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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 2004 (vol 56 pp. 843)
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 |
3. CJM 1999 (vol 51 pp. 850)
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 theorem Categories:46L05, 46L40, 46L89, 47D15, 47D25, 46M10, 46M99, 47A20, 47A45, 47B35 |