caa a22 4500 606160833 CHVBK 20210128100631.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s10468-014-9513-8 doi (NATIONALLICENCE)springer-10.1007/s10468-014-9513-8 Vaš Lia Department of Mathematics, Physics and Statistics, University of the Sciences, 19104, Philadelphia, PA, USA aut Canonical Traces and Directly Finite Leavitt Path Algebras [Elektronische Daten] [Lia Vaš] Motivated by the study of traces on graph C ∗-algebras, we consider traces (additive, central maps) on Leavitt path algebras, the algebraic counterparts of graph C ∗-algebras. In particular, we consider traces which vanish on nonzero graded components of a Leavitt path algebra and refer to them as canonical since they are uniquely determined by their values on the vertices. A desirable property of a ℂ $\mathbb {C}$ -valued trace on a C ∗-algebra is that the trace of an element of the positive cone is nonnegative. We adapt this property to traces on a Leavitt path algebra L K (E) with values in any involutive ring. We refer to traces with this property as positive. If a positive trace is injective on positive elements, we say that it is faithful. We characterize when a canonical, K-linear trace is positive and when it is faithful in terms of its values on the vertices. As a consequence, we obtain a bijective correspondence between the set of faithful, gauge invariant, ℂ $\mathbb {C}$ -valued (algebra) traces on L ℂ ( E ) $L_{\mathbb {C}}(E)$ of a countable graph E and the set of faithful, semifinite, lower semicontinuous, gauge invariant (operator theory) traces on the corresponding graph C ∗-algebra C ∗(E). With the direct finite condition (i.e x y=1 implies y x=1) for unital rings adapted to rings with local units, we characterize directly finite Leavitt path algebras as exactly those having the underlying graphs in which no cycle has an exit. Our proof involves consideration of "local” Cohn-Leavitt subalgebras of finite subgraphs. Lastly, we show that, while related, the class of locally noetherian, the class of directly finite, and the class of Leavitt path algebras which admit a faithful trace are different in general. Springer Science+Business Media Dordrecht, 2015 Trace nationallicence Leavitt path algebra nationallicence Directly finite nationallicence Involution nationallicence Graph trace nationallicence Gauge invariant nationallicence Positive nationallicence Faithful nationallicence Cohn-Leavitt algebra nationallicence Algebras and Representation Theory Springer Netherlands 18/3(2015-06-01), 711-738 1386-923X 18:3<711 2015 18 10468 https://doi.org/10.1007/s10468-014-9513-8 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10468-014-9513-8 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Vaš Lia Department of Mathematics, Physics and Statistics, University of the Sciences, 19104, Philadelphia, PA, USA aut NATIONALLICENCE 773 0- Algebras and Representation Theory Springer Netherlands 18/3(2015-06-01), 711-738 1386-923X 18:3<711 2015 18 10468