Is $UL\neq PSPACE$ known? $L\neq DSPACE[\omega(\log n)]$ is known.
Is $UL\neq DSPACE[\omega(\log n)]$ and $UL\neq PSPACE$ known?
$\mathbf{UL}$ is contained in $\mathbf{NL}$, which is contained in $\mathbf{DSPACE}(\log^2 n)$ by Savitch's theorem, which is strictly contained in $\mathbf{PSPACE}$ by the space hierarchy theorem, so $\mathbf{UL}$ is strictly contained in $\mathbf{PSPACE}$. It would be surprising if e.g. $\mathbf{UL} = \mathbf{DSPACE}(\log^2 n)$, but I don't think it's known to be false.