In operator algebras, the Toeplitz algebra is the C*-algebra generated by the unilateral shift on the Hilbert space l2(N). Taking l2(N) to be the Hardy space H2, the Toeplitz algebra consists of elements of the form
Tf+K
where Tf is a Toeplitz operator with continuous symbol and K is a compact operator.
Toeplitz operators with continuous symbols commute modulo the compact operators. So the Toeplitz algebra can be viewed as the C*-algebra extension of continuous functions on the circle by the compact operators. This extension is called the Toeplitz extension.
By Atkinson's theorem, an element of the Toeplitz algebra Tf + K is a Fredholm operator if and only if the symbol f of Tf is invertible. 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. From this, together with properties of Toeplitz operators, one can conclude that the Toeplitz algebra is the universal C*-algebra generated by a proper isometry; this is Coburn's theorem.