Toeplitz algebra

Description

Let $\ell^2(\N)$ be the Hilbert space of square-summable sequences, with orthonormal basis $e_0, e_1, e_2, \ldots$. The unilateral shift is the operator $S$ with $S e_n = e_{n+1}$ for all $n$. The Toeplitz algebra $\mathcal{T}$ is the C*-subalgebra of the algebra of bounded operators $B(\ell^2(\N))$ generated by $S$. It is the norm closure of the set of polynomials in $S$ and $S^*$. It is a C*-algebra by construction. Identify $\ell^2(\N)$ with the Hardy space $H^2$ of the unit circle. Then $\mathcal{T}$ is the C*-algebra generated by the Toeplitz operators $T_f$ with continuous symbol $f$.

Check that $\mathcal{T}$ contains the compact operators: $S^* S = 1$, and $P = 1 - S S^*$ is the projection onto $\C e_0$. Then $S^m P (S^*)^n$ is the rank-one operator sending $e_n$ to $e_m$ and the other basis vectors to $0$. These matrix units lie in $\mathcal{T}$, and the norm closure of their span is the ideal of compact operators $K$.