Ranges of positive contractive projections in Nakano spaces

We show that in any nontrivial Nakano space X=Lp(·) (Ω, Σ, μ) with essentially bounded random exponent function p(·), the range Y = R(P) of a positive contractive projection P is itself representable as a Nakano space LpY(·) (ΩY ΣY, νY), for a certain measurable set ΩY⊆Ω (the support of the range), a certain sub-sigma ring ΣY⊆Σ (with maximal element ΩY) naturally determined by the lattice structure of Y, and a semi-finite measure νY, namely the restriction of some measure Ω on E which is equivalent to μ. Furthermore, we show that the random exponent pY(·) associated with such a range can be taken to be the restriction to ΩY of the random exponent p(·) (this restriction turns out to be ΣY-measurable). As an application of this result, we find Banach lattice isometric characterizations of suitable classes of Nakano spaces. These classes are defined in terms of an important lattice-isometric invariant of Nakano spaces, the essential range of the variable exponent.