Locally Compact Quantum Metric Spaces and Spectral Triples

Abstract

In this chapter, we start by giving an overview of quantum (locally) compact metric spaces. Then, we show that we can associate quantum compact metric spaces to some Markov semigroups of Fourier multipliers satisfying additional conditions: an injectivity and a gap condition on the cocycle which represents the semigroup, and the finite dimensionality (with explicit control on p) of the cocycle Hilbert space. We show a similar result for semigroups of Schur multipliers and obtain a quantum locally compact metric space. We further explore the connections of our gap condition between Fourier multipliers and Schur multipliers with some examples. In the sequel, we introduce spectral triples (= noncommutative manifolds) associated to Markov semigroups of Fourier multipliers or Schur multipliers satisfying again some technical conditions, and in all we investigate four different settings. Along the way, we introduce a Banach space variant of the notion of spectral triple suitable for our context. Finally, we investigate the bisectoriality and the functional calculus of some Hodge-Dirac operators which are crucial in the noncommutative geometries which we introduce here.