PlanetPhysics/Nuclear C Algebra

From Wikiversity
Jump to navigation Jump to search

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

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

Examples of nuclear C*-algebras[edit | edit source]

  • All commutative 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 C*-algebras.

Exact C*-algebra[edit | edit source]

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

Let be a matrix space, let be a general operator space, and also let be a C*-algebra. A -algebra is exact if it is `finitely representable' in , that is, if for every finite dimensional subspace in and quantity , there exists a subspace of some , and also a linear isomorphism such that the -norm

Counter-example[edit | edit source]

The group C*-algebras for the free groups on two or more generators are not nuclear. Furthermore, a -subalgebra of a nuclear C*-algebra need not be nuclear.

All Sources[edit | edit source]

[1] [2]

References[edit | edit source]

  1. E. C. Lance. 1981. Tensor Products and nuclear C*-algebras., in {\em 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 -algebras, Hilbert -Modules and Quantum Mechanics", pp. 89 a graduate level preprint discussing general C*-algebras in Postscript format.