Search: MSC category 46L89
( Other ``noncommutative'' mathematics based on $C^$algebra theory [See also 58B32, 58B34, 58J22] *$algebra theory [See also 58B32, 58B34, 58J22] * )
1. CMB Online first
 Józiak, Paweł

Remarks on Hopf images and quantum permutation groups $S_n^+$
Motivated by a question of A. Skalski and P.M. SoÅtan (2016)
about inner faithfulness of the S. Curran's map of extending
a quantum increasing sequence to a quantum permutation, we revisit
the results and techniques of T. Banica and J. Bichon (2009)
and study some grouptheoretic properties of the quantum permutation
group on $4$ points. This enables us not only to answer the aforementioned
question in positive in case $n=4, k=2$, but also to classify
the automorphisms of $S_4^+$, describe all the embeddings $O_{1}(2)\subset
S_4^+$ and show that all the copies of $O_{1}(2)$ inside $S_4^+$
are conjugate. We then use these results to show that the converse
to the criterion we applied to answer the aforementioned question
is not valid.
Keywords:Hopf image, quantum permutation group, compact quantum group Categories:20G42, 81R50, 46L89, 16W35 

2. CMB 2016 (vol 60 pp. 122)
 Ghanei, Mohammad Reza; NasrIsfahani, Rasoul; Nemati, Mehdi

A Homological Property and Arens Regularity of Locally Compact Quantum Groups
We characterize two important notions of amenability and compactness
of
a locally compact quantum group ${\mathbb G}$ in terms of certain
homological
properties. For this, we show that ${\mathbb G}$ is character
amenable if and only if it is both amenable and coamenable.
We finally apply our results to
Arens regularity problems of the quantum group algebra
$L^1({\mathbb G})$; in particular, we improve an interesting result
by Hu, Neufang and Ruan.
Keywords:amenability, Arens regularity, coamenability, locally compact quantum group, homological property Categories:46L89, 43A07, 46H20, 46M10, 58B32 

3. CMB 2013 (vol 57 pp. 546)
 Kalantar, Mehrdad

Compact Operators in Regular LCQ Groups
We show that a regular locally compact quantum group $\mathbb{G}$ is discrete
if and only if $\mathcal{L}^{\infty}(\mathbb{G})$ contains nonzero compact operators on
$\mathcal{L}^{2}(\mathbb{G})$.
As a corollary we classify all discrete quantum groups among
regular locally compact quantum groups $\mathbb{G}$ where
$\mathcal{L}^{1}(\mathbb{G})$ has the RadonNikodym property.
Keywords:locally compact quantum groups, regularity, compact operators Category:46L89 

4. CMB 2012 (vol 57 pp. 424)
 Sołtan, Piotr M.; Viselter, Ami

A Note on Amenability of Locally Compact Quantum Groups
In this short note we introduce a notion called ``quantum injectivity''
of locally compact quantum groups, and prove that it is equivalent
to amenability of the dual. Particularly, this provides a new characterization
of amenability of locally compact groups.
Keywords:amenability, conditional expectation, injectivity, locally compact quantum group, quantum injectivity Categories:20G42, 22D25, 46L89 

5. CMB 2009 (vol 52 pp. 39)
 Cimpri\v{c}, Jakob

A Representation Theorem for Archimedean Quadratic Modules on $*$Rings
We present a new approach to noncommutative real algebraic geometry
based on the representation theory of $C^\ast$algebras.
An important result in commutative real algebraic geometry is
Jacobi's representation theorem for archimedean quadratic modules
on commutative rings.
We show that this theorem is a consequence of the
GelfandNaimark representation theorem for commutative $C^\ast$algebras.
A noncommutative version of GelfandNaimark theory was studied by
I. Fujimoto. We use his results to generalize
Jacobi's theorem to associative rings with involution.
Keywords:Ordered rings with involution, $C^\ast$algebras and their representations, noncommutative convexity theory, real algebraic geometry Categories:16W80, 46L05, 46L89, 14P99 
