Infinite index subfactors and the GICAR categories

Given a II$_1$-subfactor $A\subset B$ 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 $\bigotimes_A^n L^2(B)$. As a corollary, we get a lower bound on the dimension of the centralizer algebras $A_0'\cap A_{2n}$ for infinite index subfactors, and we also get that $A_0'\cap A_{2n}$ is nonabelian for $n\geq 2$, where $(A_n)_{n\geq 0}$ is the Jones tower for $A_0=A\subset B=A_1$. We also show that the annular GICAR/planar rook category acts as maps amongst the $A$-central vectors in $\bigotimes_A^n L^2(B)$, 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.