Nuclear C*-algebra

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 A⊗B and the completion of A⊗B 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

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.
• (For separable algebras) It is isomorphic to a C*-subalgebra B of the Cuntz algebra 𝒪₂ with the property that there exists a conditional expectation from 𝒪₂ to B.

Examples

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.

See also

• Injective tensor product
• Nuclear space – A generalization of finite dimensional Euclidean spaces different from Hilbert spaces
• Projective tensor product

References

• Connes, Alain (1976), "Classification of injective factors.", Annals of Mathematics, Second Series 104 (1): 73–115, doi:10.2307/1971057, ISSN 0003-486X 
• 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, http://www.oup.com/us/catalog/general/subject/Mathematics/PureMathematics/?ci=9780198534822 
• 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 
• Pisier, Gilles (2003), Introduction to operator space theory, London Mathematical Society Lecture Note Series, 294, Cambridge University Press, ISBN 978-0-521-81165-1 
• 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 
• 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 
• 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 

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