*Simon Scott*

- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780198568360
- eISBN:
- 9780191594748
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568360.001.0001
- Subject:
- Mathematics, Analysis

This text provides a broad account of the theory of traces and determinants on geometric algebras of differential and pseudodifferential operators over compact manifolds. Trace and determinant ...
More

This text provides a broad account of the theory of traces and determinants on geometric algebras of differential and pseudodifferential operators over compact manifolds. Trace and determinant functionals on geometric operator algebras provide a means of constructing refined invariants in analysis, topology, differential geometry, analytic number theory and QFT. The consequent interactions around such invariants have led to significant advances both in pure mathematics and theoretical physics. As the fundamental tools of trace theory have become well understood and clear general structures have emerged, so the need for specialist texts which explain the basic theoretical principles and the computational techniques has become increasingly exigent. This text is the first to deal with the general theory of traces and determinants of operators on manifolds in a broad context, encompassing a number of the principle applications and backed up by specific computations which set out in detail to newcomers the nuts-and-bolts of the basic theory. Both the microanalytic approach to traces and determinants via pseudodifferential operator theory and the more computational approach directed by applications in geometric analysis, are developed in a general framework that will be of interest to mathematicians and physicists in a number of different fields.Less

This text provides a broad account of the theory of traces and determinants on geometric algebras of differential and pseudodifferential operators over compact manifolds. Trace and determinant functionals on geometric operator algebras provide a means of constructing refined invariants in analysis, topology, differential geometry, analytic number theory and QFT. The consequent interactions around such invariants have led to significant advances both in pure mathematics and theoretical physics. As the fundamental tools of trace theory have become well understood and clear general structures have emerged, so the need for specialist texts which explain the basic theoretical principles and the computational techniques has become increasingly exigent. This text is the first to deal with the general theory of traces and determinants of operators on manifolds in a broad context, encompassing a number of the principle applications and backed up by specific computations which set out in detail to newcomers the nuts-and-bolts of the basic theory. Both the microanalytic approach to traces and determinants via pseudodifferential operator theory and the more computational approach directed by applications in geometric analysis, are developed in a general framework that will be of interest to mathematicians and physicists in a number of different fields.

*Simon Scott*

- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780198568360
- eISBN:
- 9780191594748
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568360.003.0003
- Subject:
- Mathematics, Analysis

Determinants as abstract invariants have received less study in the mathematical literature than traces, on which there is an extensive literature. However, determinants can be treated in a general ...
More

Determinants as abstract invariants have received less study in the mathematical literature than traces, on which there is an extensive literature. However, determinants can be treated in a general framework. The significant new object here is a logarithm operator from semigroups to tracial algebras. The character of the logarithm operator defined by the algebra trace is then the log-determinant. Determinants may therefore be understood in general terms as characters of logarithmic representations of semigroups. This chapter introduces and elaborates these ideas and provides numerous examples of such structures. In the latter part of the chapter the specific case of logarithms and determinant structures on pseudodifferential operators is presented, outlining the basic structures that have been identified and how logarithms and geometric index theory are closely related. A general folklore principle here is that there is one class of logarithms for each higher K theory.Less

Determinants as abstract invariants have received less study in the mathematical literature than traces, on which there is an extensive literature. However, determinants can be treated in a general framework. The significant new object here is a logarithm operator from semigroups to tracial algebras. The character of the logarithm operator defined by the algebra trace is then the log-determinant. Determinants may therefore be understood in general terms as characters of logarithmic representations of semigroups. This chapter introduces and elaborates these ideas and provides numerous examples of such structures. In the latter part of the chapter the specific case of logarithms and determinant structures on pseudodifferential operators is presented, outlining the basic structures that have been identified and how logarithms and geometric index theory are closely related. A general folklore principle here is that there is one class of logarithms for each higher K theory.