In operator algebras, the Toeplitz algebra is the C*-algebra generated by the unilateral shift on the Hilbert space l<sup>2</sup>(N). Identifying l<sup>2</sup>(N) with the Hardy space H<sup>2</sup>, the Toeplitz algebra consists of elements of the form
where ' 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 'is a Fredholm operator if and only if the symbol of ' is invertible. In that case, the Fredholm index of '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.