  We prove that any nef b-divisor class on a projective variety defined over an
algebraically closed field of characteristic 0 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. We show that any big and
basepoint free curve class is a power of a nef b-divisor, and relate this
statement to Zariski decompositions 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.5
