Vol. 5, No. 2, 2020

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Tate tame symbol and the joint torsion of commuting operators

Jens Kaad and Ryszard Nest

Vol. 5 (2020), No. 2, 181–211
Abstract

We investigate determinants of Koszul complexes of holomorphic functions of a commuting tuple of bounded operators acting on a Hilbert space. Our main result shows that the analytic joint torsion, which compares two such determinants, can be computed by a local formula which involves a tame symbol of the involved holomorphic functions. As an application we are able to extend the classical tame symbol of meromorphic functions on a Riemann surface to the more involved setting of transversal functions on a complex analytic curve. This follows by spelling out our main result in the case of Toeplitz operators acting on the Hardy space over the polydisc.

Keywords
determinant functors, Koszul complexes, holomorphic functional calculus, joint torsion, tame symbols
Mathematical Subject Classification 2010
Primary: 32A10, 47A13
Secondary: 19C20, 19K56, 32C15
Milestones
Received: 8 February 2018
Revised: 26 July 2019
Accepted: 7 October 2019
Published: 20 June 2020
Authors
Jens Kaad
Department of Mathematics and Computer Science
University of Southern Denmark
Odense
Denmark
Ryszard Nest
Department of Mathematical Sciences
University of Copenhagen
Copenhagen
Denmark