Automorphisms of some subalgebras of the Toeplitz algebra

We study the automorphism groups of the subalgebras $\mathscr T_m$ and $\mathscr T(m)$ of the classical Toeplitz algebra which are invariant under a finite subgroup in $S^1$. We also give description for the automorphism group of the subalgebra of compact operators in $\mathscr T_m$. The relationship between the automorphisms of $\mathscr T_m$ and $\mathscr T(m)$ is revealed.