Detecting Fourier subspaces

2015/03/24 by Charles A. Akemann, Akemann, Charles A., Nik Weaver +1
Mathematics · #05A99 #11K38 #43A25 #46L05 #94A12 #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Operator Algebras (math.OA) #math.CO #math.FA #math.GR #math.OA #msc:05A99 #msc:11K38 #msc:43A25 #msc:46L05 #msc:94A12

paper · pdf · doi:10.48550/arxiv.1503.06893

8 pages

arxiv created 2015/03/24 · arxiv updated 2015/03/25

Abstract

Let G be a finite abelian group. We examine the discrepancy between subspaces of l²(G) which are diagonalized in the standard basis and subspaces which are diagonalized in the dual Fourier basis. The general principle is that a Fourier subspace whose dimension is small compared to |G| = dim(l²(G)) tends to be far away from standard subspaces. In particular, the recent positive solution of the Kadison-Singer problem shows that from within any Fourier subspace whose dimension is small compared to |G| there is standard subspace which is essentially indistinguishable from its orthogonal complement.

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