Abstract
Assume that
A
(
G
)
A(G)
and
B
(
H
)
B(H)
are the Fourier and Fourier–Stieltjes algebras of locally compact groups 𝐺 and 𝐻, respectively.
Ilie and Spronk have shown that continuous piecewise affine maps
α
:
Y
⊆
H
→
G
\alpha\colon Y\subseteq H\to G
induce completely bounded homomorphisms
Φ
:
A
(
G
)
→
B
(
H
)
\Phi\colon A(G)\to B(H)
and that, when 𝐺 is amenable, every completely bounded homomorphism arises in this way.
This generalised work of Cohen in the abelian setting.
We believe that there is a gap in a key lemma of the existing argument, which we do not see how to repair.
We present here a different strategy to show the result, which instead of using topological arguments, is more combinatorial and makes use of measure-theoretic ideas, following more closely the original ideas of Cohen.