Purely infinite simple Leavitt path algebras.
Leavitt path algebras are the algebraic version of Cuntz-Krieger graph \(C^*\)-algebras, a class of algebras intensively investigated by analysts for more than two decades. These algebras were introduced by \textit{G. Abrams} and \textit{G. Aranda Pino} [J. Algebra 293, No. 2, 319-334 (2005; Zbl 1119.16011)]. Abrams and Aranda Pino provided a generalization of Leavitt algebras \(L(1,n)\), of type \((1,n)\), which were introduced by \textit{W. G. Leavitt} in 1962 [in Trans. Am. Math. Soc. 103, 113-130 (1962; Zbl 0112.02701)] in order to give examples of algebras not satisfying the IBN (invariant basis number) property. An idempotent \(e\) in a ring \(R\) is said to be infinite if \(eR\) is isomorphic, as a right \(R\)-module, to a proper direct summand of itself. The ring \(R\) is called purely infinite if each of its nonzero right ideals contains an infinite idempotent. This notion was introduced in the algebraic setting by \textit{P. Ara, K. R. Goodearl} and \textit{E. Pardo} [K-Theory 26, No. 1, 69-100 (2002; Zbl 1012.16013)]. In this paper, the authors establish necessary and sufficient conditions on a row-finite graph \(E\) so that the associated Leavitt path \(K\)-algebra \(L_K(E)\) (for \(K\) a field) is purely infinite and simple. These are: (i) The only hereditary and saturated subsets of \(E^0\), the set of vertices of the graph, are the trivial ones (\(E^0\) and \(\emptyset\)), (ii) every cycle in \(E\) must have an exit, and (iii) every vertex connects to a cycle. The analytic counterpart of this result can be found in [\textit{T. Bates} et al., New York J. Math. 6, 307-324 (2000; Zbl 0976.46041)]. Although the algebraic result is the same in shape, the methods used differ from the analytic ones.
purely infinite simple Leavitt path algebras
scientific article