Singular traces

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This book presents the first comprehensive study of singular traces, mathematically formalizing their role within a self-contained theory of functional analysis. It includes extensive notes on historical developments and concludes with a thorough examination of the integration half of Connes' quantum calculus. Singular traces, which vanish on finite rank operator subideals, are integral to A. Connes' interpretation of noncommutative residues, particularly the Dixmier trace. This trace generalizes the restricted Adler-Manin-Wodzicki residue of pseudo-differential operators and serves as a residue for various 'geometric' spaces, such as Connes-Chamseddine models, Yang-Mills actions for quantum differential forms, fractals, isospectral deformations, foliations, and noncommutative index theory. The theory has evolved through contributions from various authors, including recent advancements by Nigel Kalton. Singular traces relate to symmetric functionals, residues of zeta functions, and heat kernel asymptotics, characterized by Lidksii and Fredholm formulas. The text clarifies the traces and formulas used in noncommutative geometry, revealing new mathematical and physical implications. It provides essential functional analysis results and a complete theory of traces on compact operators, serving as a valuable reference for mathematical physicists and those interested in the deeper aspects of traces related to harmonic sequences, wh