---
tag: D2M4VXJ7PA
kind: concept
name: Nuclear C*-algebra
msc: [46L06, 46L05]
sources:
  - url: https://en.wikipedia.org/wiki/Nuclear_C*-algebra
same_as:
  wikidata: Q2602075
  wikipedia: Nuclear C*-algebra
---

## Summary

A nuclear C\*-algebra is a C*-algebra whose algebraic tensor product with any other C\*-algebra has only one C\*-norm.
Equivalently, the identity map of the algebra can be approximated by [[completely positive maps|CGY299QMC3]] that factor through matrix algebras.
Nuclearity is a finiteness or amenability property: for C\*-algebras it is equivalent to amenability as a Banach algebra.
It is a standing hypothesis in the classification program for simple C\*-algebras.

## Explanations

### L1 human

To combine two C\*-algebras into one, one forms their tensor product and must choose a norm on it.
In general there can be many reasonable choices, which give different completions.
Nuclear algebras are the well-behaved ones for which there is no choice to make.
All commutative C\*-algebras, all finite-dimensional ones and all AF algebras are nuclear.