Geodesic
Stieltjes perfect semigroups are perfect
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 729-753 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

An abelian $*$-semigroup $S$ is perfect (resp. Stieltjes perfect) if every positive definite (resp. completely so) function on $S$ admits a unique disintegration as an integral of hermitian multiplicative functions (resp. nonnegative such). We prove that every Stieltjes perfect semigroup is perfect. The converse has been known for semigroups with neutral element, but is here shown to be not true in general. We prove that an abelian $*$-semigroup $S$ is perfect if for each $s \in S$ there exist $t\in S$ and $m,n\in \mathbb{N}_0$ such that $m+n\ge 2$ and $s+s^ *=s^*+mt+nt^*$. This was known only with $s=mt+nt^*$ instead. The equality cannot be replaced by $s+s^*+s=s+s^*+mt+nt^*$ in general, but for semigroups with neutral element it can be replaced by $s+p(s+s^*)=p(s+s^*)+ mt+nt^*$ for arbitrary $p\in \mathbb{N}$ (allowed to depend on $s$).
An abelian $*$-semigroup $S$ is perfect (resp. Stieltjes perfect) if every positive definite (resp. completely so) function on $S$ admits a unique disintegration as an integral of hermitian multiplicative functions (resp. nonnegative such). We prove that every Stieltjes perfect semigroup is perfect. The converse has been known for semigroups with neutral element, but is here shown to be not true in general. We prove that an abelian $*$-semigroup $S$ is perfect if for each $s \in S$ there exist $t\in S$ and $m,n\in \mathbb{N}_0$ such that $m+n\ge 2$ and $s+s^ *=s^*+mt+nt^*$. This was known only with $s=mt+nt^*$ instead. The equality cannot be replaced by $s+s^*+s=s+s^*+mt+nt^*$ in general, but for semigroups with neutral element it can be replaced by $s+p(s+s^*)=p(s+s^*)+ mt+nt^*$ for arbitrary $p\in \mathbb{N}$ (allowed to depend on $s$).
Classification : 43A35, 44A60
Keywords: perfect; Stieltjes perfect; moment; positive definite; conelike; semi-$*$-divisible; $*$-semigroup 
@article{CMJ_2005_55_3_a15,
     author = {Bisgaard, Torben Maack and Sakakibara, Nobuhisa},
     title = {Stieltjes perfect semigroups are perfect},
     journal = {Czechoslovak Mathematical Journal},
     pages = {729--753},
     year = {2005},
     volume = {55},
     number = {3},
     mrnumber = {2153098},
     zbl = {1081.43002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a15/}
}
TY  - JOUR
AU  - Bisgaard, Torben Maack
AU  - Sakakibara, Nobuhisa
TI  - Stieltjes perfect semigroups are perfect
JO  - Czechoslovak Mathematical Journal
PY  - 2005
SP  - 729
EP  - 753
VL  - 55
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a15/
LA  - en
ID  - CMJ_2005_55_3_a15
ER  - 
%0 Journal Article
%A Bisgaard, Torben Maack
%A Sakakibara, Nobuhisa
%T Stieltjes perfect semigroups are perfect
%J Czechoslovak Mathematical Journal
%D 2005
%P 729-753
%V 55
%N 3
%U http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a15/
%G en
%F CMJ_2005_55_3_a15
Bisgaard, Torben Maack; Sakakibara, Nobuhisa. Stieltjes perfect semigroups are perfect. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 729-753. http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a15/