1. CJM Online first
 Runde, Volker; Viselter, Ami

On positive definiteness over locally compact quantum groups
The notion of positivedefinite functions over locally compact
quantum
groups was recently introduced and studied by Daws and Salmi.
Based
on this work, we generalize various wellknown results about
positivedefinite
functions over groups to the quantum framework. Among these are
theorems
on "square roots" of positivedefinite functions, comparison
of
various topologies, positivedefinite measures and characterizations
of amenability, and the separation property with respect to compact
quantum subgroups.
Keywords:bicrossed product, locally compact quantum group, noncommutative $L^p$space, positivedefinite function, positivedefinite measure, separation property Categories:20G42, 22D25, 43A35, 46L51, 46L52, 46L89 

2. CJM 2012 (vol 65 pp. 768)
 Fuller, Adam Hanley

Nonselfadjoint Semicrossed Products by Abelian Semigroups
Let $\mathcal{S}$ be the semigroup $\mathcal{S}=\sum^{\oplus k}_{i=1}\mathcal{S}_i$, where for each $i\in I$,
$\mathcal{S}_i$ is a countable subsemigroup of the additive semigroup $\mathbb{R}_+$ containing $0$. We consider representations
of $\mathcal{S}$ as contractions $\{T_s\}_{s\in\mathcal{S}}$ on a Hilbert space with the Nicacovariance property:
$T_s^*T_t=T_tT_s^*$ whenever $t\wedge s=0$. We show that all such representations have a unique minimal isometric Nicacovariant
dilation.
This result is used to help analyse the nonselfadjoint semicrossed product algebras formed from Nicacovariant representations of the action of $\mathcal{S}$ on an operator algebra $\mathcal{A}$ by completely contractive endomorphisms.
We conclude by calculating the $C^*$envelope of the isometric nonselfadjoint semicrossed product algebra (in the sense
of Kakariadis and Katsoulis).
Keywords:semicrossed product, crossed product, C*envelope, dilations Categories:47L55, 47A20, 47L65 

3. CJM 2011 (vol 64 pp. 705)