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Cohomology of a class of Kadison-Singer algebras

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Abstract

Let L be the complete lattice generated by a nest N on an infinite-dimensional separable Hilbert space H and a rank one projection P ζ given by a vector ζ in H. Assume that ζ is a separating vector for N″, the core of the nest algebra Alg(N). We show that L is a Kadison-Singer lattice, and hence the corresponding algebra Alg(L) is a Kadison-Singer algebra. We also describe the center of Alg(L) and its commutator modulo itself, and show that every bounded derivation from Alg(L) into itself is inner, and all n-th bounded cohomology groups H n(Alg(L), B(H)) of Alg(L) with coefficients in B(H) are trivial for all n ⩾ 1.

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References

  1. Christensen E. Derivations of nest algebras. Math Ann, 1975, 229: 208–233

    Google Scholar 

  2. Davidson K R. Nest Algebras. π Pitman Research Notes in Mathematics Series, 191. New York: Longman Scientific & Technical, 1988

    Google Scholar 

  3. Ge L, Yuan W. Kadison-Singer algebras, I — hyperfinite case. Proc Natl Acad Sci USA, 2010, 107: 1838–1843

    Article  Google Scholar 

  4. Ge L, Yuan W. Kadison-Singer algebras, II — finite case. Proc Natl Acad Sci USA, 2010, 107: 4840–4844

    Article  Google Scholar 

  5. Han D. A note on the commutants of CSL algebras modulo bimodules. Proc Amer Math Soc, 1992, 116: 707–709

    Article  MATH  MathSciNet  Google Scholar 

  6. Johnson B E. Cohomology in Banach Algebras. Memoirs of the American Mathematical Society, No 127. Providence, RI: American Mathematical Society, 1972

    Google Scholar 

  7. Kadison K R, Ringrose J. Fundamentals of the Operator Algebras, Vol I and II. Orlando: Academic Press, 1983 and 1986

    Google Scholar 

  8. Kadison R, Singer I. Triangular operator algebras. Fundamentals and hyperreducible theory. Amer J Math, 1960, 82: 227–259

    Article  MATH  MathSciNet  Google Scholar 

  9. Lance E C. Cohomology and perturbations of nest algebras. Proc London Math Soc, 1981, 43: 334–356

    Article  MATH  MathSciNet  Google Scholar 

  10. Longstaff W. Strong reflexive lattices. J London Math Soc, 1975, 11: 491–498

    Article  MATH  MathSciNet  Google Scholar 

  11. Ringrose J. On some algebras of operators. Proc London Math Soc, 1965, 3: 61–83

    Article  MathSciNet  Google Scholar 

  12. Sinclair A M, Smith R. Hochschild Cohomology of von Neumann Algebras. London Math Soc Lecture Notes Ser, vol. 203. Cambridge: Cambridge University Press, 1995

    MATH  Google Scholar 

  13. Wang L, Yuan W. On some examples of Kadison-Singer algebras. Preprint

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  1. Institute of Operations Research, Qufu Normal University, Rizhao, 276826, China

    ChengJun Hou

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  1. ChengJun Hou
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Correspondence to ChengJun Hou.

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Hou, C. Cohomology of a class of Kadison-Singer algebras. Sci. China Math. 53, 1827–1839 (2010). https://doi.org/10.1007/s11425-010-4025-4

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  • DOI: https://doi.org/10.1007/s11425-010-4025-4

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