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1. CJM 2016 (vol 68 pp. 1067)

Runde, Volker; Viselter, Ami
 On Positive Definiteness over Locally Compact Quantum Groups The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on "square roots" of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups. Keywords:bicrossed product, locally compact quantum group, non-commutative $L^p$-space, positive-definite function, positive-definite measure, separation propertyCategories:20G42, 22D25, 43A35, 46L51, 46L52, 46L89

2. CJM 2012 (vol 65 pp. 768)

 Nonself-adjoint 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 Nica-covariance property: $T_s^*T_t=T_tT_s^*$ whenever $t\wedge s=0$. We show that all such representations have a unique minimal isometric Nica-covariant dilation. This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant 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 nonself-adjoint semicrossed product algebra (in the sense of Kakariadis and Katsoulis). Keywords:semicrossed product, crossed product, C*-envelope, dilationsCategories:47L55, 47A20, 47L65
 Pure Infiniteness of the Crossed Product of an AH-Algebra by an Endomorphism It is shown that simplicity of the crossed product of a unital AH-algebra with slow dimension growth by an endomorphism implies that the algebra is also purely infinite, provided only that the endomorphism leaves no trace state invariant and takes the unit to a full projection. Keywords:purely infinite $C^*$-algebras, crossed productsCategory:46-xx