Nuclear C*-algebra

Summary

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 edit

Nuclearity admits the following equivalent characterizations:

Examples edit

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 edit

References edit

  1. ^ Argerami, Martin (20 January 2023). Answer to "The C∗ algebras of matrices, continuous functions, measures and matrix-valued measures / continuous functions and their state spaces". Mathematics StackExchange. Stack Exchange.
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  • 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
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  • 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