Narrow Results

The aim of this work is to define the infinite tensor product of a countable family {E_i}_{i ∈ 𝕀} of spatial product systems with respect to a family of normalized units, and to analyze the main properties of this construction. Among other things, we show that is an amenable product system, respectively a product system of type I, provided that every product system E_i is amenable, respectively of type I.

In this note an interpretation of Riemann's zeta function is provided in terms of an ℝ-equivariant L²-index of a Dirac–Ramond type operator, akin to the one on (mean zero) loops in flat space constructed by the present author and T. Wurzbacher. We build on the formal similarity between Euler's partitio numerorum function (the S¹-equivariant L²-index of the loop space Dirac–Ramond operator) and Riemann's zeta function. Also, a Lefschetz–Atiyah–Bott interpretation of the result together with a generalization to M. Lapidus' fractal membranes are also discussed. A fermionic Bost–Connes type statistical mechanical model is presented as well, exhibiting a "phase transition at (inverse) temperature β = 1", which also holds for some "well-behaved" g-prime systems in the sense of Hilberdink–Lapidus.