Expand all Collapse all | Results 1 - 13 of 13 |
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
Embedding theorem for inhomogeneous Besov and Triebel-Lizorkin spaces on RD-spaces In this article we prove the embedding theorem for inhomogeneous
Besov and Triebel-Lizorkin spaces on RD-spaces.
The crucial idea is to use the geometric density condition
on the measure.
Keywords:spaces of homogeneous type, test function space, distributions, CalderÃ³n reproducing formula, Besov and Triebel-Lizorkin spaces, embedding Categories:42B25, 46F05, 46E35 |
3. CMB Online first
Countable dense homogeneity in powers of zero-dimensional definable spaces We show that, for a coanalytic subspace $X$ of $2^\omega$, the
countable dense homogeneity of $X^\omega$ is equivalent to $X$
being Polish. This strengthens a result of HruÅ¡Ã¡k and Zamora
AvilÃ©s. Then, inspired by results of HernÃ¡ndez-GutiÃ©rrez,
HruÅ¡Ã¡k and van Mill, using a technique of Medvedev, we
construct a non-Polish subspace $X$ of $2^\omega$ such that $X^\omega$
is countable dense homogeneous. This gives the first $\mathsf{ZFC}$ answer
to a question of HruÅ¡Ã¡k and Zamora AvilÃ©s. Furthermore,
since our example is consistently analytic, the equivalence result
mentioned above is sharp. Our results also answer a question
of Medini and Milovich. Finally, we show that if every countable
subset of a zero-dimensional separable metrizable space $X$ is
included in a Polish subspace of $X$ then $X^\omega$ is countable
dense homogeneous.
Keywords:countable dense homogeneous, infinite power, coanalytic, Polish, $\lambda'$-set Categories:54H05, 54G20, 54E52 |
4. CMB 2014 (vol 57 pp. 673)
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 |
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 2012 (vol 56 pp. 860)
On Countable Dense and $n$-homogeneity We prove that a connected, countable dense homogeneous space is
$n$-homogeneous for every $n$, and strongly 2-homogeneous provided it
is locally connected. We also present an example of a connected and
countable dense homogeneous space which is not strongly
2-homogeneous. This answers Problem 136 of Watson in the Open Problems
in Topology Book in the negative.
Keywords:countable dense homogeneous, connected, $n$-homogeneous, strongly $n$-homogeneous, counterexample Categories:54H15, 54C10, 54F05 |
7. CMB 2011 (vol 56 pp. 225)
On the Notion of Visibility of Torsors Let $J$ be an abelian variety and
$A$ be an abelian subvariety of $J$, both defined over $\mathbf{Q}$.
Let $x$ be an element of $H^1(\mathbf{Q},A)$.
Then there are at least two definitions of $x$ being visible in $J$:
one asks that the torsor corresponding to $x$ be isomorphic over $\mathbf{Q}$
to a subvariety of $J$, and the other asks that $x$ be in the kernel
of the natural map $H^1(\mathbf{Q},A) \to H^1(\mathbf{Q},J)$. In this article, we
clarify the relation between the two definitions.
Keywords:torsors, principal homogeneous spaces, visibility, Shafarevich-Tate group Categories:11G35, 14G25 |
8. CMB 2011 (vol 55 pp. 351)
Rational Homogeneous Algebras An algebra $A$ is homogeneous if the automorphism group of $A$
acts transitively on the one-dimensional subspaces of $A$. The existence of homogeneous algebras depends critically on the choice of the scalar field. We examine the case where the scalar field is the rationals. We prove that if $A$ is a rational homogeneous algebra with $\operatorname{dim} A>1$, then $A^{2}=0$.
Keywords:non-associative algebra, homogeneous, automorphism Categories:17D99, 17A36 |
9. 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 |
10. CMB 2009 (vol 53 pp. 263)
Weighted Norm Inequalities for a Maximal Operator in Some Subspace of Amalgams We give weighted norm inequalities for the maximal fractional operator $ \mathcal M_{q,\beta }$ of HardyÂLittlewood and the fractional integral $I_{\gamma}$. These inequalities are established between $(L^{q},L^{p}) ^{\alpha }(X,d,\mu )$ spaces (which are superspaces of Lebesgue spaces $L^{\alpha}(X,d,\mu)$ and subspaces of amalgams $(L^{q},L^{p})(X,d,\mu)$) and in the setting of space of homogeneous type $(X,d,\mu)$. The conditions on the weights are stated in terms of Orlicz norm.
Keywords:fractional maximal operator, fractional integral, space of homogeneous type Categories:42B35, 42B20, 42B25 |
11. CMB 2009 (vol 53 pp. 218)
Restriction of the Tangent Bundle of $G/P$ to a Hypersurface Let P be a maximal proper parabolic subgroup of a connected simple linear algebraic group G, defined over $\mathbb C$, such that $n := \dim_{\mathbb C} G/P \geq 4$. Let $\iota \colon Z \hookrightarrow G/P$ be a reduced smooth hypersurface of degree at least $(n-1)\cdot \operatorname{degree}(T(G/P))/n$. We prove that the restriction of the tangent bundle $\iota^*TG/P$ is semistable.
Keywords:tangent bundle, homogeneous space, semistability, hypersurface Categories:14F05, 14J60, 14M15 |
12. CMB 1999 (vol 42 pp. 463)
A Generalized Characterization of Commutators of Parabolic Singular Integrals Let $x=(x_1, \dots, x_n)\in\rz$ and $\dz_\lz x=(\lz^{\az_1}x_1,
\dots,\lz^{\az_n}x_n)$, where $\lz>0$ and $1\le \az_1\le\cdots
\le\az_n$. Denote $|\az|=\az_1+\cdots+\az_n$. We characterize those
functions $A(x)$ for which the parabolic Calder\'on commutator
$$
T_{A}f(x)\equiv \pv \int_{\mathbb{R}^n}
K(x-y)[A(x)-A(y)]f(y)\,dy
$$
is bounded on $L^2(\mathbb{R}^n)$, where $K(\dz_\lz x)=\lz^{-|\az|-1}K(x)$,
$K$ is smooth away from the origin and satisfies a certain cancellation
property.
Keywords:parabolic singular integral, commutator, parabolic $\BMO$ sobolev space, homogeneous space, T1-theorem, symbol Category:42B20 |
13. CMB 1997 (vol 40 pp. 60)
Cauchy's problem for harmonic functions with entire data on a sphere We give an elementary potential-theoretic proof of a theorem of
G.~Johnsson: all solutions of Cauchy's problems for the Laplace
equations with an entire data on a sphere extend harmonically to
the whole space ${\bf R}^N$ except, perhaps, for the center of the
sphere.
Keywords:harmonic functions, Cauchy's problem, homogeneous harmonics Categories:35B60, 31B20 |