Boidol’s conjecture on symmetric and *-regular locally compact groups

Prove that for any locally compact group G, if G is symmetric (hermitian), then G is *- regular—namely, for every closed two-sided ideal I in the full group C*-algebra C*(G), I equals the closure of I ∩ L1(G) in the C*(G)-norm—and conversely, establish that every almost connected, *- regular locally compact group G is symmetric.

Background

The paper recalls Boidol’s foundational work on -regularity and notes his conjectural characterization linking symmetry (hermitianity) and *-regularity, including a converse statement for almost connected groups. *-regularity for a locally compact group G requires that every closed ideal I of C(G) is the closure of its intersection with L1(G). Symmetry refers to hermitian groups in the sense of harmonic analysis.