In this book, the authors revisit Haagerup’s enigmatic reduction theorem, showing how it may be extended to general von Neumann algebras M equipped with an arbitrary faithful normal semifinite weight in a manner which faithfully captures the essence of the original. This theorem then proves to be vital tool for extending the known theory of noncommutative Hp spaces from the sigma-finite and semifinite contexts to general von Neumann algebras. The authors show that even in this generality, the theory of subdiagonal subalgebras allows for a robust theory of Hp spaces. Inspired by the theory of topologically ordered groups, they then go further and propose the even more general concept of approximately subdiagonal subalgebras. This concept proves to be general enough to contain all group theoretic examples. The noncommutative Hilbert transform is shown to make sense for even Hp spaces of these algebras. This then forms the context for a study of noncommutative Fredholm Toeplitz operators in the closing sections, which has applications to group algebras of all topologically ordered groups.

Louis Labuschagne obtained his PhD at the University of Cape Town in 1988 in the field of Single Linear Operator theory. In January of the same year he started his professional academic career at the University of Stellenbosch University. In 1992 he moved to the University of Pretoria in 1992 and at the same time also shifted his research focus to Operator Algebras and the quantum function spaces they engender and the suitability of the latter as a formalism for quantum physics. After spending 19 years in Pretoria - first at the University of Pretoria and then UNISA from 2001 til the end of 2010 - he then took up an appointment at North-West University in January 2011. During his tenure at North-West University he served as president of the South African Mathematical Society in the period Nov 2011- Nov 2015 and helped found the Focus Area for Pure and Applied Analytics at North-West University also serving as its first director. Upon his retirement at the end of 2025 he was appointed as an extra-ordinary professor in the self-same entity. He currently resides in Cape Town with his wife Sheila and is still active in research.

Quanhua Xu obtained his PhD at Université Pierre et Marie Curie (that has since become known as Sorbonne Université) in 1988 in the field of functional analysis. In September of the same year he started his professional academic career at Université de Lille 1. In 1991 he moved back to Université Pierre et Marie Curie and stayed there until 1996. Since then he has been professor at Université Marie et Louis Pasteur (the former Université de Franche-Comté). He has also been holding a visiting position at the Harbin Institute of Technology where he founded an institute of research in mathematics in 2016. His training was in the geometry of Banach spaces and his current research activity is mainly focused on quantum probability and operator spaces, also incorporating (quantum) harmonic analysis. Noncommutative Lp-spaces take a central place in his activity. One important feature of his work is the omnipresence of noncommutative martingale or matricial inequalities. While having a quantum character, these inequalities also give a new perspective on the classical inequalities they generalize. On the other hand, recent discoveries in quantum information theory, which have natural applications to theoretical and mathematical physics, also depend on these noncommutative inequalities. Quanhua Xu’s research covers a large spectrum of analysis and probability.