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 $\mathcal{O}_2$.