Exercise 8, § 5

We use again the notations of the preceding exercise.

a) We denote by $T(\mathbf{S}_1)$ the closed subalgebra of $\mathscr{L}(F)$ generated by the Toeplitz operators $T_{e_1}$ and $T_{e_{-1}}$. Now prove that $T(\mathbf{S}_1)$ is a star algebra of $\mathscr{L}(F)$.

b) Let $f\in \mathscr{C}(\mathbf{S}_1)$. Now prove that $T_f$ belongs to $T(\mathbf{S}_1)$.

c) The representation of $T(\mathbf{S}_1)$ in F is irreducible; the star algebra $T(\mathbf{S}_1)$ contains $\mathscr{L}^c(F)$. d) We have $u\in T(\mathbf{S}_1)$ if and only if there exist $f\in \mathscr{C}(\mathbf{S}_1)$ and $h\in \mathscr{L}^c(F)$ such that $u= T_f+h$. (Now prove that the set of endomorphisms of the form $T_f+h$ forms a closed star algebra in $\mathscr{L}(F).$)

e) There exists an exact sequence of morphisms of involutive algebras

0 $\leftarrow \rightarrow \mathscr{L}^c(F)\leftarrow \rightarrow T(\mathbf{S}_1)\leftarrow_{\varrho}\rightarrow \mathscr{C}(\mathbf{S}_1)\leftarrow \rightarrow 0$

such that $\varrho (T_f) =f$ for every function $f\in \mathscr{C}(\mathbf{S}_1)$.