520

Т . М . Bisgaard and P. Ressel

Propositions . Let G be an Abelian group carrying the involution x*= -x, and let Shea subsemigroup o/Q+ x G. Define P = {pe(i^+ \(p, 0)eS}. Assume (i) There is a sequence {dj)JL I ofintegers ^Isuchthat Р = <^{(а^...а^Г^\кеЩ}. (ii) For each peP there exists se S such that (p, 0) = s + s*. (iii) If{4^ ^l {r, x)eS with q<r, then r-qeP. (iv) //(r, x)eS andx>0, then r>0. Then S is perfect.

Proof . We may assume G = 712(8), where 7^2: Q+ x G->G is the projection. We consider (7:=Q+ x G with the product of the usual topology on Q+ and the discrete topology on G. Let œ denote the image of Haar measure on G* under

Ty - ^liO } ®'c : G * - ^U * . Define C = {l^o}®t|tgG*}ç(7*, ^{и)о = {фе^{и)\ф is continuous except possibly at (0, 0)}, and E^ (L/*)o = {àeE+ (С/*) IЯ| С is tional to со}.

We have to show that fi^fi: Е^{8*)-^^{8) is bijective; this will follow from the next three lemmas.

Lemma 2. Яь^Я"*: E+ ([/*)o ^ £+ {8% where hiS^Uis the inclusion, is bijective.

Proof Clearly, h*{C) = {l^o}}^8*. Now let аб5*\{1(0}}. By (ii), the character р^(т{р, 0) on P is non-negative, hence of the form p|P for a uniquely determined pe(S^%. Since d+l^o}, there exists s = {r, x)e8\{0} such that (t(s) + 0; by (iv) it follows that r>0 and so 0<|(j(s)p = (T(2r, 0) = p(2r). Hence, p is everywhere positive. Using (iii) one easily shows that there is a unique tgG* such that (T = h*{p(S)T). Thus, h* is onto. Moreover, h*\{U*\C) is a homeomorphism tween C/*\C and 5*\{l^o}}.

It is now clear that Ян-^Д"*: M+([/*)o->M+(S*), where М+(и*)о :={ЯбМ+ (С/*) IЯ| С is proportional to со}, is bijective. For ЯбМ+ (U*) we have

ЯеЕ + ( С / * ) iff ^^{r,0)dÀ{i)<oo forall ге(1^^ iff J^(r, 0)^Я((^)<оо for all reP iff Я''*е£,.(5*). П

Lemma 3. Ян^Х: E+ {U*)o-^^(U)o is a bijection.

Proof Since и is perfect, À\-^l is a bijection between E+(L/*) and ^(U). It remains to be shown that if àeE^{U*) then Яе£:;(С/*)о iff Xg^(L/)o. For each X6G define ö^6JE:((Q*) by о^ = (т(х)Я)"^ where tti! (7*-Q* xG*-Q* is the projection. Then X(r, x) = ^,(r), reQ^, xeG. Let aGM+(G*) be the image of Я|С under the inverse of тн->1^о}®т: G*^C(ç(7*). Then ö,({l(o}}) = ^W, ^eG. Note that discontinuity of 1 at (r, x) is possible only if r = 0. Hence,

Xe^ { U ) o iff >f(-, x) is continuous for each JC6G\{0}

iffö^ { l { o } } ) = OforeachxGG\{0}iff'a(x) = OforeachxeG\{0}

iff à is proportional to l^o} iff a is proportional to Haar measure on G*

iffЯG£ + ( [ / * ) o . D

Lemma 4. ф\-^ф\8: 0>{U)o-^^{8) is bijective.