
TL;DR
This paper characterizes algebraic Cuntz-Krieger algebras via graph properties and K-theory, establishing isomorphisms with Leavitt path algebras and $C^*$-algebras, and explores their structural stability.
Contribution
It provides new criteria for identifying algebraic Cuntz-Krieger algebras using graph conditions and K-theoretic invariants, including Morita equivalence and corner stability.
Findings
Finite graphs with no sinks correspond to algebraic Cuntz-Krieger algebras.
Isomorphism conditions involve unitality and K-theory rank equalities.
Corners of these algebras are also algebraic Cuntz-Krieger algebras.
Abstract
We show that is a finite graph with no sinks if and only if the Leavitt path algebra is isomorphic to an algebraic Cuntz-Krieger algebra if and only if the -algebra is unital and . When is a field and , we show that the Leavitt path algebra is isomorphic to an algebraic Cuntz-Krieger algebra if and only if is unital and . We also show that any unital -algebra which is Morita equivalent or stably isomorphic to an algebraic Cuntz-Krieger algebra, is isomorphic to an algebraic Cuntz-Krieger algebra. As a consequence, corners of algebraic Cuntz-Krieger algebras are algebraic Cuntz-Krieger algebras.
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