Amplified graph C*-algebras II: Reconstruction

Abstract

Let E E be a countable directed graph that is amplified in the sense that whenever there is an edge from v v to w w , there are infinitely many edges from v v to w w . We show that E E can be recovered from C ∗ ( E ) C^*(E) together with its canonical gauge-action, and also from L K ( E ) L_\mathbb {K}(E) together with its canonical grading.

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“…In this paper, we consider amplified graphs. The classification of amplified graph C * -algebras was the starting point in the classification of unital graph C * -algebras via moves (see [12] and [13]).…”

“…A similar result holds for the relations x0z (not requiring the isomorphisms to commute with the canonical gauge action). Recall from [12] that if E is an amplified graph then its amplified transitive closure tE is the amplified graph with tE 0 = E 0 and v(tE 1 )w = ∅ if and only if vE * w \ {v} = ∅. Theorem 1.1 of [12] shows that for amplified graphs, if (E, F ) ∈ 000, then tE ∼ = tF .…”

“…Recall from [12] that if E is an amplified graph then its amplified transitive closure tE is the amplified graph with tE 0 = E 0 and v(tE 1 )w = ∅ if and only if vE * w \ {v} = ∅. Theorem 1.1 of [12] shows that for amplified graphs, if (E, F ) ∈ 000, then tE ∼ = tF . By the construction (3.1)…”

“…The geometric classification (that is, classification by the underlying graph modulo the equivalence relation generated by a list of allowable graph moves) of the C * -algebras of finite-vertex amplified graph C * -algebras was completed in [12], and was an important precursor to the eventual geometric classification of all finite graph C * -algebras [13]. But there has been increasing recent interest in understanding isomorphisms of graph C * -algebras that preserve additional structure: for example the canonical gauge action of the circle; or the canonical diagonal subalgebra isomorphic to the algebra of continuous functions vanishing at infinity on the infinite path space of the graph; or the smaller coefficient algebra generated by the vertex projections; or some combination of these (see, for example, [5][6][7][8][9][10]19]).…”