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.
Keywords: Logarithm operators, characters, determinants, zeta determinant, residue determinant, eta invariant, regularized determinants, odd Chern character, determinant structures, log‐classical, log polyhomogeneous
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