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1. CJM Online first
The C*-algebras of Compact Transformation Groups We investigate the representation theory of the
crossed-product $C^*$-algebra associated to a compact group $G$
acting on a locally compact space $X$ when the stability subgroups
vary discontinuously.
Our main result applies when $G$ has a principal stability subgroup or
$X$ is locally of finite $G$-orbit type. Then the upper multiplicity
of the representation of the crossed product induced from an
irreducible representation $V$ of a stability subgroup is obtained by
restricting $V$ to a certain closed subgroup of the stability subgroup
and taking the maximum of the multiplicities of the irreducible
summands occurring in the restriction of $V$. As a corollary we obtain
that when the trivial subgroup is a principal stability subgroup, the
crossed product is a Fell algebra if and only if every stability
subgroup is abelian. A second corollary is that the $C^*$-algebra of
the motion group $\mathbb{R}^n\rtimes \operatorname{SO}(n)$ is a Fell algebra. This uses
the classical branching theorem for the special orthogonal group
$\operatorname{SO}(n)$ with respect to $\operatorname{SO}(n-1)$. Since proper transformation
groups are locally induced from the actions of compact groups, we
describe how some of our results can be extended to transformation
groups that are locally proper.
Keywords:compact transformation group, proper action, spectrum of a C*-algebra, multiplicity of a representation, crossed-product C*-algebra, continuous-trace C*-algebra, Fell algebra Categories:46L05, 46L55 |
2. CJM 2008 (vol 60 pp. 975)
An AF Algebra Associated with the Farey Tessellation We associate with the Farey tessellation of the upper
half-plane an
AF algebra $\AA$ encoding the ``cutting sequences'' that define
vertical geodesics.
The Effros--Shen AF algebras arise as quotients
of $\AA$. Using the path algebra model for AF algebras we construct, for
each $\tau \in \big(0,\frac{1}{4}\big]$, projections $(E_n)$ in
$\AA$ such that $E_n E_{n\pm 1}E_n \leq \tau E_n$.
Categories:46L05, 11A55, 11B57, 46L55, 37E05, 82B20 |
3. CJM 2006 (vol 58 pp. 39)
$C^*$-Algebras of Irreversible Dynamical Systems We show that certain $C^*$-algebras which have been studied by,
among others, Arzumanian, Vershik, Deaconu, and Renault, in
connection with a measure-preserving transformation of a measure space
or a covering map of a compact space, are special cases of the
endomorphism crossed-product construction recently introduced by the
first named author. As a consequence these algebras are given
presentations in terms of generators and relations. These results
come as a consequence of a general theorem on faithfulness of
representations which are covariant with respect to certain circle
actions. For the case of topologically free covering maps we prove a
stronger result on faithfulness of representations which needs no
covariance. We also give a necessary and sufficient condition for
simplicity.
Categories:46L55, 37A55 |
4. CJM 2005 (vol 57 pp. 983)
A Symmetric Imprimitivity Theorem for Commuting Proper Actions We prove a symmetric imprimitivity theorem for commuting proper
actions of locally compact groups $H$ and $K$ on a $C^*$-algebra.
Categories:46L05, 46L08, 46L55 |
5. CJM 2003 (vol 55 pp. 1302)
The Ideal Structures of Crossed Products of Cuntz Algebras by Quasi-Free Actions of Abelian Groups We completely determine the ideal structures of the crossed
products of Cuntz algebras by quasi-free actions of abelian groups
and give another proof of A.~Kishimoto's result on the simplicity
of such crossed products. We also give a necessary and sufficient
condition that our algebras become primitive, and compute the
Connes spectra and $K$-groups of our algebras.
Categories:46L05, 46L55, 46L45 |
6. CJM 1999 (vol 51 pp. 745)
Induced Coactions of Discrete Groups on $C^*$-Algebras Using the close relationship between coactions of discrete groups and
Fell bundles, we introduce a procedure for inducing a $C^*$-coaction
$\delta\colon D\to D\otimes C^*(G/N)$ of a quotient group $G/N$ of a
discrete group $G$ to a $C^*$-coaction $\Ind\delta\colon\Ind D\to \Ind
D\otimes C^*(G)$ of $G$. We show that induced coactions behave in many
respects similarly to induced actions. In particular, as an analogue of
the well known imprimitivity theorem for induced actions we prove that
the crossed products $\Ind D\times_{\Ind\delta}G$ and $D\times_{\delta}G/N$
are always Morita equivalent. We also obtain nonabelian analogues of a
theorem of Olesen and Pedersen which show that there is a duality between
induced coactions and twisted actions in the sense of Green. We further
investigate amenability of Fell bundles corresponding to induced coactions.
Category:46L55 |