Nuclear C*-algebra
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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.[1]
See also
- Exact C*-algebra
- Injective tensor product
- Nuclear space – Generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
- Projective tensor product
References
- ↑ Argerami, Martin (20 January 2023). Answer to "The C∗ algebras of matrices, continuous functions, measures and matrix-valued measures / continuous functions and their state spaces". Mathematics StackExchange. Stack Exchange.
- 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
Template:Topological tensor products and nuclear spaces
it:C*-algebra#C*-algebra nucleare
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