  We prove an alternate Toeplitz corona theorem for the algebras of pointwise
kernel multipliers of Besov-Sobolev spaces on the unit ball in
$\mathbb{C}^{n}$, and for the algebra of bounded analytic functions on certain
strictly pseudoconvex domains and polydiscs in higher dimensions as well. This
alternate Toeplitz corona theorem extends to more general Hilbert function
spaces where it does not require the complete Pick property. Instead, the
kernel functions $k_{x}\left(y\right)$ of certain Hilbert function spaces
$\mathcal{H}$ are assumed to be invertible multipliers on $\mathcal{H}$, and
then we continue a research thread begun by Agler and McCarthy in 1999, and
continued by Amar in 2003, and most recently by Trent and Wick in 2009. In
dimension $n=1$ we prove the corona theorem for the kernel multiplier algebras
of Besov-Sobolev Banach spaces in the unit disk, extending the result for
Hilbert spaces $H^\infty\cap Q_p$ by A. Nicolau and J. Xiao.
