Cuntz algebra

In mathematics, the Cuntz algebra ${\displaystyle {\mathcal {O}}_{n}}$, named after Joachim Cuntz, is the universal C*-algebra generated by ${\displaystyle n}$ isometries of an infinite-dimensional Hilbert space ${\displaystyle {\mathcal {H}}}$ satisfying certain relations.^[1] These algebras were introduced as the first concrete examples of a separable infinite simple C*-algebra, meaning as a Hilbert space, ${\displaystyle {\mathcal {O}}_{n}}$ is isometric to the sequence space

${\displaystyle l^{2}(\mathbb {N} )}$

and it has no nontrivial closed ideals. These algebras are fundamental to the study of simple infinite C*-algebras since any such algebra contains, for any given n, a subalgebra that has ${\displaystyle {\mathcal {O}}_{n}}$ as quotient.