Diagonal-preserving ring ⁎-isomorphisms of Leavitt path algebras

Abstract

We study graph C *-algebras equipped with generalised gauge actions, and characterise in terms of groupoids and groupoid cocycles when two graph C *-algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the generalised gauge actions. We apply this characterisation to show that two Cuntz-Krieger algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the gauge actions if and only if the corresponding one-sided subshifts are eventually conjugate, and that the stabilisation of two Cuntz-Krieger algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the gauge actions if and only if the corresponding two-sided subshifts are conjugate.

Diagonal-preserving graded isomorphisms of Steinberg algebras

“…In this paper, we use techniques and ideas of [3] and [6] to extend [3,Theorem 3.1] to a larger class of ample Hausdorff groupoids by significantly relaxing the assumption that the kernel of the cocycle is topologically principal (Theorem 3.1). We also prove a "stabilised version" of this result (Theorem 3.11) and by combining this with [12,Theorem 3.2] we are able to relate groupoid equivalence and Kakutani equivalence with diagonal-preserving isomorphism of stabilised Steinberg algebras (Corollary 3.12).…”

“…Working with the Steinberg algebra model of a Leavitt path algebra, Brown, Clark and an Huef showed in [6] that if E is a row-finite directed graph without sinks (sources using their convention) and R is a commutative integral domain with identity, then the graph groupoid G E can be reconstructed from the data (L R (E), D R (E)) ([6, Theorem 4.9]). They used this reconstruction result to show that, for row-finite directed graphs without sinks, there is a diagonal-preserving * -isomorphism between Leavitt path algebras if and only if there is an isomorphism between the corresponding graph groupoiods ([6, Theorem 6.2]), and deduced that this is equivalent to there being a diagonal-preserving isomorphism between the corresponding graph C * -algebras ( [6,Corollary 6.3]).…”

“…They used this reconstruction result to show that, for row-finite directed graphs without sinks, there is a diagonal-preserving * -isomorphism between Leavitt path algebras if and only if there is an isomorphism between the corresponding graph groupoiods ([6, Theorem 6.2]), and deduced that this is equivalent to there being a diagonal-preserving isomorphism between the corresponding graph C * -algebras ( [6,Corollary 6.3]). …”

“…For (G , c) ∈ C (R,Γ) , we construct from A R (G ) and D R (G ) a graded ample Hausdorff groupoid W G such that W G is isomorphic to G (Proposition 3.6), and then show that a diagonal-preserving graded ring-isomorphism between A R (G 1 ) and A R (G 2 ) induces a graded isomorphism between W G 1 and W G 2 (Proposition 3.10). Since we are working with isomorphisms that are not necessarily * -isomorphisms, and since we also need to recover the grading of G given by the cocycle c, we use a definition of normalisers similar to the one used in [3] instead of the definition used in [6].…”

“…We now generalise [6,Proposition 5.2] and show that a diagonal-preserving ring-isomorphism of Steinberg algebras induces a homeomorphism of the unit spaces of the groupoids.…”