---
tag: 36Q7XH9P7W
kind: example
name: Cuntz algebra
msc: [46L05, 46L80]
sources:
  - url: https://en.wikipedia.org/wiki/Cuntz_algebra
same_as:
  wikidata: Q1144125
  wikipedia: Cuntz algebra
---

## Example

Let $n \ge 2$ be an integer.
The *Cuntz algebra* $\mathcal{O}_n$ is the universal C*-algebra generated by elements $s_1, \ldots, s_n$ subject to the relations
$$s_i^* s_i = 1 \quad (1 \le i \le n), \qquad \sum_{i=1}^{n} s_i s_i^* = 1.$$
So the $s_i$ are isometries, and the projections $s_i s_i^*$ onto their ranges add up to $1$.
The relations imply $s_i^* s_j = 0$ for $i \ne j$: the ranges are mutually orthogonal.

Concrete check for $n = 2$: on the Hilbert space $\ell^2(\N)$ with orthonormal basis $(e_k)_{k \ge 0}$, let $s_1 e_k = e_{2k}$ and $s_2 e_k = e_{2k+1}$.
Each $s_i$ maps an orthonormal basis to an orthonormal family, so it is an isometry.
The range of $s_1$ is spanned by the even basis vectors and that of $s_2$ by the odd ones, so $s_1 s_1^* + s_2 s_2^* = 1$.
The C\*-algebra generated by $s_1$ and $s_2$ is a copy of $\mathcal{O}_2$.

## Remarks

Cuntz showed that $\mathcal{O}_n$ is simple, so any $n$ isometries with these relations generate an isomorphic C\*-algebra.
It is also infinite: $s_1$ is an isometry with $s_1 s_1^* \ne 1$.
Its K-theory is $K_0(\mathcal{O}_n) \cong \Z / (n-1)\Z$ and $K_1(\mathcal{O}_n) = 0$, so the algebras $\mathcal{O}_n$ for different $n$ are not isomorphic.
Each $\mathcal{O}_n$ is separable and [[nuclear|D2M4VXJ7PA]].

## Explanations

### L1 human

In an infinite-dimensional space there is room to fit two full copies of the whole space side by side, as the even and odd positions of a sequence.
The Cuntz algebra is the algebra generated by such copying maps.
It is a basic example of a simple C\*-algebra that is very far from the matrix algebras.