Search: MSC category 46L89 ( Other noncommutative'' mathematics based on $C^$-algebra theory [See also 58B32, 58B34, 58J22] *$-algebra theory [See also 58B32, 58B34, 58J22] * )  Expand all Collapse all Results 1 - 6 of 6 1. CJM Online first Runde, Volker; Viselter, Ami  On positive definiteness over locally compact quantum groups The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on "square roots" of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups. Keywords:bicrossed product, locally compact quantum group, non-commutative$L^p$-space, positive-definite function, positive-definite measure, separation propertyCategories:20G42, 22D25, 43A35, 46L51, 46L52, 46L89 2. CJM 2016 (vol 68 pp. 698) Skalski, Adam; Sołtan, Piotr  Quantum Families of Invertible Maps and Related Problems The notion of families of quantum invertible maps (C$^*$-algebra homomorphisms satisfying PodleÅ' condition) is employed to strengthen and reinterpret several results concerning universal quantum groups acting on finite quantum spaces. In particular Wang's quantum automorphism groups are shown to be universal with respect to quantum families of invertible maps. Further the construction of the Hopf image of Banica and Bichon is phrased in the purely analytic language and employed to define the quantum subgroup generated by a family of quantum subgroups or more generally a family of quantum invertible maps. Keywords:quantum families of invertible maps, Hopf image, universal quantum groupCategories:46L89, 46L65 3. CJM 2016 (vol 68 pp. 309) Daws, Matthew  Categorical Aspects of Quantum Groups: Multipliers and Intrinsic Groups We show that the assignment of the (left) completely bounded multiplier algebra$M_{cb}^l(L^1(\mathbb G))$to a locally compact quantum group$\mathbb G$, and the assignment of the intrinsic group, form functors between appropriate categories. Morphisms of locally compact quantum groups can be described by Hopf$*$-homomorphisms between universal$C^*$-algebras, by bicharacters, or by special sorts of coactions. We show that the whole theory of completely bounded multipliers can be lifted to the universal$C^*$-algebra level, and that then the different pictures of both multipliers (reduced, universal, and as centralisers) and morphisms interact in extremely natural ways. The intrinsic group of a quantum group can be realised as a class of multipliers, and so our techniques immediately apply. We also show how to think of the intrinsic group using the universal$C^*$-algebra picture, and then, again, show how the differing views on the intrinsic group interact naturally with morphisms. We show that the intrinsic group is the maximal classical'' quantum subgroup of a locally compact quantum group, show that it is even closed in the strong Vaes sense, and that the intrinsic group functor is an adjoint to the inclusion functor from locally compact groups to quantum groups. Keywords:locally compact quantum group, morphism, intrinsic group, multiplier, centraliserCategories:20G42, 22D25, 43A22, 43A35, 43A95, 46L52, 46L89, 47L25 4. CJM 2013 (vol 65 pp. 1073) Kalantar, Mehrdad; Neufang, Matthias  From Quantum Groups to Groups In this paper we use the recent developments in the representation theory of locally compact quantum groups, to assign, to each locally compact quantum group$\mathbb{G}$, a locally compact group$\tilde {\mathbb{G}}$which is the quantum version of point-masses, and is an invariant for the latter. We show that quantum point-masses" can be identified with several other locally compact groups that can be naturally assigned to the quantum group$\mathbb{G}$. This assignment preserves compactness as well as discreteness (hence also finiteness), and for large classes of quantum groups, amenability. We calculate this invariant for some of the most well-known examples of non-classical quantum groups. Also, we show that several structural properties of$\mathbb{G}$are encoded by$\tilde {\mathbb{G}}$: the latter, despite being a simpler object, can carry very important information about$\mathbb{G}$. Keywords:locally compact quantum group, locally compact group, von Neumann algebraCategory:46L89 5. CJM 2004 (vol 56 pp. 843) Ruan, Zhong-Jin  Type Decomposition and the Rectangular AFD Property for$W^*$-TRO's We study the type decomposition and the rectangular AFD property for$W^*$-TRO's. Like von Neumann algebras, every$W^*$-TRO can be uniquely decomposed into the direct sum of$W^*$-TRO's of type$I$, type$II$, and type$III$. We may further consider$W^*$-TRO's of type$I_{m, n}$with cardinal numbers$m$and$n$, and consider$W^*$-TRO's of type$II_{\lambda, \mu}$with$\lambda, \mu = 1$or$\infty$. It is shown that every separable stable$W^*$-TRO (which includes type$I_{\infty,\infty}$, type$II_{\infty, \infty}$and type$III$) is TRO-isomorphic to a von Neumann algebra. We also introduce the rectangular version of the approximately finite dimensional property for$W^*$-TRO's. One of our major results is to show that a separable$W^*$-TRO is injective if and only if it is rectangularly approximately finite dimensional. As a consequence of this result, we show that a dual operator space is injective if and only if its operator predual is a rigid rectangular${\OL}_{1, 1^+}$space (equivalently, a rectangular Categories:46L07, 46L08, 46L89 6. CJM 1999 (vol 51 pp. 850) Muhly, Paul S.; Solel, Baruch  Tensor Algebras, Induced Representations, and the Wold Decomposition Our objective in this sequel to \cite{MSp96a} is to develop extensions, to representations of tensor algebras over$C^{*}\$-correspondences, of two fundamental facts about isometries on Hilbert space: The Wold decomposition theorem and Beurling's theorem, and to apply these to the analysis of the invariant subspace structure of certain subalgebras of Cuntz-Krieger algebras. Keywords:tensor algebras, correspondence, induced representation, Wold decomposition, Beurling's theoremCategories:46L05, 46L40, 46L89, 47D15, 47D25, 46M10, 46M99, 47A20, 47A45, 47B35