Expand all Collapse all | Results 1 - 25 of 72 |
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 Online first
On two-faced families of non-commutative random variables We demonstrate that the notions of bi-free independence and combinatorial-bi-free
independence of two-faced families are equivalent using a diagrammatic
view of bi-non-crossing partitions.
These diagrams produce an operator model on a Fock space suitable
for representing any two-faced family of non-commutative random
variables.
Furthermore, using a Kreweras complement on bi-non-crossing partitions
we establish the expected formulas for the multiplicative convolution
of a bi-free pair of two-faced families.
Keywords:free probability, operator algebras, bi-free Category:46L54 |
3. CJM Online first
Minimal Dynamical Systems on Connected Odd Dimensional Spaces Let $\beta\colon S^{2n+1}\to S^{2n+1}$ be a minimal homeomorphism ($n\ge 1$). We show that
the crossed product $C(S^{2n+1})\rtimes_\beta \mathbb{Z}$ has rational tracial rank at most one.
Let $\Omega$ be a connected compact metric space with finite covering dimension and
with $H^1(\Omega, \mathbb{Z})=\{0\}.$ Suppose that $K_i(C(\Omega))=\mathbb{Z}\oplus G_i,$ where $G_i$ is a finite abelian group, $i=0,1.$
Let $\beta\colon \Omega\to\Omega$ be a minimal homeomorphism. We also show that
$A=C(\Omega)\rtimes_\beta\mathbb{Z}$ has rational tracial rank at most one and is
classifiable.
In particular, this applies to the minimal dynamical systems on
odd dimensional real projective spaces.
This is done by studying minimal homeomorphisms on $X\times \Omega,$ where
$X$ is the Cantor set.
Keywords:minimal dynamical systems Categories:46L35, 46L05 |
4. CJM Online first
The Category of Bratteli Diagrams A category structure for Bratteli diagrams is proposed and a
functor from
the category of AF algebras to the category of Bratteli diagrams
is
constructed. Since isomorphism of Bratteli diagrams in this
category coincides
with Bratteli's notion of equivalence, we obtain in particular
a functorial formulation of Bratteli's
classification of AF algebras (and at the same time, of Glimm's
classification of UHF~algebras).
It is shown that the three approaches
to classification of AF~algebras, namely, through Bratteli diagrams,
K-theory, and
abstract classifying categories, are essentially the same
from a categorical point of view.
Keywords:C$^{*}$-algebra, category, functor, AF algebra, dimension group, Bratteli diagram Categories:46L05, 46L35, 46M15 |
5. CJM 2014 (vol 67 pp. 404)
Rotation Algebras and the Exel Trace Formula We found that if $u$ and $v$ are any two unitaries in
a unital $C^*$-algebra with $\|uv-vu\|\lt 2$ and $uvu^*v^*$ commutes with
$u$ and $v,$ then the $C^*$-subalgebra $A_{u,v}$ generated by $u$ and
$v$ is isomorphic to a quotient of some rotation algebra $A_\theta$
provided that $A_{u,v}$ has a unique tracial state.
We also found that the Exel trace formula holds in any unital
$C^*$-algebra.
Let $\theta\in (-1/2, 1/2)$ be a real number. We prove the
following:
For any $\epsilon\gt 0,$ there exists $\delta\gt 0$ satisfying the following:
if $u$ and $v$ are two unitaries in any unital simple $C^*$-algebra
$A$ with tracial rank zero such that
\[
\|uv-e^{2\pi i\theta}vu\|\lt \delta
\text{ and }
{1\over{2\pi i}}\tau(\log(uvu^*v^*))=\theta,
\]
for all tracial state $\tau$ of $A,$ then there exists a pair
of unitaries $\tilde{u}$ and $\tilde{v}$ in $A$
such that
\[
\tilde{u}\tilde{v}=e^{2\pi i\theta} \tilde{v}\tilde{u},\,\,
\|u-\tilde{u}\|\lt \epsilon
\text{ and }
\|v-\tilde{v}\|\lt \epsilon.
\]
Keywords:rotation algebras, Exel trace formula Category:46L05 |
6. CJM 2013 (vol 65 pp. 1287)
$K$-theory of Furstenberg Transformation Group $C^*$-algebras The paper studies the $K$-theoretic invariants of the crossed product
$C^{*}$-algebras associated with an important family of homeomorphisms
of the tori $\mathbb{T}^{n}$ called Furstenberg transformations.
Using the Pimsner-Voiculescu theorem, we prove that given $n$, the
$K$-groups of those crossed products, whose corresponding $n\times n$
integer matrices are unipotent of maximal degree, always have the same
rank $a_{n}$. We show using the theory developed here that a claim
made in the literature about the torsion subgroups of these $K$-groups
is false. Using the representation theory of the simple Lie algebra
$\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a
combinatorial significance. For example, every $a_{2n+1}$ is just the
number of ways that $0$ can be represented as a sum of integers
between $-n$ and $n$ (with no repetitions). By adapting an argument
of van Lint (in which he answered a question of ErdÅs), a simple,
explicit formula for the asymptotic behavior of the sequence
$\{a_{n}\}$ is given. Finally, we describe the order structure of the
$K_{0}$-groups of an important class of Furstenberg crossed products,
obtaining their complete Elliott invariant using classification
results of H. Lin and N. C. Phillips.
Keywords:$K$-theory, transformation group $C^*$-algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism Categories:19K14, 19K99, 46L35, 46L80, , 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20 |
7. CJM 2013 (vol 66 pp. 596)
The Ordered $K$-theory of a Full Extension Let $\mathfrak{A}$ be a $C^{*}$-algebra with real rank zero which has
the stable weak cancellation property. Let $\mathfrak{I}$ be an ideal
of $\mathfrak{A}$ such that $\mathfrak{I}$ is stable and satisfies the
corona factorization property. We prove that
$
0 \to \mathfrak{I} \to \mathfrak{A} \to \mathfrak{A} / \mathfrak{I} \to 0
$
is a full extension if and only if the extension is stenotic and
$K$-lexicographic. {As an immediate application, we extend the
classification result for graph $C^*$-algebras obtained by Tomforde
and the first named author to the general non-unital case. In
combination with recent results by Katsura, Tomforde, West and the
first author, our result may also be used to give a purely
$K$-theoretical description of when an essential extension of two
simple and stable graph $C^*$-algebras is again a graph
$C^*$-algebra.}
Keywords:classification, extensions, graph algebras Categories:46L80, 46L35, 46L05 |
8. CJM 2013 (vol 65 pp. 783)
Generalised Triple Homomorphisms and Derivations We introduce generalised triple homomorphism between Jordan Banach
triple systems as a concept which extends the notion of generalised homomorphism between
Banach algebras given by K. Jarosz and B.E. Johnson in 1985 and 1987, respectively.
We prove that every generalised triple homomorphism between JB$^*$-triples
is automatically continuous. When particularised to C$^*$-algebras, we rediscover
one of the main theorems established by B.E. Johnson. We shall also consider generalised
triple derivations from a Jordan Banach triple $E$ into a Jordan Banach triple $E$-module,
proving that every generalised triple derivation from a JB$^*$-triple $E$ into itself or into $E^*$
is automatically continuous.
Keywords:generalised homomorphism, generalised triple homomorphism, generalised triple derivation, Banach algebra, Jordan Banach triple, C$^*$-algebra, JB$^*$-triple Categories:46L05, 46L70, 47B48, 17C65, 46K70, 46L40, 47B47, 47B49 |
9. 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 |
10. CJM 2012 (vol 65 pp. 863)
Cumulants of the $q$-semicircular Law, Tutte Polynomials, and Heaps The $q$-semicircular distribution is a probability law that
interpolates between the Gaussian law and the semicircular law. There
is a combinatorial interpretation of its moments in terms of matchings
where $q$ follows the number of crossings, whereas for the free
cumulants one has to restrict the enumeration to connected matchings.
The purpose of this article is to describe combinatorial properties of
the classical cumulants. We show that like the free cumulants, they
are obtained by an enumeration of connected matchings, the weight
being now an evaluation of the Tutte polynomial of a so-called
crossing graph. The case $q=0$ of these cumulants was studied by
Lassalle using symmetric functions and hypergeometric series. We show
that the underlying combinatorics is explained through the theory of
heaps, which is Viennot's geometric interpretation of the
Cartier-Foata monoid. This method also gives a general formula for
the cumulants in terms of free cumulants.
Keywords:moments, cumulants, matchings, Tutte polynomials, heaps Categories:05A18, 05C31, 46L54 |
11. CJM 2012 (vol 65 pp. 481)
Correction of Proofs in "Purely Infinite Simple $C^*$-algebras Arising from Free Product Constructions'' and a Subsequent Paper |
Correction of Proofs in "Purely Infinite Simple $C^*$-algebras Arising from Free Product Constructions'' and a Subsequent Paper The proofs of Theorem 2.2 of K. J. Dykema and M. RÃ¸rdam, Purely infinite simple
$C^*$-algebras arising from free product constructions}, Canad. J.
Math. 50 (1998), 323--341 and
of Theorem 3.1 of K. J. Dykema, Purely infinite simple
$C^*$-algebras arising from free product constructions, II, Math.
Scand. 90 (2002), 73--86 are corrected.
Keywords:C*-algebras, purely infinite Category:46L05 |
12. CJM 2012 (vol 65 pp. 52)
C$^*$-algebras Nearly Contained in Type $\mathrm{I}$ Algebras In this paper we consider near inclusions $A\subseteq_\gamma B$ of C$^*$-algebras. We show that if $B$ is a separable type $\mathrm{I}$ C$^*$-algebra and $A$ satisfies Kadison's similarity problem, then $A$ is also type $\mathrm{I}$ and use this to obtain an embedding of $A$ into $B$.
Keywords:C$^*$-algebras, near inclusions, perturbations, type I C$^*$-algebras, similarity problem Category:46L05 |
13. CJM 2012 (vol 65 pp. 485)
Filters in C*-Algebras In this paper we analyze states on C*-algebras and
their relationship to filter-like structures of projections and
positive elements in the unit ball. After developing the basic theory
we use this to investigate the Kadison-Singer conjecture, proving its
equivalence to an apparently quite weak paving conjecture and the
existence of unique maximal centred extensions of projections coming
from ultrafilters on the natural numbers. We then prove that Reid's
positive answer to this for q-points in fact also holds for rapid
p-points, and that maximal centred filters are obtained in this case.
We then show that consistently such maximal centred filters do not
exist at all meaning that, for every pure state on the Calkin algebra,
there exists a pair of projections on which the state is 1, even
though the state is bounded strictly below 1 for projections below
this pair. Lastly we investigate towers, using cardinal invariant
equalities to construct towers on the natural numbers that do and do
not remain towers when canonically embedded into the Calkin algebra.
Finally we show that consistently all towers on the natural numbers
remain towers under this embedding.
Keywords:C*-algebras, states, Kadinson-Singer conjecture, ultrafilters, towers Categories:46L03, 03E35 |
14. CJM 2011 (vol 64 pp. 755)
Homotopy Classification of Projections in the Corona Algebra of a Non-simple $C^*$-algebra We study projections in the corona algebra of $C(X)\otimes K$, where K
is the $C^*$-algebra of compact operators on a separable infinite
dimensional Hilbert space and $X=[0,1],[0,\infty),(-\infty,\infty)$,
or $[0,1]/\{ 0,1 \}$. Using BDF's essential codimension, we determine
conditions for a projection in the corona algebra to be liftable to a
projection in the multiplier algebra. We also determine the
conditions for two projections to be equal in $K_0$, Murray-von
Neumann equivalent, unitarily equivalent, or homotopic. In light of
these characterizations, we construct examples showing that the
equivalence notions above are all distinct.
Keywords:essential codimension, continuous field of Hilbert spaces, Corona algebra Categories:46L05, 46L80 |
15. CJM 2011 (vol 64 pp. 544)
On the Simple Inductive Limits of Splitting Interval Algebras with Dimension Drops A K-theoretic classification is given of the simple inductive limits
of finite direct sums of the
type I $C^*$-algebras known as splitting interval algebras with
dimension drops. (These are the subhomogeneous $C^*$-algebras, each
having spectrum a finite union
of points and an open interval, and torsion $\textrm{K}_1$-group.)
Categories:46L05, 46L35 |
16. CJM 2011 (vol 64 pp. 805)
Quantum Random Walks and Minors of Hermitian Brownian Motion Considering quantum random walks, we construct discrete-time
approximations of the eigenvalues processes of minors of Hermitian
Brownian motion. It has been recently proved by Adler, Nordenstam, and
van Moerbeke that the process of eigenvalues of
two consecutive minors of a Hermitian Brownian motion is a Markov
process; whereas, if one considers more than two consecutive minors,
the Markov property fails. We show that there are analog results in
the noncommutative counterpart and establish the Markov property of
eigenvalues of some particular submatrices of Hermitian Brownian
motion.
Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor process Categories:46L53, 60B20, 14L24 |
17. CJM 2011 (vol 64 pp. 573)
Fundamental Group of Simple $C^*$-algebras with Unique Trace III We introduce the fundamental group ${\mathcal F}(A)$ of
a simple $\sigma$-unital $C^*$-algebra $A$ with unique (up to scalar multiple)
densely defined lower semicontinuous trace.
This is a generalization of ``Fundamental Group of Simple
$C^*$-algebras with Unique Trace I and II'' by Nawata and Watatani.
Our definition in this paper makes sense for stably projectionless $C^*$-algebras.
We show that there exist separable stably projectionless $C^*$-algebras such that
their fundamental groups are equal to $\mathbb{R}_+^\times$
by using the classification theorem of Razak and Tsang.
This is a contrast to the unital case in Nawata and Watatani.
This study is motivated by the work of Kishimoto and Kumjian.
Keywords:fundamental group, Picard group, Hilbert module, countable basis, stably projectionless algebra, dimension function Categories:46L05, 46L08, 46L35 |
18. CJM 2011 (vol 64 pp. 455)
On Cardinal Invariants and Generators for von Neumann Algebras We demonstrate how most common cardinal invariants associated with a von
Neumann algebra $\mathcal M$ can be computed from the decomposability number,
$\operatorname{dens}(\mathcal M)$, and the minimal cardinality of a generating
set, $\operatorname{gen}(\mathcal M)$.
Applications include the equivalence of the well-known generator
problem, ``Is every separably-acting von Neumann algebra
singly-generated?", with the formally stronger questions, ``Is every
countably-generated von Neumann algebra singly-generated?" and ``Is
the $\operatorname{gen}$ invariant monotone?" Modulo the generator problem, we
determine the range of the invariant $\bigl( \operatorname{gen}(\mathcal M),
\operatorname{dens}(\mathcal M) \bigr)$,
which is mostly governed by the inequality $\operatorname{dens}(\mathcal M) \leq
\mathfrak C^{\operatorname{gen}(\mathcal M)}$.
Keywords:von Neumann algebra, cardinal invariant, generator problem, decomposability number, representation density Category:46L10 |
19. CJM 2011 (vol 63 pp. 798)
Representing Multipliers of the Fourier Algebra on Non-Commutative $L^p$ Spaces We show that the multiplier algebra of the Fourier algebra on a
locally compact group $G$ can be isometrically represented on a direct
sum on non-commutative $L^p$ spaces associated with the right von
Neumann algebra of $G$. The resulting image is the idealiser of the
image of the Fourier algebra. If these spaces are given their
canonical operator space structure, then we get a completely isometric
representation of the completely bounded multiplier algebra. We make
a careful study of the non-commutative $L^p$ spaces we construct and
show that they are completely isometric to those considered recently
by Forrest, Lee, and Samei. We improve a result of theirs about module
homomorphisms. We suggest a definition of a Figa-Talamanca-Herz
algebra built out of these non-commutative $L^p$ spaces, say
$A_p(\widehat G)$. It is shown that $A_2(\widehat G)$ is isometric to
$L^1(G)$, generalising the abelian situation.
Keywords:multiplier, Fourier algebra, non-commutative $L^p$ space, complex interpolation Categories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52 |
20. CJM 2011 (vol 63 pp. 551)
Topological Free Entropy Dimensions in Nuclear C$^*$-algebras and in Full Free Products of Unital C$^*$-algebras |
Topological Free Entropy Dimensions in Nuclear C$^*$-algebras and in Full Free Products of Unital C$^*$-algebras In the paper, we introduce a new concept,
topological orbit dimension of an $n$-tuple of elements in a unital
C$^{\ast}$-algebra. Using this concept, we conclude that Voiculescu's
topological free
entropy dimension of every finite family of self-adjoint generators of a
nuclear C$^{\ast}$-algebra is less than or equal to $1$. We also show that the
Voiculescu's topological free entropy dimension is additive in the full free
product of some unital C$^{\ast}$-algebras. We show that the unital full free
product of Blackadar and Kirchberg's unital MF
algebras is also an MF algebra. As an application, we obtain that
$\mathop{\textrm{Ext}}(C_{r}^{\ast}(F_{2})\ast_{\mathbb{C}}C_{r}^{\ast}(F_{2}))$ is not a group.
Keywords: topological free entropy dimension, unital C$^{*}$-algebra Categories:46L10, 46L54 |
21. CJM 2011 (vol 63 pp. 381)
A Complete Classification of AI Algebras with the Ideal Property Let $A$ be an AI algebra; that is, $A$ is the $\mbox{C}^{*}$-algebra inductive limit
of a sequence
$$
A_{1}\stackrel{\phi_{1,2}}{\longrightarrow}A_{2}\stackrel{\phi_{2,3}}{\longrightarrow}A_{3}
\longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots,
$$
where
$A_{n}=\bigoplus_{i=1}^{k_n}M_{[n,i]}(C(X^{i}_n))$,
$X^{i}_n$ are $[0,1]$, $k_n$, and
$[n,i]$ are positive integers.
Suppose that $A$ has the
ideal property: each closed two-sided ideal of $A$ is generated by
the projections inside the ideal, as a closed two-sided ideal.
In this article, we give a complete classification of AI algebras with the ideal property.
Keywords:AI algebras, K-group, tracial state, ideal property, classification Categories:46L35, 19K14, 46L05, 46L08 |
22. CJM 2011 (vol 63 pp. 500)
One-Parameter Continuous Fields of Kirchberg Algebras. II Parallel to the first two authors' earlier classification of separable, unita
one-parameter, continuous fields of Kirchberg algebras with torsion free
$\mathrm{K}$-groups supported in one dimension, one-parameter, separable, uni
continuous fields of AF-algebras are classified by their ordered
$\mathrm{K}_0$-sheaves. Effros-Handelman-Shen type theorems are pr separable
unital one-parameter continuous fields of AF-algebras and Kirchberg algebras.
Keywords:continuous fields of C$^*$-algebras, $\mathrm{K}_0$-presheaves, Effros--Handeen type theorem Category:46L35 |
23. CJM 2010 (vol 63 pp. 222)
Limit Theorems for Additive Conditionally Free Convolution
In this paper we determine the limiting distributional behavior for
sums of infinitesimal conditionally free random variables. We show that the weak
convergence of classical convolution and that of conditionally free convolution
are equivalent for measures in an infinitesimal triangular array,
where the measures may have unbounded support. Moreover, we use these
limit theorems to study the conditionally free infinite divisibility. These results
are obtained by complex analytic methods without reference to the
combinatorics of c-free convolution.
Keywords:additive c-free convolution, limit theorems, infinitesimal arrays Categories:46L53, 60F05 |
24. 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 |
25. CJM 2010 (vol 62 pp. 845)
Biflatness and Pseudo-Amenability of Segal Algebras We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, $L^1(G)$, and the Fourier algebra, $A(G)$, of a locally compact group~$G$.
Keywords:Segal algebra, pseudo-amenable Banach algebra, biflat Banach algebra Categories:43A20, 43A30, 46H25, 46H10, 46H20, 46L07 |