I will talk about indecomposability results for von Neumann algebras. First I will describe Ozawa's fundamental contribution: the notion of solidity and how it is implied by the Akemann-Ostrand property. I will also discuss the notion of strong solidity introduced by Ozawa and Popa; it can be obtained using a strengthening of the Akemann-Ostrand combined with completely bounded approximation property. The Akemann-Ostrand property for group von Neumann algebras is typically proved using boundary actions. I plan to present a more analytic approach utilising quantum Markov semigroups. It turns out that existence of a sufficiently nice quantum Markov semigroup allows to prove the Akemann-Ostrand property. I will give an outline of a proof and provide examples to which this result can be applied. I will conclude with some related open problems.