Simplicity of Cuntz algebra

Question

It is well known that the Cuntz algebra $\mathcal{O}_N$ (for fixed $N$) is simple. Is there any easy way to exhibit it (as a Banach algebra only) as a quotient of a Banach algebra modulo some maximal ideal? Of course, I am not interested in the quotient of the form $\mathcal{O}_N/ \{0\}$.

Answer 1

Cuntz introduced the algebras $\mathcal O_n$ in the paper "Simple C*-algebras generated by isometries", 1977. Section 3 of the paper is called "Extensions of $\mathcal O_n$." It is proved there that if $n$ is finite, and if $V_1,\ldots V_n$ are isometries on Hilbert space such that $\sum\limits_{k=1}^n V_k V_k^*\leq I$ and $\sum\limits_{k=1}^n V_k V_k^*\neq I$ (i.e., if $V_1,\ldots, V_n$ are isometries with orthogonal ranges that do not add up to the whole space), then $\mathcal O_n$ is a quotient of $C^*(V_1,\ldots,V_n)$ by the ideal generated by the projection $I-\sum\limits_{k=1}^n V_k V_k^*$, and this ideal is isomorphic to the algebra of compact operators.

Since it is easy to give examples of such isometries, this might be the easy way you're looking for. Here is one particular construction that may not be the easiest, but that has an important role in the subject, and appears in D.E. Evans's "On $O_n$", 1980. Let $H$ be an $n$-dimensional Hilbert space with orthonormal basis $\{e_1,\ldots,e_n\}$. Let $F(H)=\mathbb C\oplus H\oplus (H\otimes H)\oplus (H\otimes H\otimes H) \oplus \cdots=\sum\limits_{k=0}^\infty H^{\otimes k}$ be the full Fock space of $H$, and for $i\in\{1,2,\ldots,n\}$, let $V_i\in B(F(H))$ be defined by $V_i x=e_i\otimes x$. Then each $V_i$ is an isometry, $\sum\limits_{k=1}^n V_kV_k^*\leq I$, and $I-\sum\limits_{k=1}^n V_kV_k^*$ is the projection onto the first summand of $F(H)$. In this case the ideal in question is precisely the set of compact operators on $F(H)$, $K(F(H))$, and $C^*(V_1,\ldots,V_n)/K(F(H))\cong \mathcal O_n$.