math.FA — Functional Analysis

Cowling’s conjecture: the almost connected case

Contributed by Cédric ARHANCET

We prove that a locally compact group $G$ with amenable group $G/G^0$ of components has the Haagerup property if and only if it is weakly amenable with constant one. This proves Cowling's conjecture for connected Lie groups and almost connected locally compact groups. We also characterize weak amenability for arbitrary connected Lie groups and prove the product formula for the Cowling--Haagerup constant in terms of the simple factors of a Levi decomposition. Consequently, the constant is invariant under local isomorphism and decreases under quotients of connected locally compact groups. For groups with amenable group of components, it is unchanged by every closed central quotient. We deduce applications to lattices and their central quotients. The proof combines Ozawa's invariant-state obstruction with averaging over central characters and the tensor identity for induced representations.