We prove that any nef b-divisor class on a projective variety is a decreasing limit of nef Cartier classes. Building on this technical result, we construct an intersection theory of nef b-divisors, and prove several variants of the Hodge index theorem inspired by the work of Dinh and Sibony in this context. We show that any big and basepoint free curve class is a power of a nef b-divisor, and relate this statement to the Zariski decomposition of curves classes introduced by Lehmann and Xiao. Our construction allows us to relate various Banach spaces contained in the space of b-divisors which were defined in our previous work. We further discuss their properties in the case of toric varieties and connect them with classical functional objects in convex integral geometry.