Banach-valued graph limits: Graphon representability and Banach-space structure

Abstract

We study a graph-limit problem for Banach-decorated graphs. Given a sequence of $X$-decorated graphs whose homomorphism densities converge against all $X^{\ast }$-decorated test graphs, we ask whether the limiting densities are represented by an $X$-valued graphon. The results connect this graph-limit problem with Banach-space structure. If $X^{\ast }$ is separable, then the graphon representation property for graph sequences uniformly bounded in $L^p$ for every finite $p$ holds if and only if $X$ is reflexive. For Banach lattices, it is equivalent to the Radon--Nikodým property. For dual Banach spaces, it is equivalent to the conjunction of the Radon--Nikodým property and weak sequential completeness. In the bounded setting, the same characterization extends to arbitrary Banach spaces: for every Banach space $X$, the representation property for uniformly $L^{\infty }$-bounded graph sequences holds if and only if $X$ has the Radon--Nikodým property and is weakly sequentially complete.

31 pages, 3 figures

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