tracial ultraproduct 

Mathematics Subject Classification

46L05 47L10 46M07 46L10 

Notes

Acknowledgements

We would like to thank the reviewer(s) for the very careful review of our paper and many valuable suggestions that improved the quality of the paper.

References

  1. 1.
    Ando, H., Kirchberg, E.: Non-commutativity of the central sequence algebra for separable non-type I \(C^*\)-algebras. J. Lond. Math. Soc. 94(1), 280–294 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Connes, A.: Classification of injective factors. Cases \( II_{1}\), \(II_{\infty }\), \(III_{\lambda }\), \(\lambda \ne 1\). Ann. Math. 104(1), 73–115 (1976)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cuntz, J., Pedersen, G.K.: Equivalence and traces on \(C^*\)-algebras. J. Funct. Anal. 33(2), 135–164 (1979)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Haagerup, U.: Solution of the similarity problem for cyclic representations of \(C^*\)-algebras. Ann. Math. 118, 215–240 (1983)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hadwin, D.W.: A general view of reflexivity. Trans. Am. Math. Soc. 344(1), 325–360 (1994)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hadwin, D.W., Li, W.: A note on approximate liftings. Oper. Matrices 3(1), 125–143 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hadwin, D.W., Nordgren, E.A.: A general view of multipliers and composition operators. Linear Algebra Appl. 383, 187–211 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hadwin, D.W., Nordgren, E. A.: A general view of multipliers and composition operators II, Banach spaces of analytic functions, pp. 63–73. In: Contemp. Math. 454, Amer. Math. Soc., Providence, RI (2008)Google Scholar
  9. 9.
    Helson, H.: Lectures on Invariant Subspaces. Academic Press, New York, London (1964)zbMATHGoogle Scholar
  10. 10.
    Kadison, R.V.: On the orthogonalization of operator representations. Am. J. Math. 77, 600–622 (1955)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kadison, R.V.: The trace in finite operator algebras. Proc. Am. Math. Soc. 12, 973–977 (1961)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kadison, R. V.: Normal states and unitary equivalence of von Neumann algebras, \(C^*\)-algebras and their applications to statistical mechanics and quantum field theory, pp.1–18. North-Holland, Amsterdam (1976)Google Scholar
  13. 13.
    Nelson, E.: Notes on non-commutative integration. J. Funct. Anal. 15, 103–116 (1974)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Read, C.J.: A solution to the invariant subspace problem on the space \(\ell _{1}\). Bull. Lond. Math. Soc. 17(4), 305–317 (1985)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Russo, B.: Trace preserving mappings of matrix algebras. Duke Math. J. 36, 297–300 (1969)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Sakai, S.: The theory of \(W^*\) -algebras, Lecture Notes, Yale University (1962)Google Scholar

Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.Department of Mathematics & StatisticsUniversity of New HampshireDurhamUSA
  2. 2.Department of MathematicsCalifornia State University FullertonFullertonUSA

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