Nuclear C*-algebra
In mathematics, a nuclear C*-algebra is a C*-algebra A such that the injective and projective C*-cross norms on A⊗B are the same for every C*-algebra B. This property was first studied by Takesaki (1964) under the name "Property T", which is not related to Kazhdan's property T.
Characterizations
Nuclearity admits the following equivalent characterizations:
- The identity map, as a completely positive map, approximately factors through matrix algebras. By this equivalence, nuclearity can be considered a noncommutative analogue of the existence of partitions of unity.
- The enveloping von Neumann algebra is injective.
- It is amenable as a Banach algebra.
- It is isomorphic to a C*-subalgebra B of the Cuntz algebra with the property that there exists a conditional expectation from to B. This condition is only equivalent to the others for separable C*-algebras.
See also
References
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- Lance, E. Christopher (1982), "Tensor products and nuclear C*-algebras", Operator algebras and applications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math., 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, 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., 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, 127, Berlin, New York: Springer-Verlag, pp. 153–204, ISBN 978-3-540-42913-5, MR 1943007
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