HIGHER CONNECTED STABLE RANKS AND THEIR RATIONAL VARIANTS OF AF ALGEBRAS

We study the k(≥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.