scholarly journals The Index Theorem for Toeplitz Operators as a Corollary of Bott Periodicity

2021 ◽  
Author(s):  
Paul F Baum ◽  
Erik Van Erp
Keyword(s):  
Toeplitz Operators ◽  
Index Theorem ◽  
Bott Periodicity ◽  
Expository Paper

Abstract This is an expository paper about the index of Toeplitz operators, and in particular Boutet de Monvel’s theorem [5]. We prove Boutet de Monvel’s theorem as a corollary of Bott periodicity, and independently of the Atiyah-Singer index theorem.

scholarly journals An index theorem for Toeplitz operators on totally ordered groups

1998 ◽  
Vol 126 (10) ◽  
pp. 2993-2998 ◽  
Author(s):  
Sriwulan Adji ◽  
Iain Raeburn ◽  
Anton Ströh
Keyword(s):  
Toeplitz Operators ◽  
Index Theorem ◽  
Ordered Groups

The Index Theory Associated to a Non-Finite Trace on a C*-Algebra

2005 ◽  
Vol 48 (2) ◽  
pp. 251-259 ◽  
Author(s):  
G. J. Murphy
Keyword(s):  
Compact Group ◽  
Representation Theorem ◽  
Toeplitz Operators ◽  
Von Neumann Algebra ◽  
Index Theory ◽  
Index Theorem ◽  
Fredholm Index ◽  
Von Neumann ◽  
C Algebra ◽  
Finite Trace

AbstractThe index theory considered in this paper, a generalisation of the classical Fredholm index theory, is obtained in terms of a non-finite trace on a unitalC*-algebra. We relate it to the index theory of M. Breuer, which is developed in a von Neumann algebra setting, by means of a representation theorem. We show how our new index theory can be used to obtain an index theorem for Toeplitz operators on the compact group U(2), where the classical index theory does not give any interesting result.

A NOTE ON TOEPLITZ OPERATORS

1996 ◽  
Vol 07 (04) ◽  
pp. 501-513 ◽  
Author(s):  
ERIK GUENTNER ◽  
NIGEL HIGSON
Keyword(s):  
Riemannian Manifolds ◽  
Toeplitz Operators ◽  
Boundary Value ◽  
Bergman Spaces ◽  
Index Theorem ◽  
Value Theory ◽  
Point Of View ◽  
Dirac Type ◽  
Complete Manifolds ◽  
Complete Riemannian Manifolds

We study Toeplitz operators on Bergman spaces using techniques from the analysis of Dirac-type operators on complete Riemannian manifolds, and prove an index theorem of Boutet de Monvel from this point of view. Our approach is similar to that of Baum and Douglas [2], but we replace boundary value theory for the Dolbeaut operator with much simpler estimates on complete manifolds.

scholarly journals Centre-valued Index for Toeplitz Operators with Noncommuting Symbols

2016 ◽  
Vol 68 (5) ◽  
pp. 1023-1066
Author(s):  
John Phillips ◽  
Iain Raeburn
Keyword(s):  
Toeplitz Operators ◽  
Adjoint Operator ◽  
Index Theorem ◽  
Scalar Case ◽  
Fredholm Operators ◽  
Negative Spectrum ◽  
Von Neumann ◽  
C Algebra ◽  
Making Sense ◽  
Algebra Theory

AbstractWe formulate and prove a “winding number” index theorem for certain “Toeplitz” operators in the same spirit as Gohberg–Krein, Lesch and others. The “number” is replaced by a self-adjoint operator in a subalgebra Z ⊆ Z(A) of a unital C*-algebra, A. We assume a faithful Z-valued trace τ on A left invariant under an action α:R → Aut(A) leaving Z pointwise fixed. If δ is the infinitesimal generator of α and u is invertible in dom(δ), then the “winding operator” of u is . By a careful choice of representations we extend (A, Z, τ, α) to a von Neumann setting where and . Then , the von Neumann crossed product, and there is a faithful, dual -trace on . If P is the projection in corresponding to the non-negative spectrum of the generator of R inside and is the embedding, then we define and show it is Fredholm in an appropriate sense and the -valued index of Tu is the negative of the winding operator. In outline the proof follows that of the scalar case done previously by the authors. The main difficulty is making sense of the constructions with the scalars replaced by in the von Neumann setting. The construction of the dual -trace on requires the nontrivial development of a -Hilbert algebra theory. We show that certain of these Fredholm operators fiber as a “section” of Fredholm operators with scalar-valued index and the centre-valued index fibers as a section of the scalar-valued indices.

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