Automorphisms of Bowen–Franks groups of shifts of finite type

Abstract

There are four Bowen-Franks groups associated to each shift of finite type. For an irreducible shift of finite type, we show that a 4-tuple of automorphisms corresponding to the four Bowen-Franks groups can be induced simultaneously by a specific path of flow equivalence from the shift to itself, if and only if it is F-compatible. The Fcompatibleness defined in this paper describes completely the intrinsic relations among the four automorphisms induced by a flow equivalence. This result is one of the key ingredients in classifying reducible shifts of finite type up to flow equivalence. In the mean time, it also discloses a new and sharp difference between the invariants of flow equivalence for an irreducible shift of finite type, and the invariants of stable isomorphism for the associated simple Cuntz-Krieger algebra.

Classification of unital simple Leavitt path algebras of infinite graphs

“…This was accomplished by their work in [4,Proposition 2.4] where they apply a result of Huang to the shift spaces of the graphs in order to find a flow equivalence that induces a given automorphism of the K 0 -group. Huang's result can be found in [19,Theorem 1.1] with details given in [18,Theorem 2.15]. Unfortunately, when our graphs have an infinite number of edges, the "shift spaces" of the graphs have an infinite alphabet and therefore are not shift spaces in the traditional sense, so Huang's result does not apply.…”

Flow invariants in the classification of Leavitt path algebras

“…It is worth remarking that the proof of Proposition 2.4, and therefore the proof of Theorem 2.5, hinges on results of Huang ([16] and [17]). Huang's results are constructive; specifically, the flow equivalence from E t to itself which induces α (as in the proof of Proposition 2.4) is explicitly described.…”

“…We will show that, in the case of Morita equivalent purely infinite simple Leavitt path algebras over finite graphs, if the unitary Franks pair of their graphs are equivalent, then the algebras are isomorphic. The argument relies on the adaptation to our context of the deep result of Huang [18,Theorem 1.1].…”

“…starts and ends with E.By[17, Lemma 3.7], for any graphs E and F , any standard graph transformation m : E → F yields the so-called induced isomorphism…”

Equivariant flow equivalence for shifts of finite type, by matrix equivalence over group rings

“…The Huang classication of reducible SFTs up to ow equivalence is much more complicated. (Huang's original arguments are developed in [16,17,18,19] and an almost complete unpublished manuscript, 'The K-web invariant and ow equivalence of reducible shifts of nite type.' A complete alternate development is contained in [6,8].)…”