[1] Taking l2(N) to be the Hardy space H2, the Toeplitz algebra consists of elements of the form where Tf is a Toeplitz operator with continuous symbol and K is a compact operator.
So the Toeplitz algebra can be viewed as the C*-algebra extension of continuous functions on the circle by the compact operators.
In that case, the Fredholm index of Tf + K is precisely the winding number of f, the equivalence class of f in the fundamental group of the circle.
This is a special case of the Atiyah-Singer index theorem.
Wold decomposition characterizes proper isometries acting on a Hilbert space.