Expand all Collapse all | Results 1 - 20 of 20 |
1. CMB Online first
A short note on the continuous Rokhlin property and the universal coefficient theorem in E-theory Let $G$ be a metrizable compact group, $A$ a separable $\mathrm{C}^*$-algebra
and $\alpha\colon G\to\operatorname{Aut}(A)$ a strongly continuous action.
Provided that $\alpha$ satisfies the continuous Rokhlin property,
we show that the property of satisfying the UCT in $E$-theory
passes from $A$ to the crossed product $\mathrm{C}^*$-algebra $A\rtimes_\alpha
G$ and the fixed point algebra $A^\alpha$. This extends a similar
result by Gardella for $KK$-theory in the case of unital
$\mathrm{C}^*$-algebras,
but with a shorter and less technical proof. For circle actions
on separable, unital $\mathrm{C}^*$-algebras with the continuous Rokhlin
property, we establish a connection between the $E$-theory equivalence
class of $A$ and that of its fixed point algebra $A^\alpha$.
Keywords:Rokhlin property, UCT, KK-theory, E-theory, circle actions Categories:46L55, 19K35 |
2. CMB Online first
$L$-functions for Quadratic Characters and Annihilation of Motivic Cohomology Groups Let $n$ be a positive even integer, and let $F$ be a totally real
number field and $L$ be an abelian Galois extension which is totally
real or CM.
Fix a finite set $S$ of primes of $F$ containing the infinite primes
and all those which ramify in
$L$, and let $S_L$ denote the primes of $L$ lying above those in
$S$. Then $\mathcal{O}_L^S$ denotes the ring of $S_L$-integers of $L$.
Suppose that $\psi$ is a quadratic character of the Galois group of
$L$ over $F$. Under the assumption of the motivic Lichtenbaum
conjecture, we obtain a non-trivial annihilator of the motivic
cohomology group
$H_\mathcal{M}^2(\mathcal{O}_L^S,\mathbb{Z}(n))$ from the lead term of the Taylor series for the
$S$-modified Artin $L$-function $L_{L/F}^S(s,\psi)$ at $s=1-n$.
Keywords:motivic cohomology, regulator, Artin L-functions Categories:11R42, 11R70, 14F42, 19F27 |
3. CMB 2014 (vol 58 pp. 51)
Spectral Flows of Dilations of Fredholm Operators Given an essentially unitary contraction and an arbitrary unitary
dilation of it, there is a naturally associated spectral flow which is
shown to be equal to the index of the operator. This result is
interpreted in terms of the $K$-theory of an associated mapping
cone. It is then extended to connect $\mathbb{Z}_2$ indices of odd symmetric
Fredholm operators to a $\mathbb{Z}_2$-valued spectral flow.
Keywords:spectral flow, Fredholm operators, Z2 indices Categories:19K56, 46L80 |
4. CMB Online first
An Explicit Formula for the Generalized Cyclic Shuffle Map We provide an explicit formula for the generalized cyclic shuffle map for cylindrical modules.
Using this formula we give a combinatorial proof of the generalized
cyclic Eilenberg-Zilber theorem.
Keywords:generalized Cyclic shuffle map, Cylindrical module, Eilenberg-Zilber theorem, Cyclic homology Categories:19D55, 05E45 |
5. CMB 2013 (vol 57 pp. 210)
An Explicit Formula for the Generalized Cyclic Shuffle Map We provide an explicit formula for the generalized cyclic shuffle map for cylindrical modules.
Using this formula we give a combinatorial proof of the generalized
cyclic Eilenberg-Zilber theorem.
Keywords:generalized Cyclic shuffle map, Cylindrical module, Eilenberg-Zilber theorem, Cyclic homology Categories:19D55, 05E45 |
6. CMB 2012 (vol 56 pp. 551)
Real Dimension Groups Dimension groups (not countable) that are also real ordered vector
spaces can be obtained as direct limits (over directed sets) of
simplicial real vector spaces (finite dimensional vector spaces with
the coordinatewise ordering), but the directed set is not as
interesting as one would like, i.e., it is not true that a
countable-dimensional real vector space that has interpolation can be
represented as such a direct limit over the a countable directed
set. It turns out this is the case when the group is additionally
simple, and it is shown that the latter have an ordered tensor product
decomposition. In the Appendix, we provide a huge class of polynomial
rings that, with a pointwise ordering, are shown to satisfy
interpolation, extending a result outlined by Fuchs.
Keywords:dimension group, simplicial vector space, direct limit, Riesz interpolation Categories:46A40, 06F20, 13J25, 19K14 |
7. CMB 2011 (vol 55 pp. 762)
Smooth Approximation of Lipschitz Projections We show that any Lipschitz projection-valued function
$p$ on a connected closed Riemannian manifold
can be approximated uniformly by smooth
projection-valued functions $q$ with Lipschitz constant
close to that of $p$.
This answers a question of Rieffel.
Keywords:approximation, Lipschitz constant, projection Category:19K14 |
8. CMB 2010 (vol 54 pp. 82)
Lefschetz Numbers for $C^*$-Algebras
Using Poincar\'e duality, we obtain a formula of Lefschetz type
that computes the Lefschetz number of an endomorphism of a separable
nuclear $C^*$-algebra satisfying Poincar\'e duality and the Kunneth
theorem. (The Lefschetz number of an endomorphism is the graded trace
of the induced map on $\textrm{K}$-theory tensored with $\mathbb{C}$, as in the
classical case.) We then examine endomorphisms of Cuntz--Krieger
algebras $O_A$. An endomorphism has an invariant, which is a
permutation of an infinite set, and the contracting and expanding
behavior of this permutation describes the Lefschetz number of the
endomorphism. Using this description, we derive a closed polynomial
formula for the Lefschetz number depending on the matrix $A$ and the
presentation of the endomorphism.
Categories:19K35, 46L80 |
9. CMB 2007 (vol 50 pp. 227)
AF-Skeletons and Real Rank Zero Algebras with the Corona Factorization Property Let $A$ be a stable, separable, real rank zero $C^{*}$-algebra, and
suppose that $A$ has an AF-skeleton with only finitely many extreme
traces.
Then the corona algebra ${\mathcal M}(A)/A$ is
purely infinite in the sense of Kirchberg and R\o rdam, which implies that
$A$ has the corona factorization property.
Categories:46L80, 46L85, 19K35 |
10. CMB 2007 (vol 50 pp. 268)
On the Lack of Inverses to $C^*$-Extensions Related to Property T Groups Using ideas of S. Wassermann on non-exact $C^*$-algebras and
property T groups, we show that one of his examples of non-invertible
$C^*$-extensions is not semi-invertible. To prove this, we
show that a certain element vanishes in the asymptotic tensor
product. We also show that a modification of the example gives
a $C^*$-extension which is not even invertible up to homotopy.
Keywords:$C^*$-algebra extension, property T group, asymptotic tensor $C^*$-norm, homotopy Categories:19K33, 46L06, 46L80, 20F99 |
11. CMB 2005 (vol 48 pp. 237)
Indecomposable Higher Chow Cycles Let $X$ be a projective smooth variety over a field $k$.
In the first part we show that
an indecomposable element in $CH^2(X,1)$ can be lifted
to an indecomposable element in $CH^3(X_K,2)$ where $K$ is the function
field of 1 variable over $k$. We also show that if $X$ is the self-product
of an elliptic curve over $\Q$ then the $\Q$-vector space of
indecomposable cycles
$CH^3_{ind}(X_\C,2)_\Q$ is infinite dimensional.
In the second part we give a new
definition of the group of indecomposable cycles
of $CH^3(X,2)$ and give an example of non-torsion
cycle in this group.
Categories:14C25, 19D45 |
12. CMB 2005 (vol 48 pp. 221)
An Elementary Proof of Suslin Reciprocity We state and prove an important special case of Suslin reciprocity
that has found significant use in the study of algebraic cycles. An
introductory account is provided of the regulator and norm maps on Milnor
$K_2$-groups (for function fields) employed in the proof.
Categories:19D45, 19E15 |
13. CMB 2005 (vol 48 pp. 50)
Injectivity of the Connecting Maps in AH Inductive Limit Systems Let $A$ be the inductive limit of a system
$$A_{1}\xrightarrow{\phi_{1,2}}A_{2}
\xrightarrow{\phi_{2,3}} A_{3}\longrightarrow \cd
$$
with $A_n =
\bigoplus_{i=1}^{t_n} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$, where
$~X_{n,i}$ is a finite simplicial complex, and $P_{n,i}$ is a
projection in $M_{[n,i]}(C(X_{n,i}))$. In this paper, we will
prove that $A$ can be written as another inductive limit
$$B_1\xrightarrow{\psi_{1,2}} B_2
\xrightarrow{\psi_{2,3}} B_3\longrightarrow \cd $$
with $B_n =
\bigoplus_{i=1}^{s_n} Q_{n,i}M_{\{n,i\}}(C(Y_{n,i}))Q_{n,i}$,
where $Y_{n,i}$ is a finite simplicial complex, and $Q_{n,i}$ is a
projection in $M_{\{n,i\}}(C(Y_{n,i}))$, with the extra condition
that all the maps $\psi_{n,n+1}$ are \emph{injective}. (The
result is trivial if one allows the spaces $Y_{n,i}$ to be
arbitrary compact metrizable spaces.) This result is important
for the classification of simple AH algebras (see
\cite{G5,G6,EGL}. The special case that the spaces $X_{n,i}$ are
graphs is due to the third named author \cite{Li1}.
Categories:46L05, 46L35, 19K14 |
14. CMB 2004 (vol 47 pp. 431)
A Note on $4$-Rank Densities For certain real quadratic number fields, we prove density results concerning
$4$-ranks of tame kernels. We also discuss a relationship between $4$-ranks of
tame kernels and %% $4$-class ranks of narrow ideal class groups. Additionally,
we give a product formula for a local Hilbert symbol.
Categories:11R70, 19F99, 11R11, 11R45 |
15. CMB 2003 (vol 46 pp. 509)
Symmetries of Kirchberg Algebras Let $G_0$ and $G_1$ be countable abelian groups. Let $\gamma_i$ be an
automorphism of $G_i$ of order two. Then there exists a unital
Kirchberg algebra $A$ satisfying the Universal Coefficient Theorem and
with $[1_A] = 0$ in $K_0 (A)$, and an automorphism $\alpha \in
\Aut(A)$ of order two, such that $K_0 (A) \cong G_0$, such that $K_1
(A) \cong G_1$, and such that $\alpha_* \colon K_i (A) \to K_i (A)$ is
$\gamma_i$. As a consequence, we prove that every
$\mathbb{Z}_2$-graded countable module over the representation ring $R
(\mathbb{Z}_2)$ of $\mathbb{Z}_2$ is isomorphic to the equivariant
$K$-theory $K^{\mathbb{Z}_2} (A)$ for some action of $\mathbb{Z}_2$ on
a unital Kirchberg algebra~$A$.
Along the way, we prove that every not necessarily finitely generated
$\mathbb{Z} [\mathbb{Z}_2]$-module which is free as a
$\mathbb{Z}$-module has a direct sum decomposition with only three
kinds of summands, namely $\mathbb{Z} [\mathbb{Z}_2]$ itself and
$\mathbb{Z}$ on which the nontrivial element of $\mathbb{Z}_2$ acts
either trivially or by multiplication by $-1$.
Categories:20C10, 46L55, 19K99, 19L47, 46L40, 46L80 |
16. CMB 2003 (vol 46 pp. 457)
Strongly Perforated $K_{0}$-Groups of Simple $C^{*}$-Algebras In the sequel we construct simple, unital, separable, stable, amenable
$C^{*}$-algebras for which the ordered $K_{0}$-group is strongly
perforated and group isomorphic to $Z$. The particular order structures
to be constructed will be described in detail below, and all
known results of this type will be generalised.
Categories:46, 19 |
17. CMB 2002 (vol 45 pp. 180)
On the Exponent of the ${\nk}_0$-Groups of Virtually Infinite Cyclic Groups It is known that the $K$-theory of a large class of groups can be
computed from the $K$-theory of their virtually infinite cyclic
subgroups. On the other hand, Nil-groups appear to be the obstacle in
calculations involving the $K$-theory of the latter. The main
difficulty in the calculation of Nil-groups is that they are
infinitely generated when they do not vanish. We develop methods for
computing the exponent of ${\nk}_0$-groups that appear in the
calculation of the $K_0$-groups of virtually infinite cyclic groups.
Categories:18F25, 19A31 |
18. CMB 2000 (vol 43 pp. 69)
Type II Spectral Flow and the Eta Invariant The relative eta invariant of Atiyah-Patodi-Singer will be shown to be
expressible in terms of the notion of Type~I and Type~II spectral flow.
Categories:19K56, 46L80 |
19. CMB 2000 (vol 43 pp. 37)
Multiplicative Structure of the Ring $K \bigl( S(T^*\R P^{2n+1}) \bigr)$ We calculate the additive and multiplicative structure of the
ring $K\bigl(S(T^*\R P^{2n+1})\bigr)$ using the eta invariant.
Categories:19L64, 19K56, 55C35 |
20. CMB 1999 (vol 42 pp. 274)
The Bockstein Map is Necessary We construct two non-isomorphic nuclear, stably finite,
real rank zero $C^\ast$-algebras $E$ and $E'$ for which
there is an isomorphism of ordered groups
$\Theta\colon \bigoplus_{n \ge 0} K_\bullet(E;\ZZ/n) \to
\bigoplus_{n \ge 0} K_\bullet(E';\ZZ/n)$ which is compatible
with all the coefficient transformations. The $C^\ast$-algebras
$E$ and $E'$ are not isomorphic since there is no $\Theta$
as above which is also compatible with the Bockstein operations.
By tensoring with Cuntz's algebra $\OO_\infty$ one obtains a pair
of non-isomorphic, real rank zero, purely infinite $C^\ast$-algebras
with similar properties.
Keywords:$K$-theory, torsion coefficients, natural transformations, Bockstein maps, $C^\ast$-algebras, real rank zero, purely infinite, classification Categories:46L35, 46L80, 19K14 |