RT Conference Proceedings
T1 Primitive graph algebras
A1 Abrams, Gene
AB 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).
YR 2014
FD 2014-06-18
LK http://hdl.handle.net/10630/7688
UL http://hdl.handle.net/10630/7688
LA eng
NO Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech.
DS RIUMA. Repositorio Institucional de la Universidad de Málaga
RD 21 ene 2026