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).
