Expand all Collapse all | Results 26 - 50 of 68 |
26. 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 |
27. CJM 2008 (vol 60 pp. 703)
$\mathcal{Z}$-Stable ASH Algebras The Jiang--Su algebra $\mathcal{Z}$ has come to prominence in the
classification program for nuclear $C^*$-algebras of late, due
primarily to the fact that Elliott's classification conjecture in its
strongest form predicts that all simple, separable, and nuclear
$C^*$-algebras with unperforated $\mathrm{K}$-theory will absorb
$\mathcal{Z}$ tensorially, i.e., will be $\mathcal{Z}$-stable. There
exist counterexamples which suggest that the conjecture will only hold
for simple, nuclear, separable and $\mathcal{Z}$-stable
$C^*$-algebras. We prove that virtually all classes of nuclear
$C^*$-algebras for which the Elliott conjecture has been confirmed so
far consist of $\mathcal{Z}$-stable $C^*$-algebras. This
follows in large part from the following result, also proved herein:
separable and approximately divisible $C^*$-algebras are
$\mathcal{Z}$-stable.
Keywords:nuclear $C^*$-algebras, K-theory, classification Categories:46L85, 46L35 |
28. CJM 2008 (vol 60 pp. 189)
Furstenberg Transformations and Approximate Conjugacy Let $\alpha$ and
$\beta$ be two Furstenberg transformations on $2$-torus associated
with irrational numbers $\theta_1,$ $\theta_2,$ integers $d_1, d_2$ and Lipschitz functions
$f_1$ and $f_2$. It is shown that $\alpha$ and $\beta$ are approximately conjugate in a
measure theoretical sense if (and only
if) $\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z.$ Closely related to the classification of simple
amenable \CAs, it is shown that $\af$ and $\bt$ are approximately $K$-conjugate if (and only if)
$\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z$ and $|d_1|=|d_2|.$ This
is also shown to be equivalent to the condition that the associated crossed product \CAs are isomorphic.
Keywords:Furstenberg transformations, approximate conjugacy Categories:37A55, 46L35 |
29. CJM 2007 (vol 59 pp. 966)
Operator Amenability of the Fourier Algebra in the $\cb$-Multiplier Norm Let $G$ be a locally compact group, and let $A_{\cb}(G)$ denote the
closure of $A(G)$, the Fourier algebra of $G$, in the space of completely
bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group
such that $\cstar(G)$ is residually finite-dimensional, we show that
$A_{\cb}(G)$ is operator amenable. In particular,
$A_{\cb}(\free_2)$ is operator amenable even though $\free_2$, the free
group in two generators, is not an amenable group. Moreover, we show that
if $G$ is a discrete group such that $A_{\cb}(G)$ is operator amenable,
a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$
if and only if it has an approximate identity bounded in the $\cb$-multiplier
norm.
Keywords:$\cb$-multiplier norm, Fourier algebra, operator amenability, weak amenability Categories:43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25 |
30. CJM 2007 (vol 59 pp. 343)
Weak Semiprojectivity in Purely Infinite Simple $C^*$-Algebras Let $A$ be a separable amenable purely infinite simple \CA which
satisfies the Universal Coefficient Theorem. We prove that $A$ is
weakly semiprojective if and only if $K_i(A)$ is a countable
direct sum of finitely generated groups ($i=0,1$). Therefore, if
$A$ is such a \CA, for any $\ep>0$ and any finite subset ${\mathcal
F}\subset A$ there exist $\dt>0$ and a finite subset ${\mathcal
G}\subset A$ satisfying the following: for any contractive
positive linear map $L: A\to B$ (for any \CA $B$) with $
\|L(ab)-L(a)L(b)\|<\dt$ for $a, b\in {\mathcal G}$
there exists a homomorphism $h\from A\to B$ such that
$ \|h(a)-L(a)\|<\ep$ for $a\in {\mathcal F}$.
Keywords:weakly semiprojective, purely infinite simple $C^*$-algebras Categories:46L05, 46L80 |
31. CJM 2006 (vol 58 pp. 1144)
Partial $*$-Automorphisms, Normalizers, and Submodules in Monotone Complete $C^*$-Algebras For monotone complete $C^*$-algebras
$A\subset B$ with $A$ contained in $B$ as a monotone closed
$C^*$-subalgebra, the relation $X = AsA$
gives a bijection between the set of all
monotone closed linear subspaces $X$ of $B$ such that
$AX + XA \subset X$
and
$XX^* + X^*X \subset A$
and a set of certain partial
isometries $s$ in the ``normalizer" of $A$ in $B$,
and similarly for the map $s \mapsto \Ad s$
between the latter set and a set of certain ``partial $*$-automorphisms"
of $A$.
We introduce natural inverse semigroup
structures in the set of such $X$'s and the set of
partial $*$-automorphisms of $A$, modulo a certain relation, so that
the composition of these maps induces an inverse semigroup
homomorphism between them.
For a large enough $B$ the homomorphism becomes surjective and
all the partial $*$-automorphisms of
$A$ are realized via partial isometries in $B$.
In particular, the inverse semigroup associated with
a type ${\rm II}_1$ von Neumann factor,
modulo the outer automorphism group,
can be viewed as the fundamental group of the factor.
We also consider the $C^*$-algebra version of these results.
Categories:46L05, 46L08, 46L40, 20M18 |
32. CJM 2006 (vol 58 pp. 1268)
Gauge-Invariant Ideals in the $C^*$-Algebras of Finitely Aligned Higher-Rank Graphs We produce a complete description of the lattice of gauge-invariant
ideals in $C^*(\Lambda)$ for a finitely aligned $k$-graph
$\Lambda$. We provide a condition on $\Lambda$ under which every ideal
is gauge-invariant. We give conditions on $\Lambda$ under which
$C^*(\Lambda)$ satisfies the hypotheses of the Kirchberg--Phillips
classification theorem.
Keywords:Graphs as categories, graph algebra, $C^*$-algebra Category:46L05 |
33. CJM 2006 (vol 58 pp. 768)
Decomposability of von Neumann Algebras and the Mazur Property of Higher Level The decomposability
number of a von Neumann algebra $\m$ (denoted by $\dec(\m)$) is the
greatest cardinality of a family of pairwise orthogonal non-zero
projections in $\m$. In this paper, we explore the close
connection between $\dec(\m)$ and the cardinal level of the Mazur
property for the predual $\m_*$ of $\m$, the study of which was
initiated by the second author. Here, our main focus is on
those von Neumann algebras whose preduals constitute such
important Banach algebras on a locally compact group $G$ as the
group algebra $\lone$, the Fourier algebra $A(G)$, the measure
algebra $M(G)$, the algebra $\luc^*$, etc. We show that for
any of these von Neumann algebras, say $\m$, the cardinal number
$\dec(\m)$ and a certain cardinal level of the Mazur property of $\m_*$
are completely encoded in the underlying group structure. In fact,
they can be expressed precisely by two dual cardinal
invariants of $G$: the compact covering number $\kg$ of $G$ and
the least cardinality $\bg$ of an open basis at the identity of
$G$. We also present an application of the Mazur property of higher
level to the topological centre problem for the Banach algebra
$\ag^{**}$.
Keywords:Mazur property, predual of a von Neumann algebra, locally compact group and its cardinal invariants, group algebra, Fourier algebra, topological centre Categories:22D05, 43A20, 43A30, 03E55, 46L10 |
34. 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 |
35. 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 |
36. CJM 2005 (vol 57 pp. 1056)
Hyperbolic Group $C^*$-Algebras and Free-Product $C^*$-Algebras as Compact Quantum Metric Spaces Let $\ell$ be a length function on a group $G$, and let $M_{\ell}$
denote the
operator of pointwise multiplication by $\ell$ on $\bell^2(G)$.
Following Connes,
$M_{\ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It defines a
Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of
$C_r^*(G)$. We show that if $G$ is a hyperbolic group and if $\ell$ is
a word-length function on $G$, then the topology from this metric
coincides with the
weak-$*$ topology (our definition of a ``compact quantum metric
space''). We show that a convenient framework is that of filtered
$C^*$-algebras which satisfy a suitable ``Haagerup-type'' condition. We
also use this
framework to prove an analogous fact for certain reduced
free products of $C^*$-algebras.
Categories:46L87, 20F67, 46L09 |
37. CJM 2005 (vol 57 pp. 351)
Extensions by Simple $C^*$-Algebras: Quasidiagonal Extensions Let $A$ be an amenable separable $C^*$-algebra and $B$ be a non-unital
but $\sigma$-unital simple $C^*$-algebra with continuous scale.
We show that two essential extensions
$\tau_1$ and $\tau_2$ of $A$ by $B$ are approximately
unitarily equivalent if and only if
$$
[\tau_1]=[\tau_2] \text{ in } KL(A, M(B)/B).
$$
If $A$ is assumed to satisfy the Universal Coefficient Theorem,
there is a bijection from approximate unitary equivalence
classes of the above mentioned extensions to
$KL(A, M(B)/B)$.
Using $KL(A, M(B)/B)$, we compute exactly when an essential extension
is quasidiagonal. We show that quasidiagonal extensions
may not be approximately trivial.
We also study the approximately trivial extensions.
Keywords:Extensions, Simple $C^*$-algebras Categories:46L05, 46L35 |
38. 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 |
39. CJM 2004 (vol 56 pp. 1237)
Central Sequence Algebras of a Purely Infinite Simple $C^{*}$-algebra We are concerned with a unital separable nuclear purely infinite
simple $C^{*}$-algebra\ $A$ satisfying UCT with a Rohlin flow, as a
continuation of~\cite{Kismh}. Our first result (which is
independent of the Rohlin flow) is to characterize when two {\em
central} projections in $A$ are equivalent by a {\em central}
partial isometry. Our second result shows that the K-theory of
the central sequence algebra $A'\cap A^\omega$ (for an $\omega\in
\beta\N\setminus\N$) and its {\em fixed point} algebra under the
flow are the same (incorporating the previous result). We will
also complete and supplement the characterization result of the
Rohlin property for flows stated in~ \cite{Kismh}.
Category:46L40 |
40. CJM 2004 (vol 56 pp. 926)
K-Homology of the Rotation Algebras $A_{\theta}$ We study the K-homology of the rotation algebras
$A_{\theta}$ using the six-term cyclic sequence
for the K-homology of a crossed product by
${\bf Z}$. In the case that $\theta$ is irrational,
we use Pimsner and Voiculescu's work on AF-embeddings
of the $A_{\theta}$ to search for the missing
generator of the even K-homology.
Categories:58B34, 19K33, 46L |
41. CJM 2004 (vol 56 pp. 983)
Fubini's Theorem for Ultraproducts \\of Noncommutative $L_p$-Spaces Let $(\M_i)_{i\in I}$, $(\N_j)_{j\in J}$ be families of von
Neumann algebras and $\U$, $\U'$ be ultrafilters in $I$, $J$,
respectively. Let $1\le p<\infty$ and $\nen$. Let $x_1$,\dots,$x_n$ in
$\prod L_p(\M_i)$ and $y_1$,\dots,$y_n$ in $\prod L_p(\N_j)$ be
bounded families. We show the following equality
$$
\lim_{i,\U} \lim_{j,\U'} \Big\| \summ_{k=1}^n x_k(i)\otimes
y_k(j)\Big\|_{L_p(\M_i\otimes \N_j)} = \lim_{j,\U'} \lim_{i,\U}
\Big\| \summ_{k=1}^n x_k(i)\otimes y_k(j)\Big\|_{L_p(\M_i\otimes \N_j)} .
$$
For $p=1$ this Fubini type result is related to the local
reflexivity of duals of $C^*$-algebras. This fails for $p=\infty$.
Keywords:noncommutative $L_p$-spaces, ultraproducts Categories:46L52, 46B08, 46L07 |
42. 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 |
43. CJM 2004 (vol 56 pp. 225)
Complex Uniform Convexity and Riesz Measure The norm on a Banach space gives rise to a subharmonic function on the
complex plane for which the distributional Laplacian gives a Riesz measure.
This measure is calculated explicitly here for Lebesgue $L^p$ spaces and the
von~Neumann-Schatten trace ideals. Banach spaces that are $q$-uniformly
$\PL$-convex in the sense of Davis, Garling and Tomczak-Jaegermann are
characterized in terms of the mass distribution of this measure. This gives
a new proof that the trace ideals $c^p$ are $2$-uniformly $\PL$-convex for
$1\leq p\leq 2$.
Keywords:subharmonic functions, Banach spaces, Schatten trace ideals Categories:46B20, 46L52 |
44. CJM 2004 (vol 56 pp. 3)
Locally Compact Pro-$C^*$-Algebras Let $X$ be a locally compact non-compact Hausdorff topological space. Consider
the algebras $C(X)$, $C_b(X)$, $C_0(X)$, and $C_{00}(X)$ of respectively arbitrary,
bounded, vanishing at infinity, and compactly supported continuous functions on $X$.
Of these, the second and third are $C^*$-algebras, the fourth is a normed algebra,
whereas the first is only a topological algebra (it is indeed a pro-$C^\ast$-algebra).
The interesting fact about these algebras is that if one of them is given, the
others can be obtained using functional analysis tools. For instance, given the
$C^\ast$-algebra $C_0(X)$, one can get the other three algebras by
$C_{00}(X)=K\bigl(C_0(X)\bigr)$, $C_b(X)=M\bigl(C_0(X)\bigr)$, $C(X)=\Gamma\bigl(
K(C_0(X))\bigr)$, where the right hand sides are the Pedersen ideal, the
multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of
$C_0(X)$, respectively. In this article we consider the possibility of these
transitions for general $C^\ast$-algebras. The difficult part is to start with a
pro-$C^\ast$-algebra $A$ and to construct a $C^\ast$-algebra $A_0$ such that
$A=\Gamma\bigl(K(A_0)\bigr)$. The pro-$C^\ast$-algebras for which this is
possible are called {\it locally compact\/} and we have characterized them using
a concept similar to that of an approximate identity.
Keywords:pro-$C^\ast$-algebras, projective limit, multipliers of Pedersen's ideal Categories:46L05, 46M40 |
45. 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 |
46. CJM 2002 (vol 54 pp. 1100)
The Operator Biprojectivity of the Fourier Algebra In this paper, we investigate projectivity in the category of operator
spaces. In particular, we show that the Fourier algebra of a locally
compact group $G$ is operator biprojective if and only if $G$ is
discrete.
Keywords:locally compact group, Fourier algebra, operator space, projective Categories:13D03, 18G25, 43A95, 46L07, 22D99 |
47. CJM 2002 (vol 54 pp. 694)
Cuntz Algebra States Defined by Implementers of Endomorphisms of the $\CAR$ Algebra We investigate representations of the Cuntz algebra $\mathcal{O}_2$
on antisymmetric Fock space $F_a (\mathcal{K}_1)$ defined by
isometric implementers of certain quasi-free endomorphisms of the
CAR algebra in pure quasi-free states $\varphi_{P_1}$. We pay
corresponding to these representations and the Fock special
attention to the vector states on $\mathcal{O}_2$ vacuum, for which
we obtain explicit formulae. Restricting these states to the
gauge-invariant subalgebra $\mathcal{F}_2$, we find that for
natural choices of implementers, they are again pure quasi-free and
are, in fact, essentially the states $\varphi_{P_1}$. We proceed to
consider the case for an arbitrary pair of implementers, and deduce
that these Cuntz algebra representations are irreducible, as are their
restrictions to $\mathcal{F}_2$.
The endomorphisms of $B \bigl( F_a (\mathcal{K}_1) \bigr)$ associated
with these representations of $\mathcal{O}_2$ are also considered.
Categories:46L05, 46L30 |
48. CJM 2002 (vol 54 pp. 138)
On the Classification of Simple Stably Projectionless $\C^*$-Algebras It is shown that simple stably projectionless $\C^S*$-algebras which
are inductive limits of certain specified building blocks with trivial
$\K$-theory are classified by their cone of positive traces with
distinguished subset. This is the first example of an isomorphism
theorem verifying the conjecture of Elliott for a subclass of the
stably projectionless algebras.
Categories:46L35, 46L05 |
49. CJM 2001 (vol 53 pp. 1223)
Classification of Certain Simple $C^*$-Algebras with Torsion in $K_1$ We show that the Elliott invariant is a classifying invariant for the
class of $C^*$-algebras that are simple unital infinite dimensional
inductive limits of finite direct sums of building blocks of the form
$$
\{f \in C(\T) \otimes M_n : f(x_i) \in M_{d_i}, i = 1,2,\dots,N\},
$$
where $x_1,x_2,\dots,x_N \in \T$, $d_1,d_2,\dots,d_N$ are integers
dividing $n$, and $M_{d_i}$ is embedded unitally into $M_n$.
Furthermore we prove existence and uniqueness theorems for
$*$-homomorphisms between such algebras and we identify the range of
the invariant.
Categories:46L80, 19K14, 46L05 |
50. CJM 2001 (vol 53 pp. 979)
Ranks of Algebras of Continuous $C^*$-Algebra Valued Functions We prove a number of results about the stable and particularly the
real ranks of tensor products of \ca s under the assumption that one
of the factors is commutative. In particular, we prove the following:
{\raggedright
\begin{enumerate}[(5)]
\item[(1)] If $X$ is any locally compact $\sm$-compact Hausdorff space
and $A$ is any \ca, then\break
$\RR \bigl( C_0 (X) \otimes A \bigr) \leq
\dim (X) + \RR(A)$.
\item[(2)] If $X$ is any locally compact Hausdorff space and $A$ is
any \pisca, then $\RR \bigl( C_0 (X) \otimes A \bigr) \leq 1$.
\item[(3)] $\RR \bigl( C ([0,1]) \otimes A \bigr) \geq 1$ for any
nonzero \ca\ $A$, and $\sr \bigl( C ([0,1]^2) \otimes A \bigr) \geq 2$
for any unital \ca\ $A$.
\item[(4)] If $A$ is a unital \ca\ such that $\RR(A) = 0$, $\sr (A) =
1$, and $K_1 (A) = 0$, then\break
$\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$.
\item[(5)] There is a simple separable unital nuclear \ca\ $A$ such
that $\RR(A) = 1$ and\break
$\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$.
\end{enumerate}}
Categories:46L05, 46L52, 46L80, 19A13, 19B10 |