2026-05-30T19:55:43Zhttps://riuma.uma.es/rest/oai/requestoai:riuma.uma.es:10630/76882026-02-03T11:46:33Zcom_10630_2254col_10630_37959
RIUMA. Repositorio Institucional de la Universidad de Málaga
author
Abrams, Gene
2014-06-18T08:15:56Z
2014-06-18T08:15:56Z
2014-06-18
http://hdl.handle.net/10630/7688
Let $E = (E^0, E^1, s,r)$ be an arbitrary directed graph (i.e., no restriction is placed on the cardinality of $E^0$, or of $E^1$, or of $s^{-1}(v)$ for $v\in E^0$). Let $L_K(E)$ denote the Leavitt path algebra of $E$ with coefficients in a field $K$, and let $C^*(E)$ denote the graph C$^*$-algebra of $E$.
% (Note: here $C^*(E)$ need not be separable.)
We give necessary and sufficient conditions on $E$ so that $L_K(E)$ is primitive. (This is joint work with Jason Bell and K.M. Rangaswamy.) We then show that these same conditions are precisely the necessary and sufficient conditions on $E$ so that $C^*(E)$ is primitive. (This is joint work with Mark Tomforde.)
This situation gives yet another example of algebraic / analytic properties of the graph algebras $L_K(E)$ and $C^*(E)$ for which the graph conditions equivalent to said property are identical, but for which the proof / techniques used are significantly different. In the Leavitt path algebra setting, we show how this result allows for the easy construction of a large collection of prime, non-primitive von Neumann regular algebras (thereby giving a systematic answer to a decades-old question of Kaplansky). In the graph C$^*$-algebra setting, we show how this result allows for the easy construction of a large collection of prime, non-primitive C$^*$-algebras (thereby giving a systematic answer to a decades-old question of Dixmier).
eng
Primitive graph algebras
conference output
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URL
https://riuma.uma.es/bitstreams/e98cd150-4002-4fdd-a261-f3f1d361380a/download
File
MD5
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application/pdf
CartelConfAbrams2014.pdf
URL
https://riuma.uma.es/bitstreams/d007c474-31b2-4dc0-903f-aa979d3d05ab/download
File
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text/plain
CartelConfAbrams2014.pdf.txt