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 finitedimensional 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 ${C}^{\mathrm{*}}$algebra follows.
Definition 0.2.
Let ${M}_{n}$ be a matrix space, let $\mathcal{A}$ be a general operator space, and also let $\u2102$ be a C*algebra. A ${C}^{*}$algebra $\u2102$ 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
$$ 
0.3 Note: A counterexample
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: 379399, 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  20130322 18:12:25 
Last modified on  20130322 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 