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 $s$).
Stiff-person syndrome (SPS) is a rare disorder characterized by muscle stiffness and painful spasms. Misdiagnosis may occur due to the fact that the clinical picture of SPS is often atypical. The main pathophysiologic mechanism underlying the development of SPS is insufficient inhibition at the cortical and spinal levels. There is good evidence for a primary autoimmune etiology. A 61-year-old man was admitted to a neurological department due to muscle hypertonia with episodic attacks of painful spasms predominantly affecting axial muscles. The symptoms developed shortly after tickborne meningoencephalitis. Electromyography (EMG) revealed signs of continuous motor unit activity. Antibodies against glutamate decarboxylase (anti-GAD) were highly elevated. We present a case of a man who developed clinically severe anti-GAD positive SPS, provoked by tick-borne encephalitis. After therapeutic plasma exchange (TPE) a rapid, temporary improvement of the clinical and neurophysiological findings was noted. Only after being placed on long-term immunosuppression did the patient achieve stable recovery. This case supports the importance of EMG findings and demonstrates the effect of TPE as well as the need for chronic immunosuppression in severe cases of SPS., Edvard Ehler, Jan Latta, Petra Mandysová, Jana Havlasová, Milan Mrklovský, and Literatura 22