The Reduction Theorem and Noncommutative Hp-Spaces

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.