Criterion for factoriality of L(G) for non-discrete locally compact groups

Determine necessary and sufficient conditions on a locally compact non-discrete group G for the group von Neumann algebra L(G), generated by the left regular representation of G, to be a factor (i.e., to have trivial center).

Background

In the discrete case, several structural properties of crossed products are understood via Fourier decompositions. Notably, for M = ℂ, one has ℂ ⋊ G = L(G), and L(G) is a factor if and only if G has infinite non-trivial conjugacy classes (the ICC condition).

For locally compact non-discrete groups, these tools no longer apply directly, and even the most basic case M = ℂ becomes subtle. The authors explicitly note the absence of any known criterion to determine when L(G) is a factor, highlighting a foundational gap in the understanding of factoriality in the non-discrete setting.