# 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

• 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 $1$ 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 $cb$-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