# PlanetPhysics/Nuclear C Algebra

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


### 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 ${\displaystyle 1}$ C*-algebras.

### Exact C*-algebra

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

Let ${\displaystyle M_{n}}$ be a matrix space, let ${\displaystyle {\mathcal {A}}}$ be a general operator space, and also let ${\displaystyle \mathbb {C} }$ be a C*-algebra. A ${\displaystyle C^{*}}$-algebra ${\displaystyle \mathbb {C} }$ is exact if it is `finitely representable' in ${\displaystyle M_{n}}$, that is, if for every finite dimensional subspace ${\displaystyle E}$ in ${\displaystyle {\mathcal {A}}}$ and quantity ${\displaystyle epsilon>0}$, there exists a subspace ${\displaystyle F}$ of some ${\displaystyle M_{n}}$, and also a linear isomorphism ${\displaystyle T:E\to F}$ such that the ${\displaystyle cb}$-norm ${\displaystyle |T|_{cb}|T^{-1}|_{cb}<1+epsilon.}$

### Counter-example

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

## References

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 ${\displaystyle C^{*}}$ -algebras, Hilbert ${\displaystyle C^{*}}$ -Modules and Quantum Mechanics", pp. 89 a graduate level preprint discussing general C*-algebras in Postscript format.