  This article is an introduction to our recent work in harmonic analysis
associated with semigroups of operators, in the effort of finding a
noncommutative Calder\'on-Zygmund theory for von Neumann algebras. The
classical CZ theory has been traditionally developed on metric measure spaces
satisfying additional regularity properties. In the lack of such metrics -or
with very little information on the metric- Markov semigroups of operators
appear to be the right substitutes of classical metric/geometric tools in
harmonic analysis. Our approach is particularly useful in the noncommutative
setting but it is also valid in classical/commutative frameworks.
