Ranges of positive contractive projections in Nakano spaces

Abstract

We show that in any nontrivial Nakano space X= L p(·) (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 L pY(·) ( Ω Y Σ Y , ν Y ), for a certain measurable set Ω Y ⊆ Ω (the support of the range), a certain sub-sigma ring Σ Y ⊆ Σ (with maximal element Ω Y ) naturally determined by the lattice structure of Y, and a semi-finite measure ν Y , namely the restriction of some measure Ω on E which is equivalent to μ. Furthermore, we show that the random exponent p Y (·) associated with such a range can be taken to be the restriction to Ω Y of the random exponent p(·) (this restriction turns out to be Σ Y -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.