Expand all Collapse all | Results 51 - 72 of 72 |
51. 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 |
52. 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 |
53. 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 |
54. 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 |
55. CJM 2001 (vol 53 pp. 809)
Asymptotic $K$-Theory for Groups Acting on $\tA_2$ Buildings Let $\Gamma$ be a torsion free lattice in $G=\PGL(3, \mathbb{F})$ where
$\mathbb{F}$ is a nonarchimedean local field. Then $\Gamma$ acts freely
on the affine Bruhat-Tits building $\mathcal{B}$ of $G$ and there is an
induced action on the boundary $\Omega$ of $\mathcal{B}$. The crossed
product $C^*$-algebra $\mathcal{A}(\Gamma)=C(\Omega) \rtimes \Gamma$
depends only on $\Gamma$ and is classified by its $K$-theory. This article
shows how to compute the $K$-theory of $\mathcal{A}(\Gamma)$ and of the
larger class of rank two Cuntz-Krieger algebras.
Keywords:$K$-theory, $C^*$-algebra, affine building Categories:46L80, 51E24 |
56. CJM 2001 (vol 53 pp. 631)
K-Theory of Non-Commutative Spheres Arising from the Fourier Automorphism For a dense $G_\delta$ set of real parameters $\theta$ in $[0,1]$
(containing the rationals) it is shown that the group $K_0 (A_\theta
\rtimes_\sigma \mathbb{Z}_4)$ is isomorphic to $\mathbb{Z}^9$, where
$A_\theta$ is the rotation C*-algebra generated by unitaries $U$, $V$
satisfying $VU = e^{2\pi i\theta} UV$ and $\sigma$ is the Fourier
automorphism of $A_\theta$ defined by $\sigma(U) = V$, $\sigma(V) =
U^{-1}$. More precisely, an explicit basis for $K_0$ consisting of
nine canonical modules is given. (A slight generalization of this
result is also obtained for certain separable continuous fields of
unital C*-algebras over $[0,1]$.) The Connes Chern character $\ch
\colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to H^{\ev} (A_\theta
\rtimes_\sigma \mathbb{Z}_4)^*$ is shown to be injective for a dense
$G_\delta$ set of parameters $\theta$. The main computational tool in
this paper is a group homomorphism $\vtr \colon K_0 (A_\theta
\rtimes_\sigma \mathbb{Z}_4) \to \mathbb{R}^8 \times \mathbb{Z}$
obtained from the Connes Chern character by restricting the
functionals in its codomain to a certain nine-dimensional subspace of
$H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)$. The range of $\vtr$
is fully determined for each $\theta$. (We conjecture that this
subspace is all of $H^{\ev}$.)
Keywords:C*-algebras, K-theory, automorphisms, rotation algebras, unbounded traces, Chern characters Categories:46L80, 46L40, 19K14 |
57. CJM 2001 (vol 53 pp. 592)
Ideal Structure of Multiplier Algebras of Simple $C^*$-algebras With Real Rank Zero We give a description of the monoid of Murray-von Neumann equivalence
classes of projections for multiplier algebras of a wide class of
$\sigma$-unital simple $C^\ast$-algebras $A$ with real rank zero and stable
rank one. The lattice of ideals of this monoid, which is known to be
crucial for understanding the ideal structure of the multiplier
algebra $\mul$, is therefore analyzed. In important cases it is shown
that, if $A$ has finite scale then the quotient of $\mul$ modulo any
closed ideal $I$ that properly contains $A$ has stable rank one. The
intricacy of the ideal structure of $\mul$ is reflected in the fact
that $\mul$ can have uncountably many different quotients, each one
having uncountably many closed ideals forming a chain with respect to
inclusion.
Keywords:$C^\ast$-algebra, multiplier algebra, real rank zero, stable rank, refinement monoid Categories:46L05, 46L80, 06F05 |
58. CJM 2001 (vol 53 pp. 546)
Multi-Sided Braid Type Subfactors We generalise the two-sided construction of examples of pairs of
subfactors of the hyperfinite II$_1$ factor $R$ in [E1]---which arise
by considering unitary braid representations with certain
properties---to multi-sided pairs. We show that the index for the
multi-sided pair can be expressed as a power of that for the
two-sided pair. This construction can be applied to the natural
examples---where the braid representations are obtained in connection
with the representation theory of Lie algebras of types $A$, $B$, $C$,
$D$. We also compute the (first) relative commutants.
Category:46L37 |
59. CJM 2001 (vol 53 pp. 325)
Ext and OrderExt Classes of Certain Automorphisms of $C^*$-Algebras Arising from Cantor Minimal Systems |
Ext and OrderExt Classes of Certain Automorphisms of $C^*$-Algebras Arising from Cantor Minimal Systems Giordano, Putnam and Skau showed that the transformation group
$C^*$-algebra arising from a Cantor minimal system is an $AT$-algebra,
and classified it by its $K$-theory. For approximately inner
automorphisms that preserve $C(X)$, we will determine their classes in
the Ext and OrderExt groups, and introduce a new invariant for the
closure of the topological full group. We will also prove that every
automorphism in the kernel of the homomorphism into the Ext group is
homotopic to an inner automorphism, which extends Kishimoto's result.
Categories:46L40, 46L80, 54H20 |
60. CJM 2001 (vol 53 pp. 355)
$R$-Diagonal Elements and Freeness With Amalgamation The concept of $R$-diagonal element was introduced in \cite{NS2},
and was subsequently found to have applications to several problems
in free probability. In this paper we describe a new approach to
$R$-diagonality, which relies on freeness with amalgamation.
The class of $R$-diagonal elements is enlarged to contain examples
living in non-tracial $*$-probability spaces, such as the
generalized circular elements of \cite{Sh1}.
Category:46L54 |
61. CJM 2001 (vol 53 pp. 51)
A Continuous Field of Projectionless $C^*$-Algebras We use some results about stable relations to show that some of the
simple, stable, projectionless crossed products of $O_2$ by $\bR$
considered by Kishimoto and Kumjian are inductive limits of type I
$C^*$-algebras. The type I $C^*$-algebras that arise are pullbacks
of finite direct sums of matrix algebras over the continuous
functions on the unit interval by finite dimensional $C^*$-algebras.
Categories:46L35, 46L57 |
62. CJM 2001 (vol 53 pp. 161)
Classification of Simple Tracially AF $C^*$-Algebras We prove that pre-classifiable (see 3.1) simple nuclear tracially AF
\CA s (TAF) are classified by their $K$-theory. As a consequence all
simple, locally AH and TAF \CA s are in fact AH algebras (it is known
that there are locally AH algebras that are not AH). We also prove
the following Rationalization Theorem. Let $A$ and $B$ be two unital
separable nuclear simple TAF \CA s with unique normalized traces
satisfying the Universal Coefficient Theorem. If $A$ and $B$ have the
same (ordered and scaled) $K$-theory and $K_0 (A)_+$ is locally
finitely generated, then $A \otimes Q \cong B \otimes Q$, where $Q$ is
the UHF-algebra with the rational $K_0$. Classification results (with
restriction on $K_0$-theory) for the above \CA s are also obtained.
For example, we show that, if $A$ and $B$ are unital nuclear separable
simple TAF \CA s with the unique normalized trace satisfying the UCT
and with $K_1(A) = K_1(B)$, and $A$ and $B$ have the same rational
(scaled ordered) $K_0$, then $A \cong B$. Similar results are also
obtained for some cases in which $K_0$ is non-divisible such as
$K_0(A) = \mathbf{Z} [1/2]$.
Categories:46L05, 46L35 |
63. CJM 2000 (vol 52 pp. 1164)
Perforated Ordered $\K_0$-Groups A simple $\C^*$-algebra is constructed for which the Murray-von
Neumann equivalence classes of projections, with the usual
addition---induced by addition of orthogonal projections---form the
additive semi-group
$$
\{0,2,3,\dots\}.
$$
(This is a particularly simple instance of the phenomenon of
perforation of the ordered $\K_0$-group, which has long been known in
the commutative case---for instance, in the case of the
four-sphere---and was recently observed by the second author in the
case of a simple $\C^*$-algebra.)
Categories:46L35, 46L80 |
64. CJM 2000 (vol 52 pp. 849)
Operator Estimates for Fredholm Modules We study estimates of the type
$$
\Vert \phi(D) - \phi(D_0) \Vert_{\emt} \leq C \cdot \Vert D - D_0
\Vert^{\alpha}, \quad \alpha = \frac12, 1
$$
where $\phi(t) = t(1 + t^2)^{-1/2}$, $D_0 = D_0^*$ is an unbounded
linear operator affiliated with a semifinite von Neumann algebra
$\calM$, $D - D_0$ is a bounded self-adjoint linear operator from
$\calM$ and $(1 + D_0^2)^{-1/2} \in \emt$, where $\emt$ is a symmetric
operator space associated with $\calM$. In particular, we prove that
$\phi(D) - \phi(D_0)$ belongs to the non-commutative $L_p$-space for
some $p \in (1,\infty)$, provided $(1 + D_0^2)^{-1/2}$ belongs to the
non-commutative weak $L_r$-space for some $r \in [1,p)$. In the case
$\calM = \calB (\calH)$ and $1 \leq p \leq 2$, we show that this
result continues to hold under the weaker assumption $(1 +
D_0^2)^{-1/2} \in \calC_p$. This may be regarded as an odd
counterpart of A.~Connes' result for the case of even Fredholm
modules.
Categories:46L50, 46E30, 46L87, 47A55, 58B15 |
65. CJM 2000 (vol 52 pp. 633)
Chern Characters of Fourier Modules Let $A_\theta$ denote the rotation algebra---the universal $C^\ast$-algebra
generated by unitaries $U,V$ satisfying $VU=e^{2\pi i\theta}UV$, where
$\theta$ is a fixed real number. Let $\sigma$ denote the Fourier
automorphism of $A_\theta$ defined by $U\mapsto V$, $V\mapsto U^{-1}$,
and let $B_\theta = A_\theta \rtimes_\sigma \mathbb{Z}/4\mathbb{Z}$ denote
the associated $C^\ast$-crossed product. It is shown that there is a
canonical inclusion $\mathbb{Z}^9 \hookrightarrow K_0(B_\theta)$ for each
$\theta$ given by nine canonical modules. The unbounded trace functionals
of $B_\theta$ (yielding the Chern characters here) are calculated to obtain
the cyclic cohomology group of order zero $\HC^0(B_\theta)$ when
$\theta$ is irrational. The Chern characters of the nine modules---and more
importantly, the Fourier module---are computed and shown to involve techniques
from the theory of Jacobi's theta functions. Also derived are explicit
equations connecting unbounded traces across strong Morita equivalence, which
turn out to be non-commutative extensions of certain theta function equations.
These results provide the basis for showing that for a dense $G_\delta$ set
of values of $\theta$ one has $K_0(B_\theta)\cong\mathbb{Z}^9$ and is
generated by the nine classes constructed here.
Keywords:$C^\ast$-algebras, unbounded traces, Chern characters, irrational rotation algebras, $K$-groups Categories:46L80, 46L40 |
66. 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 |
67. 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 |
68. CJM 1998 (vol 50 pp. 673)
Fredholm modules and spectral flow An {\it odd unbounded\/} (respectively, $p$-{\it summable})
{\it Fredholm module\/} for a unital Banach $\ast$-algebra, $A$, is a pair $(H,D)$
where $A$ is represented on the Hilbert space, $H$, and $D$ is an unbounded
self-adjoint operator on $H$ satisfying:
\item{(1)} $(1+D^2)^{-1}$ is compact (respectively, $\Trace\bigl((1+D^2)^{-(p/2)}\bigr)
<\infty$), and
\item{(2)} $\{a\in A\mid [D,a]$ is bounded$\}$ is a dense
$\ast-$subalgebra of $A$.
If $u$ is a unitary in the dense $\ast-$subalgebra mentioned in (2) then
$$
uDu^\ast=D+u[D,u^{\ast}]=D+B
$$
where $B$ is a bounded self-adjoint operator. The path
$$
D_t^u:=(1-t) D+tuDu^\ast=D+tB
$$
is a ``continuous'' path of unbounded self-adjoint ``Fredholm'' operators.
More precisely, we show that
$$
F_t^u:=D_t^u \bigl(1+(D_t^u)^2\bigr)^{-{1\over 2}}
$$
is a norm-continuous path of (bounded) self-adjoint Fredholm
operators. The {\it spectral flow\/} of this path $\{F_t^u\}$ (or $\{
D_t^u\}$) is roughly speaking the net number of eigenvalues that pass
through $0$ in the positive direction as $t$ runs from $0$ to $1$.
This integer,
$$
\sf(\{D_t^u\}):=\sf(\{F_t^u\}),
$$
recovers the pairing of the $K$-homology class $[D]$ with the $K$-theory
class [$u$].
We use I.~M.~Singer's idea (as did E.~Getzler in the $\theta$-summable
case) to consider the operator $B$ as a parameter in the Banach manifold,
$B_{\sa}(H)$, so that spectral flow can be exhibited as the integral
of a closed $1$-form on this manifold. Now, for $B$ in our manifold,
any $X\in T_B(B_{\sa}(H))$ is given by an $X$ in $B_{\sa}(H)$ as the
derivative at $B$ along the curve $t\mapsto B+tX$ in the manifold.
Then we show that for $m$ a sufficiently large half-integer:
$$
\alpha (X)={1\over {\tilde {C}_m}}\Tr \Bigl(X\bigl(1+(D+B)^2\bigr)^{-m}\Bigr)
$$
is a closed $1$-form. For any piecewise smooth path $\{D_t=D+B_t\}$ with
$D_0$ and $D_1$ unitarily equivalent we show that
$$
\sf(\{D_t\})={1\over {\tilde {C}_m}} \int_0^1\Tr \Bigl({d\over {dt}}
(D_t)(1+D_t^2)^{-m}\Bigr)\,dt
$$
the integral of the $1$-form $\alpha$. If $D_0$ and $D_1$ are not unitarily
equivalent, we must add a pair of correction terms to the right-hand
side. We also prove a bounded finitely summable version of the form:
$$
\sf(\{F_t\})={1\over C_n}\int_0^1\Tr\Bigl({d\over dt}(F_t)(1-F_t^2)^n\Bigr)\,dt
$$
for $n\geq{{p-1}\over 2}$ an integer. The unbounded case is proved by
reducing to the bounded case via the map $D\mapsto F=D(1+D^2
)^{-{1\over 2}}$. We prove simultaneously a type II version of our
results.
Categories:46L80, 19K33, 47A30, 47A55 |
69. CJM 1998 (vol 50 pp. 323)
Purely infinite, simple $C^\ast$-algebras arising from free product constructions Examples of simple, separable, unital, purely infinite
$C^\ast$-algebras are constructed, including:
\item{(1)} some that are not approximately divisible;
\item{(2)} those that arise as crossed products of any of a certain class of
$C^\ast$-algebras by any of a certain class of non-unital endomorphisms;
\item{(3)} those that arise as reduced free products of pairs of
$C^\ast$-algebras with respect to any from a certain class of states.
Categories:46L05, 46L45 |
70. CJM 1997 (vol 49 pp. 1188)
Factorization in the invertible group of a $C^*$-algebra In this paper we consider the following problem:
Given a unital \cs\ $A$ and a collection of elements $S$ in the
identity component of the invertible group of $A$, denoted \ino,
characterize the group of finite products of elements of $S$. The
particular $C^*$-algebras studied in this paper are either
unital purely infinite simple or of the form \tenp, where $A$ is
any \cs\ and $K$ is the compact operators on an infinite dimensional
separable Hilbert space. The types of elements used in the
factorizations are unipotents ($1+$ nilpotent), positive invertibles
and symmetries ($s^2=1$). First we determine the groups of finite
products for each collection of elements in \tenp. Then we give
upper bounds on the number of factors needed in these cases. The main
result, which uses results for \tenp, is that for $A$ unital purely
infinite and simple, \ino\ is generated by each of these collections
of elements.
Category:46L05 |
71. CJM 1997 (vol 49 pp. 963)
Homomorphisms from $C(X)$ into $C^*$-algebras Let $A$ be a simple $C^*$-algebra
with real rank zero, stable rank one and weakly
unperforated $K_0(A)$ of countable rank. We show that
a monomorphism $\phi\colon C(S^2) \to A$ can be approximated
pointwise by homomorphisms from $C(S^2)$ into $A$ with
finite dimensional range if and only if certain index
vanishes. In particular, we show that every homomorphism
$\phi$ from $C(S^2)$ into a UHF-algebra can be approximated
pointwise by homomorphisms from $C(S^2)$ into the UHF-algebra
with finite dimensional range. As an application, we show
that if $A$ is a simple $C^*$-algebra of real rank zero
and is an inductive limit of matrices over $C(S^2)$ then
$A$ is an AF-algebra. Similar results for tori are also
obtained. Classification of ${\bf Hom}\bigl(C(X),A\bigr)$
for lower dimensional spaces is also studied.
Keywords:Homomorphism of $C(S^2)$, approximation, real, rank zero, classification Categories:46L05, 46L80, 46L35 |
72. CJM 1997 (vol 49 pp. 160)
The Classical Limit of Dynamics for Spaces Quantized by an Action of ${\Bbb R}^{\lowercase{d}}$ We have previously shown how to construct a deformation quantization
of any locally compact space on which a vector group acts. Within this
framework we show here that, for a natural class of Hamiltonians, the
quantum evolutions will have the classical evolution as their
classical limit.
Categories:46L60, 46l55, 81S30 |