A uniformly closed -algebra of operators on a Hilbert space (a uniformly closed -algebra). Such algebras are closely connected to important fields of geometric analysis, e.g., index theory, geometric quantization and several complex variables.
In the one-dimensional case one considers the Hardy space over the one-dimensional torus (cf. also Hardy spaces), and defines the Toeplitz operator with "symbol" function by for all , where is the orthogonal projection given by the Cauchy integral theorem. The -algebra generated by all operators with continuous symbol is not commutative, but defines a -algebra extension
of the -algebra of all compact operators; in fact, this "Toeplitz extension" is the generator of the Abelian group .
-algebra extensions are the building blocks of -theory and index theory; in our case a Toeplitz operator is Fredholm (cf. also Fredholm operator) if has no zeros, and then the index is the (negative) winding number of .
In the multi-variable case, Toeplitz -algebras have been studied in several important cases, e.g. for strictly pseudo-convex domains [a1], including the unit ball [a2], [a10], for tube domains and Siegel domains over convex "symmetric" cones [a5], [a8], and for general bounded symmetric domains in having a transitive semi-simple Lie group of holomorphic automorphisms [a7]. Here, the principal new feature is the fact that Toeplitz operators (say, on the Hardy space over the Shilov boundary of a pseudo-convex domain ) with continuous symbols are not essentially commuting, i.e.
in general. Thus, the corresponding Toeplitz -algebra is not a (one-step) extension of ; instead one obtains a multi-step -filtration
of -ideals, with essentially commutative subquotients , whose maximal ideal space (its spectrum) reflects the boundary strata of the underlying domain. The length of the composition series is an important geometric invariant, called the rank of . The index theory and -theory of these multi-variable Toeplitz -algebras is more difficult to study; on the other hand one obtains interesting classes of operators arising by geometric quantization of the underlying domain , regarded as a complex Kähler manifold.
A general method for studying the structure and representations of Toeplitz -algebras, at least for Shilov boundaries arising as a symmetric space (not necessarily Riemannian), is the so-called -duality [a11], [a9]. For example, if is a Lie group with (reduced) group -algebra , then the so-called co-crossed product -algebra induced by a natural co-action can be identified with . Now the Cauchy–Szegö orthogonal projection (cf. also Cauchy operator) defines a certain -completion , and the corresponding Toeplitz -algebra can be realized as (a corner of) . In this way the well-developed representation theory of (co-) crossed product -algebras [a4] can be applied to obtain Toeplitz -representations related to the boundary . For example, the two-dimensional torus gives rise to non-type- -algebras (for cones with irrational slopes), and the underlying "Reinhardt" domains (cf. also Reinhardt domain) have interesting complex-analytic properties, such as a non-compact solution operator of the Neumann -problem [a6].
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Toeplitz C*-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Toeplitz_C*-algebra&oldid=42916