2014/11/04 by Nadir Bizi, Bizi, Nadir, Fabien Besnard +1
Mathematics · Physics and Astronomy · #06A11 #58B34 #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #gr-qc #math-ph #math.MP #math.OA #msc:06A11 #msc:58B34
paper · pdf · doi:10.48550/arxiv.1411.0878
28 pages, 4 figures 1 typo fixed. 2 references added
arxiv created 2014/11/08 · arxiv updated 2014/11/11
This paper continues the investigations of noncommutative ordered spaces put forward by one of the authors. These metaphoric spaces are defined dually by so-called isocones which generalize to the noncommutative setting the convex cones of order-preserving functions. In this paper we will consider the case of isocones inside almost-commutative algebras of the form \cal C(M)⊗ Af, with M a compact metrizable space. We will give a family of isocones in such an algebra with the property that every possible isocone is contained in exactly one member of the family. We conjecture that this family is in fact a complete classification, a hypothesis related with the noncommutative Stone-Weierstrass conjecture. We also obtain that every isocone in \cal C(M)⊗ Af, with Af noncommutative, induces an order relation on M with the property that every point in M lies in a neighbourhood of incomparable points. Thus, if the causal order relation on spacetime is induced by an isocone in an almost-commutative (but not commutative) algebra, then causality must disappear at small scale.