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1. CJM 2013 (vol 65 pp. 1073)
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 algebra Category:46L89 |
2. CJM 2012 (vol 65 pp. 1043)
Convolution of Trace Class Operators over Locally Compact Quantum Groups We study locally compact quantum groups $\mathbb{G}$ through the
convolution algebras $L_1(\mathbb{G})$ and $(T(L_2(\mathbb{G})),
\triangleright)$. We prove that the reduced quantum group
$C^*$-algebra $C_0(\mathbb{G})$ can be recovered from the convolution
$\triangleright$ by showing that the right $T(L_2(\mathbb{G}))$-module
$\langle K(L_2(\mathbb{G}) \triangleright T(L_2(\mathbb{G}))\rangle$ is
equal to $C_0(\mathbb{G})$. On the other hand, we show that the left
$T(L_2(\mathbb{G}))$-module $\langle T(L_2(\mathbb{G}))\triangleright
K(L_2(\mathbb{G})\rangle$ is isomorphic to the reduced crossed product
$C_0(\widehat{\mathbb{G}}) \,_r\!\ltimes C_0(\mathbb{G})$, and hence is
a much larger $C^*$-subalgebra of $B(L_2(\mathbb{G}))$.
We establish a natural isomorphism between the completely bounded
right multiplier algebras of $L_1(\mathbb{G})$ and
$(T(L_2(\mathbb{G})), \triangleright)$, and settle two invariance
problems associated with the representation theorem of
Junge-Neufang-Ruan (2009). We characterize regularity and discreteness
of the quantum group $\mathbb{G}$ in terms of continuity properties of
the convolution $\triangleright$ on $T(L_2(\mathbb{G}))$. We prove
that if $\mathbb{G}$ is semi-regular, then the space
$\langle T(L_2(\mathbb{G}))\triangleright B(L_2(\mathbb{G}))\rangle$ of right
$\mathbb{G}$-continuous operators on $L_2(\mathbb{G})$, which was
introduced by Bekka (1990) for $L_{\infty}(G)$, is a unital $C^*$-subalgebra
of $B(L_2(\mathbb{G}))$. In the representation framework formulated by
Neufang-Ruan-Spronk (2008) and Junge-Neufang-Ruan, we show that the
dual properties of compactness and discreteness can be characterized
simultaneously via automatic normality of quantum group bimodule maps
on $B(L_2(\mathbb{G}))$. We also characterize some commutation
relations of completely bounded multipliers of $(T(L_2(\mathbb{G})),
\triangleright)$ over $B(L_2(\mathbb{G}))$.
Keywords:locally compact quantum groups and associated Banach algebras Categories:22D15, 43A30, 46H05 |
3. CJM 2010 (vol 63 pp. 3)
Free Bessel Laws
We introduce and study a remarkable family of real probability
measures $\pi_{st}$ that we call free Bessel laws. These are related
to the free Poisson law $\pi$ via the formulae
$\pi_{s1}=\pi^{\boxtimes s}$ and ${\pi_{1t}=\pi^{\boxplus t}}$. Our
study includes definition and basic properties, analytic aspects
(supports, atoms, densities), combinatorial aspects (functional
transforms, moments, partitions), and a discussion of the relation
with random matrices and quantum groups.
Keywords:Poisson law, Bessel function, Wishart matrix, quantum group Categories:46L54, 15A52, 16W30 |
4. CJM 2005 (vol 57 pp. 17)
On Amenability and Co-Amenability of Algebraic Quantum Groups and Their Corepresentations We introduce and study several notions of amenability for unitary
corepresentations and $*$-representations of algebraic quantum groups,
which may be used to characterize amenability and co-amenability for
such quantum groups. As a background for this study, we investigate
the associated tensor C$^{*}$-categories.
Keywords:quantum group, amenability Categories:46L05, 46L65, 22D10, 22D25, 43A07, 43A65, 58B32 |