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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 {{harvtxt|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]] <math>\mathcal{O}_2</math> with the property that there exists a [[conditional expectation]] from <math>\mathcal{O}_2</math> to ''B''.

==See also==
*[[Nuclear space]]
*[[Exact C*-algebra]]

==References==
*{{Citation | last1=Connes | first1=Alain | author1-link=Alain Connes | title=Classification of injective factors.  | jstor=1971057 | mr=0454659  | year=1976 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=104 | issue=1 | pages=73–115}}
*{{Citation | last1=Effros | first1=Edward G. | last2=Ruan | first2=Zhong-Jin | title=Operator spaces | url=http://www.oup.com/us/catalog/general/subject/Mathematics/PureMathematics/?ci=9780198534822 | publisher=The Clarendon Press Oxford University Press | series=London Mathematical Society Monographs. New Series | isbn=978-0-19-853482-2 | mr=1793753  | year=2000 | volume=23}}
*{{Citation | last1=Lance | first1=E. Christopher | title=Operator algebras and applications, Part I (Kingston, Ont., 1980) | publisher=Amer. Math. Soc. | location=Providence, R.I. | series=Proc. Sympos. Pure Math. | mr=679721  | year=1982 | volume=38 | chapter=Tensor products and nuclear C*-algebras | pages=379–399}}
*{{Citation | last1=Pisier | first1=Gilles | title=Introduction to operator space theory | publisher=[[Cambridge University Press]] | series=London Mathematical Society Lecture Note Series | isbn=978-0-521-81165-1 | mr=2006539  | year=2003 | volume=294}}
*{{Citation | last1=Rørdam | first1=M. | title=Classification of nuclear C*-algebras. Entropy in operator algebras | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Encyclopaedia Math. Sci. | mr=1878882  | year=2002 | volume=126 | chapter=Classification of nuclear simple C*-algebras | pages=1–145}}
*{{Citation | last1=Takesaki | first1=Masamichi | title=On the cross-norm of the direct product of C*-algebras | mr=0165384  | year=1964 | journal=The Tohoku Mathematical Journal. Second Series | issn=0040-8735 | volume=16 | pages=111–122}}
*{{Citation | last1=Takesaki | first1=Masamichi | title=Theory of operator algebras. III | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Encyclopaedia of Mathematical Sciences | isbn=978-3-540-42913-5 | mr=1943007  | year=2003 | volume=127 | chapter=Nuclear C*-algebras | pages=153–204}}

[[Category:C*-algebras]]

[[it:C*-algebra#C*-algebra nucleare]]