A Characterization for Discrete Quantum Group

The Martin Boundary of a Discrete Quantum Group

We consider the Markov operator Pφ on a discrete quantum group given by convolution with a q-tracial state φ. In the study of harmonic elements x, Pφ(x) = x, we define the Martin boundary Aφ. It is a separable C ∗-algebra carrying canonical actions of the quantum group and its dual. We establish a representation theorem to the effect that positive harmonic elements correspond to positive linear...

A Geometric Renormalisation Group in Discrete Quantum Space-Time

We model quantum space-time on the Planck scale as dynamical networks of elementary relations or time dependent random graphs, the time dependence being an effect of the underlying dynamical network laws. We formulate a kind of geometric renormalisation group on these (random) networks leading to a hierarchy of increasingly coarse-grained networks of overlapping lumps. We provide arguments that...

Stability of additive functional equation on discrete quantum semigroups

We construct  a noncommutative analog of additive functional equations on discrete quantum semigroups and show that this noncommutative functional equation has Hyers-Ulam stability on amenable discrete quantum semigroups. The discrete quantum semigroups that we consider in this paper are in the sense of van Daele, and the amenability is in the sense of Bèdos-Murphy-Tuset. Our main result genera...

Quantum Physics: A Case for Anti-Realism?

A characterization of the symmetric group of prime degree

Let $G$ be a finite group and $Gamma(G)$ the prime graph of $G$‎. ‎Recently people have been using prime graphs to study simple groups‎. ‎Naturally we pose a question‎: ‎can we use prime graphs to study almost‎ ‎simple groups or non-simple groups? In this paper some results in‎ ‎this respect are obtained and as follows‎: ‎$Gcong S_p$ if and only‎ ‎if $|G|=|S_p|$ and $Gamma(G)=Gamma(S_p)$‎, ‎whe...