Let $G$ be a locally compact group, and let $wG$ be its weakly almost
periodic compactification. The kernel $K(wG)$ of $wG$ is then a group
isomorphic to the almost periodic compactification of $G$.

There are (non-compact) groups such that $wG = K(wG) \cup G$ (the union
is necessarily disjoint). For such groups, the closed ideal $wG
\setminus G$ of $wG$ then has an identity.

Question: Let $G$ be a locally compact group (not compact) such that $wG
\setminus G$ has an identity. Does that necessarily mean that $wG
\setminus G = K(wG)$?

Any pertinent hints are appreciated.

Volker Runde.

begin:vcard
n:Runde;Volker
tel;fax:+1 780 492 6826
tel;home:+1 780 480 1181
tel;work:+1 780 492 3526
x-mozilla-html:FALSE
url:http://www.math.ualberta.ca/~runde/
org:University of Alberta;Mathematical and Statistical Sciences
adr:;;CAB 632;Edmonton;Alberta;T6G 2G1;Canada
version:2.1

title:Associate Professor
x-mozilla-cpt:;1856
fn:Volker Runde
end:vcard