Index of $Γ$-equivariant Toeplitz operators

Abstract

Let $Γ$ be a discrete icc subgroup of PSL(2,R) of infinite covolume. and let M denote the quotient of the unit disc by $Γ$. We prove that a Toeplitz operator with $Γ$-invariant symbol f in C(M) is Brauer Fredholm if its symbol is invertible on the boundary of M and its Brauer index is equal to the winding number of f at the boundary. We construct the associated extension of the algebra of functions continuous on the boundary of M by the Brauer ideal in the C*-algebra generated by such operators.