Toeplitz algebra

In operator algebras, the Toeplitz algebra is the C*-algebra generated by the unilateral shift on the Hilbert space "l"²(N). Taking "l"²(N) to be the Hardy space "H"², the Toeplitz algebra consists of elements of the form

:$T\_f\; +\; K;$

where "T_f" 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 "T_f" + "K" is a Fredholm operator if and only if the symbol "f" of "T_f" is invertible. In that case, the Fredholm index of "T_f" + "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 conlude that the Toeplitz algebra is the universal C*-algebra generated by a proper isometry; this is "Coburn's theorem".

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