Expand all Collapse all | Results 1 - 25 of 32 |
1. CMB Online first
On the Structure of Cuntz Semigroups in (Possibly) Nonunital C*-algebras We examine the ranks of operators in semi-finite $\mathrm{C}^*$-algebras
as measured by their densely defined lower semicontinuous traces.
We first prove that a unital simple $\mathrm{C}^*$-algebra whose
extreme tracial boundary is nonempty and finite contains positive
operators of every possible rank, independent of the property
of strict comparison. We then turn to nonunital simple algebras
and establish criteria that imply that the Cuntz semigroup is
recovered functorially from the Murray-von Neumann semigroup
and the space of densely defined lower semicontinuous traces.
Finally, we prove that these criteria are satisfied by not-necessarily-unital
approximately subhomogeneous algebras of slow dimension growth.
Combined with results of the first-named author, this shows that
slow dimension growth coincides with $\mathcal Z$-stability,
for approximately subhomogeneous algebras.
Keywords:nuclear C*-algebras, Cuntz semigroup, dimension functions, stably projectionless C*-algebras, approximately subhomogeneous C*-algebras, slow dimension growth Categories:46L35, 46L05, 46L80, 47L40, 46L85 |
2. CMB Online first
On Global Dimensions of Tree Type Finite Dimensional Algebras A formula is provided to
explicitly describe global dimensions of all kinds of tree type
finite dimensional $k-$algebras for $k$ an algebraic closed field.
In particular, it is pointed out that if the underlying tree type
quiver has $n$ vertices, then the maximum of possible global
dimensions is $n-1$.
Keywords:global dimension, tree type finite dimensional $k-$algebra, quiver Categories:16D40, 16E10, , 16G20 |
3. CMB Online first
Complexifying Lie Group Actions on Homogeneous Manifolds of Non-compact Dimension Two If $X$ is a connected complex manifold with $d_X = 2$ that admits a (connected) Lie group $G$
acting transitively as a group of holomorphic transformations, then the action extends to an action of the
complexification $\widehat{G}$ of $G$ on $X$ except when
either the unit disk in the complex plane
or a strictly pseudoconcave homogeneous complex manifold is
the base or fiber of some homogeneous fibration of $X$.
Keywords:homogeneous complex manifold, non-compact dimension two, complexification Category:32M10 |
4. CMB 2013 (vol 57 pp. 245)
Assouad-Nagata Dimension of Wreath Products of Groups Consider the wreath product $H\wr G$, where $H\ne 1$ is finite and $G$ is finitely generated.
We show that the Assouad-Nagata dimension $\dim_{AN}(H\wr G)$ of $H\wr G$
depends on the growth of $G$ as follows:
\par If the growth of $G$ is not bounded by a linear function, then $\dim_{AN}(H\wr G)=\infty$,
otherwise $\dim_{AN}(H\wr G)=\dim_{AN}(G)\leq 1$.
Keywords:Assouad-Nagata dimension, asymptotic dimension, wreath product, growth of groups Categories:54F45, 55M10, 54C65 |
5. CMB 2013 (vol 57 pp. 335)
Alexandroff Manifolds and Homogeneous Continua ny homogeneous,
metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq
1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal
domain.
This implies that any homogeneous $n$-dimensional metric $ANR$-continuum is a $V^n$-continuum in the sense of Alexandroff.
We also prove that any finite-dimensional homogeneous metric continuum
$X$, satisfying $\check{H}^n(X;G)\neq 0$ for some group $G$ and $n\geq
1$, cannot be separated by
a compactum $K$ with $\check{H}^{n-1}(K;G)=0$ and $\dim_G K\leq
n-1$. This provides a partial answer to a question of
Kallipoliti-Papasoglu
whether any two-dimensional homogeneous Peano continuum cannot be separated by arcs.
Keywords:Cantor manifold, cohomological dimension, cohomology groups, homogeneous compactum, separator, $V^n$-continuum Categories:54F45, 54F15 |
6. CMB 2013 (vol 56 pp. 795)
Upper Bounds for the Essential Dimension of $E_7$ This paper gives a new upper bound for the essential dimension and the
essential 2-dimension of the split simply connected group of type
$E_7$ over a field of characteristic not 2 or 3. In particular,
$\operatorname{ed}(E_7) \leq 29$, and $\operatorname{ed}(E_7;2) \leq 27$.
Keywords:$E_7$, essential dimension, stabilizer in general position Categories:20G15, 20G41 |
7. CMB 2013 (vol 57 pp. 159)
Strongly $0$-dimensional Modules In a multiplication module, prime submodules have the property, if a prime
submodule contains a finite intersection of submodules then one of the
submodules is contained in the prime submodule. In this paper, we generalize
this property to infinite intersection of submodules and call such prime
submodules strongly prime submodule. A multiplication module in which every
prime submodule is strongly prime will be called strongly 0-dimensional
module. It is also an extension of strongly 0-dimensional rings. After
this we investigate properties of strongly 0-dimensional modules and give
relations of von Neumann regular modules, Q-modules and strongly
0-dimensional modules.
Keywords:strongly 0-dimensional rings, Q-module, Von Neumann regular module Categories:13C99, 16D10 |
8. CMB 2013 (vol 56 pp. 745)
Dimension Functions of Self-Affine Scaling Sets In this paper, the dimension function of a self-affine generalized scaling set associated with an $n\times n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK = (K+d_1) \cup (K+d_2)$, where $B=A^t$, $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$, and $d_1,d_2\in\mathbb{R}^n$. We show that the dimension function of $K$ must be constant if either $n=1$ or $2$ or one of the digits is $0$, and that it is bounded by $2\lvert K\rvert$ for any $n$.
Keywords:scaling set, self-affine tile, orthonormal multiwavelet, dimension function Category:42C40 |
9. CMB 2012 (vol 56 pp. 737)
On the Radius of Comparison of a Commutative C*-algebra Let $X$ be a compact metric space. A lower bound for the radius of
comparison of the C*-algebra $\operatorname{C}(X)$ is given in terms of
$\operatorname{dim}_{\mathbb{Q}} X$, where $\operatorname{dim}_{\mathbb{Q}} X $ is
the cohomological dimension with rational coefficients. If
$\operatorname{dim}_{\mathbb{Q}} X =\operatorname{dim} X=d$, then the
radius of comparison of the C*-algebra $\operatorname{C}(X)$ is $\max\{0, (d-1)/2-1\}$ if $d$ is odd, and must be either $d/2-1$ or $d/2-2$ if $d$ is even (the possibility of $d/2-1$ does occur, but we do not know if the possibility of $d/2-2$ also can occur).
Keywords:Cuntz semigroup, comparison radius, cohomology dimension, covering dimension |
10. CMB 2012 (vol 56 pp. 683)
Envelope Dimension of Modules and the Simplified Radical Formula We introduce and investigate the notion of envelope dimension of
commutative rings and modules over them. In particular, we show that
the envelope dimension of a ring, $R$, is equal to that of the
$R$-module $R^{(\mathbb{N})}$. Also we prove that the Krull dimension of a
ring is no more than its envelope dimension and characterize
Noetherian rings for which these two dimensions are equal. Moreover we
generalize and study the concept of simplified radical formula for
modules, which
we defined in an earlier paper.
Keywords:envelope dimension, simplified radical formula, prime submodule Categories:13A99, 13C99, 13C13, 13E05 |
11. 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 |
12. CMB 2012 (vol 56 pp. 491)
A Note on Homological Dimensions of Artinian Local Cohomology Modules Let $(R,{\frak m})$ be a non-zero commutative Noetherian local ring
(with identity), $M$ be a non-zero finitely generated $R$-module. In
this paper for any ${\frak p}\in {\rm Spec}(R)$ we show that
$
\operatorname{{\rm injdim_{_{R_{\frak p}}}}}
H^{i-\dim(R/{\frak p})}_{{\frak p}R_{\frak p}}(M_{\frak p})$ and
${\rm fd}_{R_{\p}} H^{i-\dim(R/{\frak p})}_{{\frak p}R_{\frak
p}}(M_{\frak p})$ are bounded from above by $
\operatorname{{\rm injdim_{_{R}}}}
H^i_{\frak
m}(M)$ and
$ {\rm fd}_R H^i_{\frak m}(M)$ respectively, for all integers $i\geq \dim(R/{\frak p})$.
Keywords:cofinite modules, flat dimension, injective dimension, Krull dimension, local cohomology Category:13D45 |
13. CMB 2012 (vol 56 pp. 630)
Inverse Semigroups and Sheu's Groupoid for the Odd Dimensional Quantum Spheres In this paper, we give a different proof of the fact that the odd dimensional
quantum spheres are groupoid $C^{*}$-algebras. We show that the $C^{*}$-algebra
$C(S_{q}^{2\ell+1})$ is generated by an inverse semigroup $T$ of partial
isometries. We show that the groupoid $\mathcal{G}_{tight}$ associated with the
inverse semigroup $T$ by Exel is exactly the same as the groupoid
considered by Sheu.
Keywords:inverse semigroups, groupoids, odd dimensional quantum spheres Categories:46L99, 20M18 |
14. CMB 2011 (vol 56 pp. 354)
The Sizes of Rearrangements of Cantor Sets A linear Cantor set $C$ with zero Lebesgue measure is associated with
the countable collection of the bounded complementary open intervals. A
rearrangment of $C$ has the same lengths of its complementary
intervals, but with different locations. We study the Hausdorff and packing
$h$-measures and dimensional properties of the set of all rearrangments of
some given $C$ for general dimension functions $h$. For each set of
complementary lengths, we construct a Cantor set rearrangement which has the
maximal Hausdorff and the minimal packing $h$-premeasure, up to a constant.
We also show that if the packing measure of this Cantor set is positive,
then there is a rearrangement which has infinite packing measure.
Keywords:Hausdorff dimension, packing dimension, dimension functions, Cantor sets, cut-out set Categories:28A78, 28A80 |
15. CMB 2011 (vol 56 pp. 292)
Quasisymmetrically Minimal Moran Sets M. Hu and S. Wen considered quasisymmetrically minimal uniform Cantor
sets of Hausdorff dimension $1$, where at the $k$-th set one removes
from each interval $I$ a certain number $n_{k}$ of open subintervals
of length $c_{k}|I|$, leaving $(n_{k}+1)$ closed subintervals of
equal length. Quasisymmetrically Moran sets of Hausdorff dimension $1$
considered in the paper are more general than uniform Cantor sets in
that neither the open subintervals nor the closed subintervals are
required to be of equal length.
Keywords:quasisymmetric, Moran set, Hausdorff dimension Categories:28A80, 54C30 |
16. CMB 2011 (vol 55 pp. 339)
From Matrix to Operator Inequalities We generalize LÃ¶wner's method for proving that matrix monotone
functions are operator monotone. The relation $x\leq y$ on bounded
operators is our model for a definition of $C^{*}$-relations
being residually finite dimensional.
Our main result is a meta-theorem about theorems involving relations
on bounded operators. If we can show there are residually finite dimensional
relations involved and verify a technical condition, then such a
theorem will follow from its restriction to matrices.
Applications are shown regarding norms of exponentials, the norms
of commutators, and "positive" noncommutative $*$-polynomials.
Keywords:$C*$-algebras, matrices, bounded operators, relations, operator norm, order, commutator, exponential, residually finite dimensional Categories:46L05, 47B99 |
17. CMB 2011 (vol 54 pp. 619)
Artinian and Non-Artinian Local Cohomology Modules Let $M$ be a finite module over a commutative noetherian ring $R$.
For ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, the relations between
cohomological dimensions of $M$ with respect to $\mathfrak{a},
\mathfrak{b}$,
$\mathfrak{a}\cap\mathfrak{b}$ and $\mathfrak{a}+ \mathfrak{b}$ are studied. When $R$ is local, it is
shown that $M$ is generalized Cohen-Macaulay if there exists an
ideal $\mathfrak{a}$ such that all local cohomology modules of $M$ with
respect to $\mathfrak{a}$ have finite lengths. Also, when $r$ is an integer
such that $0\leq r< \dim_R(M)$, any maximal element $\mathfrak{q}$ of the
non-empty set of ideals $\{\mathfrak{a} : \textrm{H}_\mathfrak{a}^i(M)
$ is not artinian for
some $ i, i\geq r \}$ is a prime ideal, and all Bass numbers
of $\textrm{H}_\mathfrak{q}^i(M)$ are finite for all $i\geq r$.
Keywords:local cohomology modules, cohomological dimensions, Bass numbers Categories:13D45, 13E10 |
18. CMB 2010 (vol 53 pp. 629)
Asymptotic Dimension of Proper CAT(0) Spaces that are Homeomorphic to the Plane In this paper, we investigate
a proper CAT(0) space $(X,d)$
that is homeomorphic to $\mathbb R^2$ and
we show that the asymptotic dimension $\operatorname{asdim} (X,d)$ is
equal to $2$.
Keywords:asymptotic dimension, CAT(0) space, plane Categories:20F69, 54F45, 20F65 |
19. CMB 2010 (vol 53 pp. 564)
On $6$-Dimensional Nearly KÃ¤hler Manifolds In this paper we give a sufficient condition for a complete, simply connected, and strict nearly KÃ¤hler manifold of dimension 6 to be a homogeneous nearly KÃ¤hler manifold. This result was announced in a previous paper by the first author.
Keywords:Nearly KÃ¤hler manifold, 6-dimension, Homogeneous, The 1st Chern Class, Einstein manifolds Categories:53C40, 53C15 |
20. CMB 2010 (vol 53 pp. 503)
The Time Change Method and SDEs with Nonnegative Drift Using the time change method we show how to construct a solution to the stochastic equation $dX_t=b(X_{t-})dZ_t+a(X_t)dt$ with a nonnegative drift $a$ provided there exists a solution to the auxililary equation $dL_t=[a^{-1/\alpha}b](L_{t-})d\bar Z_t+dt$ where $Z, \bar Z$ are two symmetric stable processes of the same index $\alpha\in(0,2]$. This approach allows us to prove the existence of solutions for both stochastic equations for the values $0<\alpha<1$ and only measurable coefficients $a$ and $b$ satisfying some conditions of boundedness. The existence proof for the auxililary equation uses the method of integral estimates in the sense of Krylov.
Keywords:One-dimensional SDEs, symmetric stable processes, nonnegative drift, time change, integral estimates, weak convergence Categories:60H10, 60J60, 60J65, 60G44 |
21. CMB 2010 (vol 53 pp. 438)
Near-Homeomorphisms of Nöbeling Manifolds We characterize maps between $n$-dimensional NÃ¶beling manifolds that can be approximated by homeomorphisms.
Keywords:n-dimensional Nöbeling manifold, Z-set unknotting, near-homeomorphism Categories:55M10, 54F45 |
22. CMB 2010 (vol 53 pp. 327)
Multidimensional Exponential Inequalities with Weights We establish sufficient conditions on the weight functions $u$ and $v$ for the validity of the multidimensional weighted inequality $$ \Bigl(\int_E \Phi(T_k f(x))^q u(x)\,dx\Bigr)^{1/q} \le C \Bigl (\int_E \Phi(f(x))^p v(x)\,dx\Bigr )^{1/p}, $$
where 0<$p$, $q$<$\infty$, $\Phi$ is a logarithmically convex function, and $T_k$ is an integral operator over star-shaped regions. The condition is also necessary for the exponential integral inequality. Moreover, the estimation of $C$ is given and we apply the obtained results to generalize some multidimensional Levin--Cochran-Lee type inequalities.
Keywords:multidimensional inequalities, geometric mean operators, exponential inequalities, star-shaped regions Categories:26D15, 26D10 |
23. CMB 2008 (vol 51 pp. 236)
24. CMB 2007 (vol 50 pp. 481)
Concordance des nÅuds de dimension $4$ We prove that for a simply connected closed
$4$-dimensional manifold, its embeddings
into the sphere of dimension $6$ are all
concordant to each other.
Keywords:concordance, cobordisme, n{\oe}ud de dimension $4$, chirurgie plongÃ©e Categories:57Q45, 57Q60, 57R40, 57R65, 57N13 |
25. CMB 2007 (vol 50 pp. 588)
Cohomological Dimension and Schreier's Formula in Galois Cohomology Let $p$ be a prime and $F$ a field containing a primitive $p$-th
root of unity. Then for $n\in \N$, the cohomological dimension
of the maximal pro-$p$-quotient $G$ of the absolute Galois group
of $F$ is at most $n$ if and only if the corestriction maps
$H^n(H,\Fp) \to H^n(G,\Fp)$ are surjective for all open
subgroups $H$ of index $p$. Using this result, we generalize
Schreier's formula for $\dim_{\Fp} H^1(H,\Fp)$ to $\dim_{\Fp}
H^n(H,\Fp)$.
Keywords:cohomological dimension, Schreier's formula, Galois theory, $p$-extensions, pro-$p$-groups Categories:12G05, 12G10 |