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. 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 with the property that there exists a conditional expectation from 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

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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 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