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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. These were first studied by Takesaki (1964) under the name "Property T" (this is unconnected with Kazhdan's property T).


Characterizations

Nuclearity admits the following equivalent characterizations:

A C*-algebra is nuclear if the identity map, as a completely positive map, approximately factors through matrix algebras. Abelian C*-algebras are nuclear. One might say that nuclearity means that the C*-algebra admits noncommutative "partitions of unity."
A C*-algebra is nuclear if and only if its enveloping von Neumann algebra is injective.
A separable C*-algebra is nuclear if and only if it is isomorphic to a C*-subalgebra B of the Cuntz \( algebra \( \mathcal{O}_2 \) with the property that there exists a conditional expectation from \mathcal{O}_2 to B \).
A C*-algebra is nuclear if and only if it is amenable as a Banach algebra.

See also

Nuclear space
Exact C*-algebra

References

Connes, Alain (1976), "Classification of injective factors.", Annals of Mathematics. Second Series 104 (1): 73–115, ISSN 0003-486X, JSTOR 1971057, MR 0454659
Effros, Edward G.; Ruan, Zhong-Jin (2000), Operator spaces, London Mathematical Society Monographs. New Series 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. 38, Providence, R.I.: Amer. Math. Soc., pp. 379–399, MR 679721
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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