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The Index of Toeplitz Operators on Free Transformation Group C *‐Algebras
Author(s) -
Park Efton
Publication year - 2002
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609301008554
Subject(s) - mathematics , toeplitz matrix , invertible matrix , pure mathematics , differential operator , vector bundle , group (periodic table) , operator (biology) , toeplitz operator , hermitian matrix , algebra over a field , symbol (formal) , transformation (genetics) , biochemistry , chemistry , organic chemistry , repressor , computer science , transcription factor , gene , programming language
Let Γ be a discrete group acting on a compact manifold X , let V be a Γ‐equivalent Hermitian vector bundle over X , and let D be a first‐order elliptic self‐adjoint Γ‐equivalent differential operator acting on sections of V . This data is used to define Toeplitz operators with symbols in the transformation group C *‐algebra C ( X )⋊Γ, and it is shown that if the symbol of such a Toeplitz operator is invertible, then the operator is Fredholm. In the case where Γ is finite and acts freely on X , a geometric‐topological formula for the index is stated that involves an explicitly constructed differential form associated to the symbol. 2000 Mathematics Subject Classification 47A53 (primary), 19K56, 47B35, 46L87 (secondary).