# Toeplitz C*-algebra

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].

#### References

[a1] | L. Boutet de Monvel, "On the index of Toeplitz operators of several complex variables" Invent. Math. , 50 (1979) pp. 249–272 Zbl 0398.47018 |

[a2] | L. Coburn, "Singular integral operators and Toeplitz operators on odd spheres" Indiana Univ. Math. J. , 23 (1973) pp. 433–439 MR0322595 Zbl 0271.46052 |

[a3] | R. Douglas, R. Howe, "On the -algebra of Toeplitz operators on the quarter-plane" Trans. Amer. Math. Soc. , 158 (1971) pp. 203–217 MR288591 |

[a4] | M. Landstad, J. Phillips, I. Raeburn, C. Sutherland, "Representations of crossed products by coactions and principal bundles" Trans. Amer. Math. Soc. , 299 (1987) pp. 747–784 MR0869232 Zbl 0722.46031 |

[a5] | P. Muhly, J. Renault, "-algebras of multivariable Wiener–Hopf operators" Trans. Amer. Math. Soc. , 274 (1982) pp. 1–44 MR0670916 Zbl 0509.46050 Zbl 0509.46049 |

[a6] | N. Salinas, A. Sheu, H. Upmeier, "Toeplitz operators on pseudoconvex domains and foliation algebras" Ann. Math. , 130 (1989) pp. 531–565 MR1025166 Zbl 0708.47021 |

[a7] | H. Upmeier, "Toeplitz -algebras on bounded symmetric domains" Ann. Math. , 119 (1984) pp. 549–576 MR744863 |

[a8] | H. Upmeier, "Toeplitz operators on symmetric Siegel domains" Math. Ann. , 271 (1985) pp. 401–414 MR0787189 Zbl 0565.47016 |

[a9] | H. Upmeier, "Toeplitz operators and index theory in several complex variables" , Birkhäuser (1996) MR1384981 Zbl 0957.47023 |

[a10] | U. Venugopalkrishna, "Fredholm operators associated with strongly pseudoconvex domains in " J. Funct. Anal. , 9 (1972) pp. 349–373 MR0315502 Zbl 0241.47023 |

[a11] | A. Wassermann, "Algèbres d'opérateurs de Toeplitz sur les groupes unitaires" C.R. Acad. Sci. Paris , 299 (1984) pp. 871–874 MR0777751 |

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Toeplitz C*-algebra.

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