Before posing my question, let me provide a little background since the Wikipedia page on this stuff is sorely lacking.
Let's start with the classical case of the Toeplitz index problem on the circle. We view $S^1$ as the unit circle in the complex plane, and define the Hardy space $H^2(S^1)$ of the circle to be the closed subspace of $L^2(S^1)$ generated by the functions $z^n$ for $n \geq 0$. Let $P$ denote the orthogonal projection from $L^2(S^1)$ to $H^2(S^1)$. Given a function $g \in L^\infty(S^1)$, the Toeplitz operator with symbol $g$ is the operator $T_g: H^2(S^1) \to H^2(S^1)$ given by $T_g(f) = P(gf)$. If $g$ is continuous and nowhere vanishing then $T_g$ is a Fredholm operator, and by the usual stability properties the Fredholm index depends only on the homotopy class of $g$. Thus the classical Toeplitz index problem is to compute the Fredholm index of $T_g$ in purely topological terms. It turns out that the Fredholm index is minus the winding number of $g$ (in the sense of complex analysis).
This generalizes in several directions. The starting point for many of these generalizations is the observation that $H^2(S^1)$ is the closure space of boundary values of holomorphic functions on the open unit disk. For instance, given a strongly pseudoconvex domain $\Omega$ in $\mathbb{C}^n$ one can consider the Hardy space $H^2(\partial \Omega)$ obtained by taking the closure of the space of boundary values of holomorphic functions on $\Omega$ which extend continuously to the closure. One can then form Toeplitz operators associated to continuous nonvanishing functions on the boundary and prove that they are Fredholm. In this case the Fredholm index is generally 0, so it is more interesting to consider $GL_n(\mathbb{C})$-valued functions on the boundary which, by Bott periodicity, are classified up to homotopy by the integers. As the classical case would suggest, the Fredholm index of a Toeplitz operator with symbol $F: \partial(\Omega) \to GL_n(\mathbb{C})$ is given by minus the homotopy class of $F$ (regarded as an integer).
I would like to play a similar game with a discrete group $\Gamma$. It might be too much to ask to pose this sort of problem for a general discrete group; maybe it helps to assume that $\Gamma$ is Gromov hyperbolic to get things started. The sort of index problem that I have in mind would consist of the following data:
- A compactification of (the Cayley graph of) $\Gamma$
- A ($\Gamma$ - invariant?) measure on $\partial \Gamma$
- A specially chosen space of functions $F(\Gamma)$ on $\Gamma$, some of which extend nontrivially to the boundary.
One could then form the space $H^2(\partial \Gamma)$ of boundary values of functions in $F(\Gamma)$ which extend to $L^2$ functions on the boundary and the orthogonal projection operator $P: L^2(\partial \Gamma) \to H^2(\partial \Gamma)$. My question:
Can the data above be chosen so that the obvious Toeplitz operator associated to a continuous nonvanishing (or maybe continuous $GL_n(\mathbb{C})$-valued) function on $\partial \Gamma$ is Fredholm? If so, is it known what its Fredholm index calculates?
