HIGHER CONNECTED STABLE RANKS AND THEIR RATIONAL VARIANTS OF AF ALGEBRAS

Abstract

We study the $k(\ge 1)$-connected stable rank and the $k$-homotopy stabilization rank ( J. Funct. Anal. 255 (2008), 3303–3328) and their rational homotopy variants of AF algebras. We prove that, for each odd integer $k$, the rational $k$-connected stable rank (resp. the rational $k$-homotopy stabilization rank) of an AF algebra is equal to the $k$-connected stable rank (resp. the $k$-homotopy stabilization rank) and also characterize the condition that the (rational) $k$-connected stable rank of an AF algebra $A$ is at most $m$ in terms of the Bratteli diagram of $A$. These ranks of AF algebras for even integer $k$ are also studied. They are $k$-connected stable rank-counterparts of the (rational) $K$-stability theorem for AF algebras due to Seth and Vaidyanathan ( New York J. Math. 26 (2020), 931–949). Our proof applies their proof scheme and results.