Nuclear C*-algebra - Imported Knowlpedia Content

Definition

A $C^*$ -algebra $A$ is nuclear if, for every $C^*$ -algebra $B$ , the algebraic tensor product $A\odot B$ has a unique $C^*$ -tensor norm. Equivalently, for every $B$ , the canonical quotient from the maximal tensor product $A\otimes_{\max}B$ onto the minimal tensor product $A\otimes_{\min}B$ is an isomorphism. No unitality or separability assumption is part of the definition. Nuclearity says that $A$ cannot detect the distinction between universal and spatial completions, not that $A$ is finite-dimensional or commutative.

Completely positive characterization

The Choi–Effros–Kirchberg approximation theorem says that $A$ is nuclear exactly when it has the completely positive approximation property. Concretely, the identity on $A$ can be approximated in point-norm by factorizations

A\xrightarrow{\ \phi_\lambda\ }M_{n(\lambda)}(\mathbb C) \xrightarrow{\ \psi_\lambda\ }A

through matrix algebras, where both maps are completely positive contractions Brown–Ozawa, Theorem 2.3.8. This converts a global tensor-norm condition into finite-dimensional local approximations.

Examples and permanence

Every commutative $C^*$ -algebra, every matrix algebra, the compact-operator $C^*$ -algebra, and every AF algebra are nuclear. Nuclearity passes to ideals, quotients, inductive limits, and extensions, but not to arbitrary $C^*$ -subalgebras.

For a discrete group $\Gamma$ , the [[operator-algebras/nuclearity-reduced-group-cstar-algebra-discrete-amenability|Lance theorem]] states that $C_r^*(\Gamma)$ is nuclear exactly when $\Gamma$ is amenable. Hence the reduced $C^*$ -algebra of the free group on two generators, a free group, is a standard non-example.

Structural significance

Nuclearity is an operator-algebraic regularity property. It permits tensor products to be written without choosing between minimal and maximal norms and is a standing hypothesis in much of the structure and classification theory of $C^*$ -algebras. It is stronger than exactness and stronger than ordinary Banach-space approximation properties.

The original tensor-norm theory and its completely positive formulation were developed through several equivalent characterizations; the approximation form is especially useful because it behaves well under limits and constructions Choi–Effros, pp. 61–79.

Conventions and scope

References

E. Christopher Lance, “On nuclear C-algebras,” Journal of Functional Analysis 12 (1973), 157–176. DOI record. Relevant: tensor-norm characterizations and group C-algebras.
Man-Duen Choi and Edward G. Effros, “Nuclear C-Algebras and the Approximation Property,” American Journal of Mathematics* 100 (1978), 61–79. DOI record. Relevant: completely positive approximation methods.
Nathanial P. Brown and Narutaka Ozawa, C-Algebras and Finite-Dimensional Approximations*, Graduate Studies in Mathematics 88, American Mathematical Society, 2008. DOI record. Relevant: §2.3, especially Theorem 2.3.8, on tensor products, CPAP, and nuclearity.