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Homenuclear C*algebra
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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 finitedimensional C*algebras

Group C*algebras of amenable 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 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 a graduate level preprint discussing general C*algebras in Postscript format.
Mathematics Subject Classification
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