Relating to the prime versus primitive issues, it was shown by Abrams, Bell and Rangaswamy that a Leavitt path algebra over a field defined by a countable digraph is prime if and only if it is primitive but this is not true for uncountable graphs.

More generally, I showed that if $\mathscr G$ is a locally compact and totally disconnected second countable Hausdorff etale groupoid, then its convolution algebra over a field is prime if and only if it is primitive but this again fails in the non-second countable case.