operator algebras In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study ...

, the Toeplitz algebra is the

C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continu ...

generated by the

unilateral shift In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator. Shift ...

on the

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...

''l''²(N). Taking ''l''²(N) to be the

Hardy space In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In ...

''H''², the Toeplitz algebra consists of elements of the form :

T_f + K\;

where ''T_f'' is a

Toeplitz operator In operator theory, a Toeplitz operator is the compression of a multiplication operator on the circle to the Hardy space. Details Let ''S''1 be the circle, with the standard Lebesgue measure, and ''L''2(''S''1) be the Hilbert space of square-in ...

with continuous symbol and ''K'' is a

compact operator In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact ...

. 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 In mathematics, Fredholm operators are certain Operator (mathematics), operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operat ...

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 In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of tu ...

of ''f'', the equivalence class of ''f'' in the

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

of the circle. This is a special case of the Atiyah-Singer index theorem.

Wold decomposition In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states ...

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 In mathematics, a universal C*-algebra is a C*-algebra described in terms of generators and relations. In contrast to rings or algebras, where one can consider quotients by free rings to construct universal objects, C*-algebras must be realizable ...

generated by a proper isometry; this is ''Coburn's theorem''.

References

{{DEFAULTSORT:Toeplitz Algebra C*-algebras