nuclear C*-algebra

Definition 0.1.

A C*-algebraPlanetmathPlanetmath A is called a nuclear C*-algebra if all C*-norms on every algebraic tensor productPlanetmathPlanetmathPlanetmath AX, of A with any other C*-algebra X, agree with, and also equal the spatial C*-norm (viz Lance, 1981). Therefore, there is a unique completion of AX to a C*-algebra , for any other C*-algebra X.

0.1 Examples of nuclear C*-algebras

  • All commutativePlanetmathPlanetmathPlanetmath C*-algebras and all finite-dimensional C*-algebras

  • Group C*-algebras of amenable groups

  • Crossed products of strongly amenable C*-algebras by amenable discrete groups,

  • Type 1 C*-algebras.

0.2 Exact C*-algebra

In general terms, a C*-algebra is exact if it is isomorphicPlanetmathPlanetmathPlanetmath with a C*-subalgebra of some nuclear C*-algebra. The precise definition of an exact C*-algebra follows.

Definition 0.2.

Let Mn be a matrix space, let 𝒜 be a general operator space, and also let be a C*-algebra. A C*-algebra is exact if it is ‘finitely representable’ in Mn, that is, if for every finite dimensional subspaceMathworldPlanetmathPlanetmath E in 𝒜 and quantity epsilon>0, there exists a subspace F of some Mn, and also a linear isomorphism T:EF such that the cb-norm


0.3 Note: A counter-example

A C* -subalgebra of a nuclear C*-algebra need not be nuclear.


  • 1 E. C. Lance. 1981. Tensor Products and nuclear C*-algebras., in Operator Algebras and Applications, R.V. Kadison, ed., Proceed. Symp. Pure Maths., 38: 379-399, part 1.
  • 2 N. P. Landsman. 1998. “Lecture notes on C*-algebras, Hilbert C*-Modules and Quantum Mechanics”, pp. 89 graduate level preprint discussing general C*-algebras*algebrae.psin Postscript format.
Title nuclear C*-algebra
Canonical name NuclearCalgebra
Date of creation 2013-03-22 18:12:25
Last modified on 2013-03-22 18:12:25
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 63
Author bci1 (20947)
Entry type Definition
Classification msc 81T05
Classification msc 81R50
Classification msc 81R15
Synonym quantum operator algebraPlanetmathPlanetmath
Synonym C*-algebra
Synonym C*-algebra
Related topic QuantumOperatorAlgebrasInQuantumFieldTheories
Related topic NoncommutativeGeometry
Related topic GroupoidCConvolutionAlgebra
Related topic GroupoidCDynamicalSystem
Related topic CAlgebra3
Related topic CAlgebra
Related topic QuotientsInCAlgebras
Defines generated C*-algebra
Defines exact C^*-algebra
Defines group C*-algebra