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Nuclear C*-algebra

From Wikipedia, the free encyclopedia

In the mathematical field of functional analysis, a nuclear C*-algebra is a C*-algebra A such that for every C*-algebra B the injective and projective C*-cross norms coincides on the algebraic tensor product AB and the completion of AB with respect to this norm is a C*-algebra. This property was first studied by Takesaki (1964) under the name "Property T", which is not related to Kazhdan's property T.

Characterizations

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Nuclearity admits the following equivalent characterizations:

Examples

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The commutative unital C* algebra of (real or complex-valued) continuous functions on a compact Hausdorff space as well as the noncommutative unital algebra of n×n real or complex matrices are nuclear.[1]

See also

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References

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  • Connes, Alain (1976), "Classification of injective factors.", Annals of Mathematics, Second Series, 104 (1): 73–115, doi:10.2307/1971057, ISSN 0003-486X, JSTOR 1971057, MR 0454659
  • Effros, Edward G.; Ruan, Zhong-Jin (2000), Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press Oxford University Press, ISBN 978-0-19-853482-2, MR 1793753
  • Lance, E. Christopher (1982), "Tensor products and nuclear C*-algebras", Operator algebras and applications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math., vol. 38, Providence, R.I.: Amer. Math. Soc., pp. 379–399, MR 0679721
  • Pisier, Gilles (2003), Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294, Cambridge University Press, ISBN 978-0-521-81165-1, MR 2006539
  • Rørdam, M. (2002), "Classification of nuclear simple C*-algebras", Classification of nuclear C*-algebras. Entropy in operator algebras, Encyclopaedia Math. Sci., vol. 126, Berlin, New York: Springer-Verlag, pp. 1–145, MR 1878882
  • Takesaki, Masamichi (1964), "On the cross-norm of the direct product of C*-algebras", The Tohoku Mathematical Journal, Second Series, 16: 111–122, doi:10.2748/tmj/1178243737, ISSN 0040-8735, MR 0165384
  • Takesaki, Masamichi (2003), "Nuclear C*-algebras", Theory of operator algebras. III, Encyclopaedia of Mathematical Sciences, vol. 127, Berlin, New York: Springer-Verlag, pp. 153–204, ISBN 978-3-540-42913-5, MR 1943007