A New Approach to Functional Analysis on Graphs, the Connes-Spectral Triple and its Distance Function

Details A New Approach to Functional Analysis on Graphs, the Connes-Spectral Triple and its Distance Function A New Approach to Functional Analysis on Graphs, the Connes-Spectral Triple and its Distance Function

1997-08-04

arxiv.org/abs/hep-th/9708010v2

Physics

High Energy Physics

High Energy Physics - Theory

18 pages, Latex, section 3 about functional analysis expanded, some proofs more detailed

Scientific paper

Continuing previous work we develop a certain piece of functional analysis on general graphs and use it to create what Connes calls a 'spectral triple', i.e. a Hilbert space structure, a representation of a certain (function) algebra and a socalled 'Dirac operator', encoding part of the geometric/algebraic properties of the graph. We derive in particular an explicit expression for the 'Connes-distance function' and show that it is in general bounded from above by the ordinary distance on graphs (being, typically, strictly smaller(!) than the latter). We exhibit, among other things, the underlying reason for this phenomenon.

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