In the mathematical field of
functional analysis, a nuclear C*-algebra is a
C*-algebraA 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:
(For separable algebras) It is isomorphic to a C*-subalgebra B of the
Cuntz algebra𝒪2 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
compactHausdorff space as well as the noncommutative unital algebra of n×n real or complex matrices are nuclear.[1]
Lance, E. Christopher (1982), "Tensor products and nuclear C*-algebras", Operator algebras and applications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math., vol. 38, Providence, R.I.: Amer. Math. Soc., pp. 379–399,
MR0679721
Rørdam, M. (2002), "Classification of nuclear simple C*-algebras", Classification of nuclear C*-algebras. Entropy in operator algebras, Encyclopaedia Math. Sci., vol. 126, Berlin, New York:
Springer-Verlag, pp. 1–145,
MR1878882
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,
ISSN0040-8735,
MR0165384
Takesaki, Masamichi (2003), "Nuclear C*-algebras", Theory of operator algebras. III, Encyclopaedia of Mathematical Sciences, vol. 127, Berlin, New York:
Springer-Verlag, pp. 153–204,
ISBN978-3-540-42913-5,
MR1943007