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Description

Let E = (E0,E1,s,r) be an arbitrary directed graph (i.e., no restriction is placed on the cardinality of E0, or of E1, or of s−1(v) for v ∈ E0). Let LK(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 LK(E) is primitive (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 (joint work with Mark Tomforde). This gives yet another example in a long and growing list of algebraic/analytic properties of the graph algebras LK(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).