Commutative $C^*$-algebras generated by Toeplitz operators on the super unit ball

TL;DR

This paper generalizes the theory of commutative $C^*$-algebras generated by Toeplitz operators from classical domains to supermanifolds, constructing new algebraic structures on super ball and super Siegel domain.

Contribution

It introduces the construction of commutative $C^*$-algebras of super Toeplitz operators on supermanifolds, extending classical results to the super setting.

Findings

Constructed commutative $C^*$-algebras on super ball and super Siegel domain.

Generalized classical Toeplitz algebra results to supermanifolds.

Identified specific commutative algebras for each even maximal Abelian subgroup.

Abstract

We extend known results about commutative $C^*$-algebras generated Toeplitz operators over the unit ball to the supermanifold setup. This is obtained by constructing commutative $C^*$-algebras of super Toeplitz operators over the super ball $\mathbb{B}^{p|q}$ and the super Siegel domain $\mathbb{U}^{p|q}$ that naturally generalize the previous results for the unit ball and the Siegel domain. In particular, we obtain one such commutative $C^*$-algebra for each even maximal Abelian subgroup of automorphisms of the super ball.