Abstract
Let T = (T1,…,T s ) be a commuting s-tuple of elements in the algebra of bounded operators on a complex Banach space. When the (2s –2)-dimensional Hausdorff measure of σ ess (T) is zero we introduce certain Lefschetz number ratios which give the transition functions for a holomorphic line bundle E T on ℂS \ σ ess (T). The Lefschetz numbers themselves define a meromorphic section of E T whose divisor is a complex analytic cycle carried on the Taylor spectrum; its local degree at z has a natural interpretation as a maximal ideal index for (T – z). This index has jumps on the singular locus of the spectrum and sometimes determines the K 1-homology class of T. The correspondence T → E T has the property that E T⊕T′ = E T⊗E T′, and if s is even, \( E_{T^t } = - E \) where t denotes transpose. Furthermore, when M is a Hilbert space, σ ess (T) is smooth and K is compact, E T+K = E T . The present note details the construction of the map T → E T and its relationship to the fundamental trace form of crypto-integral algebras.
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Carey, R.W., Pincus, J.D. (1988). On Local Index and the Cocycle Property for Lefschetz Numbers. In: Gohberg, I. (eds) Topics in Operator Theory and Interpolation. Operator Theory: Advances and Applications, vol 29. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9162-2_2
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DOI: https://doi.org/10.1007/978-3-0348-9162-2_2
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