Abstract: In this talk I will present a recent construction of equilibrium states at positive temperature, with and without Bose-Einstein condensation, for a non-relativistic Bosonic QFT (gas of Bose particles) in the infinite volume limit, interacting through a localised two body interaction. In order to obtain this result, we use methods of quantum field theory in the algebraic formulation and of quantum statistical mechanics in the operator algebraic setting. The convergence of the interacting correlation functions is obtained constructing an equivalent perturbative series expansion introducing an auxiliary stochastic Gaussian field which mediates the interaction. Limits where the localisation of the two-body interaction is removed are eventually discussed in combination with other regimes. This talk is based on a collaboration with Nicola Pinamonti. Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page