There is the well-known extension of $C^*$-algebras $0\rightarrow K\rightarrow\mathcal{T}\rightarrow C(S^1)\rightarrow 0$ where $\mathcal{T}$ is the Toeplitz algebra (generated by the unilateral shift on $\ell_2(\mathbb{N})$), and $K$ is the algebra of compact operators on Hilbert space. Can one make sense of this extension or some variant of it as an extension of algebras of operators on $L_p$ spaces ($p\in[1,\infty)$) if one considers the Banach algebra generated by the unilateral shift on $\ell_p(\mathbb{N})$?
I gathered some facts from section 6.2 of the book Analysis of Toeplitz Operators by Böttcher and Silbermann:
Let $\mathcal{T}_p$ ($1<p<\infty$) denote the norm closure in $B(\ell_p)$ of $\{T_f+K:f\in\mathcal{P},K\in K(\ell_p)\}$ where $\ell_p=\ell_p(\mathbb{N})$, $\mathcal{P}$ denotes the set of Laurent polynomials with complex coefficients, $T_f$ is the Toeplitz operator with symbol $f$, and $K(\ell_p)$ denotes the ideal of compact operators on $\ell_p$. Let $\pi:\mathcal{T}_p\rightarrow\mathcal{T}_p/K(\ell_p)$ be the quotient homomorphism. Then $\mathcal{T}_p/K(\ell_p)$ is the norm closure of $\{\pi(T_f):f\in\mathcal{P}\}$ in $B(\ell_p)/K(\ell_p)$ and it is a commutative Banach algebra.
The maximal ideal space of $\mathcal{T}_p/K(\ell_p)$ is the circle $S^1$, and the Gelfand transform $\Gamma$ is given by $\Gamma(\pi(T_f))(t)=f(t)$.
Is it known whether the quotient is (isometrically) isomorphic to $C(S^1)$?