nuclear C*-algebra

Definition 0.1.

A C*-algebra $A$ is called a nuclear C*-algebra if all C*-norms on every algebraic tensor product $A\otimes X$ , 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 $A\otimes X$ to a C*-algebra , for any other C*-algebra $X$ .

0.1 Examples of nuclear C*-algebras

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All commutative C*-algebras and all finite-dimensional C*-algebras
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Group C*-algebras of amenable groups
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Crossed products of strongly amenable C*-algebras by amenable discrete groups,
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Type $1$ C*-algebras.

0.2 Exact C*-algebra

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

Definition 0.2.

Let $M_{n}$ be a matrix space, let $\mathcal{A}$ be a general operator space, and also let $\mathbb{C}$ be a C*-algebra. A $C^{*}$ -algebra $\mathbb{C}$ is exact if it is ‘finitely representable’ in $M_{n}$ , that is, if for every finite dimensional subspace $E$ in $\mathcal{A}$ and quantity $epsilon>0$ , there exists a subspace $F$ of some $M_{n}$ , and also a linear isomorphism $T:E\to F$ such that the $c b$ -norm

|T|_{cb}|T^{-1}|_{cb}<1+epsilon.

0.3 Note: A counter-example

A $C^{*}$ -subalgebra of a nuclear C*-algebra need not be nuclear.

References

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 http://planetmath.org/?op=getobj&from=books&id=66a graduate level preprint discussing general C*-algebras http://aux.planetmath.org/files/books/66/C*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 algebra
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