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.

References

Even in the simplest case where $M=$, we do not have any criterion for factoriality of $\rtimes G=L(G)$.

— Strictly outer actions of locally compact groups: beyond the full factor case  (2407.11738 - Morando, 2024) in Section 1, Introduction and results, bullet list under “In the non-discrete case”

Is there a common conjugacy-dynamical criterion for locally compact groups that recovers the absence of ICC quotients in the countable discrete setting, the distal/SIN criteria in the totally disconnected setting, and the bounded-conjugacy-class condition of \Cref{main-thm}?

— Classification of Choquet-Deny Lie groups  (2609.11731 - Bénard et al., 10 Sep 2026) in Section 1, Introduction, paragraph headed “Open question”