A Toeplitz corona theorem for the pentablock and applications

We state and prove a Toeplitz corona theorem for the pentablock $\mathbb{P}$, a domain in ${\mathbb{C}}^3$ given by $$\mathbb{P}=\{(a_{21},\text{tr}(A),\det (A))\in {\mathbb{C}}^3:A=[a_{ij}]\in M_2(\mathbb{C}),\mathrm{\parallel }A\mathrm{\parallel }<1\}\mathrm{.}$$ By two different applications of this theorem, we obtain a few new characterizations in the Toeplitz corona theorems for the bidisc and the symmetrized bidisc.