Building provided with an air treatment system

Building comprising more than one separate spaces and an air treatment system, wherein the air treatment system comprises a central air drying unit fluidly connected by a network of connecting conduits to two or more local evaporative cooling units, wherein at least two local evaporative cooling units are each fluidly connected to a separate space of the building and wherein the local evaporative cooling unit comprises one or more cooling channels provided with an inlet and an outlet for air and one or more evaporating channels provided with an inlet and outlet for air and wherein the cooling channels and evaporating channels are separated by a transfer wall and the transfer wall is wet such that evaporation can take place in the evaporating channel and wherein the central air drying unit comprises an inlet for air obtained at the exterior of the building and an outlet for dried air which is fluidly connected to the local evaporative cooling units and wherein the outlet of the one or more cooling channels is fluidly connected to the interior of the separate space and the outlet of the one or more evaporating channels is fluidly connected to the exterior of the separate space.