2014/10/03 by Vaughan F. R. Jones, Jones, Vaughan F. R., David Penneys +1
Mathematics · #Algebraic structures and combinatorial models #Advanced Operator Algebra Research #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.1410.0856
Given a II1-subfactor A⊂ B of arbitrary index, we show that the rectangular GICAR category, also called the rectangular planar rook category, faithfully embeds as A-A bimodule maps among the bimodules \bigotimesAn L2(B). As a corollary, we get a lower bound on the dimension of the centralizer algebras A0'∩ A2n for infinite index subfactors, and we also get that A0'∩ A2n is nonabelian for n≥ 2, where (An)n≥ 0 is the Jones tower for A0=A⊂ B=A1. We also show that the annular GICAR/planar rook category acts as maps amongst the A-central vectors in \bigotimesAn L2(B), although this action may be degenerate. We prove these results in more generality using bimodules. The embedding of the GICAR category builds on work of Connes and Evans who originally found GICAR algebras inside Temperley-Lieb algebras with finite modulus.