Stieltjes perfect semigroups are perfect

Torben Maack Bisgaard; Nobuhisa Sakakibara

Czechoslovak Mathematical Journal (2005)

  • Volume: 55, Issue: 3, page 729-753
  • ISSN: 0011-4642

Abstract

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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 ∈ S there exist t ∈ S and m , n ∈ ℕ 0 such that m + n ≥ 2 and s + s * = s * + m t + n t * . This was known only with s = m t + n t * instead. The equality cannot be replaced by s + s * + s = s + s * + m t + n t * in general, but for semigroups with neutral element it can be replaced by s + p ( s + s * ) = p ( s + s * ) + m t + n t * for arbitrary p ∈ ℕ (allowed to depend on s ).

How to cite

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Bisgaard, Torben Maack, and Sakakibara, Nobuhisa. "Stieltjes perfect semigroups are perfect." Czechoslovak Mathematical Journal 55.3 (2005): 729-753. <http://eudml.org/doc/30984>.

@article{Bisgaard2005,
abstract = {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 $s$).},
author = {Bisgaard, Torben Maack, Sakakibara, Nobuhisa},
journal = {Czechoslovak Mathematical Journal},
keywords = {perfect; Stieltjes perfect; moment; positive definite; conelike; semi-$*$-divisible; $*$-semigroup; perfect; Stieltjes perfect; moment; positive definite; conelike; semi--divisible; -semigroup},
language = {eng},
number = {3},
pages = {729-753},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Stieltjes perfect semigroups are perfect},
url = {http://eudml.org/doc/30984},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Bisgaard, Torben Maack
AU - Sakakibara, Nobuhisa
TI - Stieltjes perfect semigroups are perfect
JO - Czechoslovak Mathematical Journal
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 3
SP - 729
EP - 753
AB - 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 $s$).
LA - eng
KW - perfect; Stieltjes perfect; moment; positive definite; conelike; semi-$*$-divisible; $*$-semigroup; perfect; Stieltjes perfect; moment; positive definite; conelike; semi--divisible; -semigroup
UR - http://eudml.org/doc/30984
ER -

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