Expand all Collapse all | Results 1 - 20 of 20 |
1. 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 |
2. CJM 2012 (vol 64 pp. 368)
C$^*$-Algebras over Topological Spaces: Filtrated K-Theory We define the filtrated K-theory of a $\mathrm{C}^*$-algebra over a finite topological space \(X\)
and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over \(X\)
in terms of filtrated K-theory.
For finite spaces with a totally ordered lattice of open subsets, this spectral sequence
becomes an exact sequence as in the Universal Coefficient Theorem,
with the same consequences for classification.
We also exhibit an example where filtrated K-theory is not yet a complete invariant.
We describe two $\mathrm{C}^*$-algebras over a space \(X\) with four points that have isomorphic
filtrated K-theory without being $\mathrm{KK}(X)$-equivalent. For this space \(X\),
we enrich filtrated K-theory by another K-theory functor to a complete invariant
up to $\mathrm{KK}(X)$-equivalence that satisfies a Universal Coefficient Theorem.
Keywords:46L35, 46L80, 46M18, 46M20 Category:19K35 |
3. 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 |
4. CJM 2009 (vol 62 pp. 646)
Reducibility in A_{R}(K), C_{R}(K), and A(K) Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let
$A_{\mathbb R}(K)$ denote the real Banach algebra of all real
symmetric continuous functions on $K$ that are analytic in the
interior $K^\circ$ of $K$, endowed with the supremum norm. We
characterize all unimodular pairs $(f,g)$ in $A_{\mathbb R}(K)^2$
which are reducible.
In addition, for an arbitrary compact $K$ in $\mathbb C$, we give a
new proof (not relying on Banach algebra theory or elementary stable
rank techniques) of the fact that the Bass stable rank of $A(K)$ is
$1$.
Finally, we also characterize all compact real symmetric sets $K$ such
that $A_{\mathbb R}(K)$, respectively $C_{\mathbb R}(K)$, has Bass
stable rank $1$.
Keywords:real Banach algebras, Bass stable rank, topological stable rank, reducibility Categories:46J15, 19B10, 30H05, 93D15 |
5. CJM 2009 (vol 62 pp. 614)
Translation Groupoids and Orbifold Cohomology We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps, giving a mechanism for transferring results from equivariant homotopy theory to the orbifold category. As an application, we use this result to define orbifold versions of a couple of equivariant cohomology theories: K-theory and Bredon cohomology for certain coefficient diagrams.
Keywords:orbifolds, equivariant homotopy theory, translation groupoids, bicategories of fractions Categories:57S15, 55N91, 19L47, 18D05, 18D35 |
6. CJM 2009 (vol 61 pp. 1073)
On the $2$-Rank of the Hilbert Kernel of Number Fields Let $E/F$ be a quadratic extension of
number fields. In this paper, we show that the genus formula for
Hilbert kernels, proved by M. Kolster and A. Movahhedi, gives the
$2$-rank of the Hilbert kernel of $E$ provided that the $2$-primary
Hilbert kernel of $F$ is trivial. However, since the original genus
formula is not explicit enough in a very particular case, we first
develop a refinement of this formula in order to employ it in the
calculation of the $2$-rank of $E$ whenever $F$ is totally real with
trivial $2$-primary Hilbert kernel. Finally, we apply our results to
quadratic, bi-quadratic, and tri-quadratic fields which include
a complete $2$-rank formula for the family of fields
$\Q(\sqrt{2},\sqrt{\delta})$ where $\delta$ is a squarefree integer.
Categories:11R70, 19F15 |
7. CJM 2008 (vol 60 pp. 1387)
On $n$-Dimensional Steinberg Symbols The aim of this work is to provide a new approach for constructing
$n$-dimensional Steinberg symbols on discrete valuation fields from
$(n+1)$-cocycles and to study reciprocity laws on curves related to
these symbols.
Keywords:Steinberg symbols, reciprocity laws, discrete valuation field, algebraic curves Categories:19F15, 19D45, 19C09 |
8. CJM 2006 (vol 58 pp. 419)
Stark's Conjecture and New Stickelberger Phenomena We introduce a new conjecture concerning the construction
of elements in the annihilator ideal
associated to a Galois action on the higher-dimensional algebraic
$K$-groups of rings of integers in number fields. Our conjecture is
motivic in the sense that it involves the (transcendental) Borel
regulator as well as being related to $l$-adic \'{e}tale
cohomology. In addition, the conjecture generalises the well-known
Coates--Sinnott conjecture. For example, for a totally real
extension when $r = -2, -4, -6, \dotsc$ the Coates--Sinnott
conjecture merely predicts that zero annihilates $K_{-2r}$ of the
ring of $S$-integers while our conjecture predicts a non-trivial
annihilator. By way of supporting evidence, we prove the
corresponding (conjecturally equivalent) conjecture for the Galois
action on the \'{e}tale cohomology of the cyclotomic extensions of
the rationals.
Categories:11G55, 11R34, 11R42, 19F27 |
9. CJM 2005 (vol 57 pp. 225)
Unbounded Fredholm Operators and Spectral Flow We study the gap (= ``projection norm'' = ``graph distance'') topology
of the space of all (not necessarily bounded) self-adjoint Fredholm
operators in a separable Hilbert space by the Cayley transform and
direct methods. In particular, we show the surprising result that
this space is connected in contrast to the bounded case. Moreover, we
present a rigorous definition of spectral flow of a path of such
operators (actually alternative but mutually equivalent definitions)
and prove the homotopy invariance. As an example, we discuss operator
curves on manifolds with boundary.
Categories:58J30, 47A53, 19K56, 58J32 |
10. CJM 2005 (vol 57 pp. 180)
On the Size of the Wild Set To every pair of algebraic number fields with isomorphic Witt rings
one can associate a number, called the {\it minimum number of wild
primes}. Earlier investigations have established lower bounds for this
number. In this paper an analysis is presented that expresses the
minimum number of wild primes in terms of the number of wild dyadic
primes. This formula not only gives immediate upper bounds, but can be
considered to be an exact formula for the minimum number of wild
primes.
Categories:11E12, 11E81, 19F15, 11R29 |
11. 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 |
12. 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 |
13. 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 |
14. 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 |
15. CJM 2001 (vol 53 pp. 3)
The Equivariant Grothendieck Groups of the Russell-Koras Threefolds The Russell-Koras contractible threefolds are the smooth affine threefolds
having a hyperbolic $\mathbb{C}^*$-action with quotient isomorphic to the
corresponding quotient of the linear action on the tangent space at the
unique fixed point. Koras and Russell gave a concrete description of all such
threefolds and determined many interesting properties they possess.
We use this description and these properties to compute the equivariant
Grothendieck groups of these threefolds. In addition, we give certain
equivariant invariants of these rings.
Categories:14J30, 19L47 |
16. CJM 2000 (vol 52 pp. 1310)
On the Homology of $\GL_n$ and Higher Pre-Bloch Groups For every integer $n>1$ and infinite field $F$ we construct a spectral
sequence converging to the homology of $\GL_n(F)$ relative to the
group of monomial matrices $\GM_n(F)$. Some entries in $E^2$-terms of
these spectral sequences may be interpreted as a natural
generalization of the Bloch group to higher dimensions. These groups
may be characterized as homology of $\GL_n$ relatively to $\GL_{n-1}$
and $\GM_n$. We apply the machinery developed to the investigation of
stabilization maps in homology of General Linear Groups.
Categories:19D55, 20J06, 18G60 |
17. CJM 2000 (vol 52 pp. 47)
Comparison of $K$-Theory Galois Module Structure Invariants We prove that two, apparently different, class-group valued Galois
module structure invariants associated to the algebraic $K$-groups
of rings of algebraic integers coincide. This comparison result is
particularly important in making explicit calculations.
Categories:11S99, 19F15, 19F27 |
18. CJM 1998 (vol 50 pp. 1048)
Localization theories for simplicial presheaves Most extant localization theories for spaces, spectra and diagrams
of such can be derived from a simple list of axioms which are verified
in broad generality. Several new theories are introduced, including
localizations for simplicial presheaves and presheaves of spectra at
homology theories represented by presheaves of spectra, and a theory
of localization along a geometric topos morphism. The
$f$-localization concept has an analog for simplicial presheaves, and
specializes to the $\hbox{\Bbbvii A}^1$-local theory of
Morel-Voevodsky. This theory answers a question of Soul\'e concerning
integral homology localizations for diagrams of spaces.
Categories:55P60, 19E08, 18F20 |
19. 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 |
20. CJM 1997 (vol 49 pp. 1265)
Hecke algebras and class-group invariant Let $G$ be a finite group. To a set of subgroups of order two we associate
a $\mod 2$ Hecke algebra and construct a homomorphism, $\psi$, from its
units to the class-group of ${\bf Z}[G]$. We show that this homomorphism
takes values in the subgroup, $D({\bf Z}[G])$. Alternative constructions of
Chinburg invariants arising from the Galois module structure of
higher-dimensional algebraic $K$-groups of rings of algebraic integers
often differ by elements in the image of $\psi$. As an application we show
that two such constructions coincide.
Categories:16S34, 19A99, 11R65 |