Expand all Collapse all | Results 26 - 50 of 195 |
26. CMB 2013 (vol 57 pp. 640)
Equilateral Sets and a SchÃ¼tte Theorem for the $4$-norm A well-known theorem of SchÃ¼tte (1963) gives a sharp lower bound for
the ratio of the maximum and minimum distances between $n+2$ points in
$n$-dimensional Euclidean space.
In this note we adapt BÃ¡rÃ¡ny's elegant proof (1994) of this theorem to the space $\ell_4^n$.
This gives a new proof that the largest cardinality of an equilateral
set in $\ell_4^n$ is $n+1$, and gives a constructive bound for an
interval $(4-\varepsilon_n,4+\varepsilon_n)$ of values of $p$ close to $4$ for which it is known that the largest cardinality of an equilateral set in $\ell_p^n$ is $n+1$.
Categories:46B20, 52A21, 52C17 |
27. CMB 2013 (vol 57 pp. 463)
Constructive Proof of Carpenter's Theorem We give a constructive proof of Carpenter's Theorem due to Kadison.
Unlike the original proof our approach also yields the
real case of this theorem.
Keywords:diagonals of projections, the Schur-Horn theorem, the Pythagorean theorem, the Carpenter theorem, spectral theory Categories:42C15, 47B15, 46C05 |
28. CMB 2013 (vol 57 pp. 598)
Interpolation of Morrey Spaces on Metric Measure Spaces In this article, via the classical complex interpolation method
and some interpolation methods traced to Gagliardo,
the authors obtain an interpolation theorem for
Morrey spaces on quasi-metric measure spaces, which generalizes
some known results on ${\mathbb R}^n$.
Keywords:complex interpolation, Morrey space, Gagliardo interpolation, CalderÃ³n product, quasi-metric measure space Categories:46B70, 46E30 |
29. CMB 2013 (vol 57 pp. 364)
How Lipschitz Functions Characterize the Underlying Metric Spaces Let $X, Y$ be metric spaces and $E, F$ be Banach spaces. Suppose that
both $X,Y$ are realcompact, or both $E,F$ are realcompact.
The zero set of a vector-valued function $f$ is denoted by $z(f)$.
A linear bijection $T$ between local or generalized Lipschitz vector-valued function spaces
is said to preserve zero-set containments or nonvanishing functions
if
\[z(f)\subseteq z(g)\quad\Longleftrightarrow\quad z(Tf)\subseteq z(Tg),\]
or
\[z(f) = \emptyset\quad \Longleftrightarrow\quad z(Tf)=\emptyset,\]
respectively.
Every zero-set containment preserver, and every nonvanishing function preserver when
$\dim E =\dim F\lt +\infty$, is a weighted composition operator
$(Tf)(y)=J_y(f(\tau(y)))$.
We show that the map $\tau\colon Y\to X$ is a locally (little) Lipschitz homeomorphism.
Keywords:(generalized, locally, little) Lipschitz functions, zero-set containment preservers, biseparating maps Categories:46E40, 54D60, 46E15 |
30. CMB 2013 (vol 57 pp. 546)
Compact Operators in Regular LCQ Groups We show that a regular locally compact quantum group $\mathbb{G}$ is discrete
if and only if $\mathcal{L}^{\infty}(\mathbb{G})$ contains non-zero compact operators on
$\mathcal{L}^{2}(\mathbb{G})$.
As a corollary we classify all discrete quantum groups among
regular locally compact quantum groups $\mathbb{G}$ where
$\mathcal{L}^{1}(\mathbb{G})$ has the Radon--Nikodym property.
Keywords:locally compact quantum groups, regularity, compact operators Category:46L89 |
31. CMB 2012 (vol 57 pp. 424)
A Note on Amenability of Locally Compact Quantum Groups In this short note we introduce a notion called ``quantum injectivity''
of locally compact quantum groups, and prove that it is equivalent
to amenability of the dual. Particularly, this provides a new characterization
of amenability of locally compact groups.
Keywords:amenability, conditional expectation, injectivity, locally compact quantum group, quantum injectivity Categories:20G42, 22D25, 46L89 |
32. CMB 2012 (vol 57 pp. 90)
Compact Subsets of the Glimm Space of a $C^*$-algebra If $A$ is a $\sigma$-unital $C^*$-algebra and $a$ is a strictly positive element of $A$ then for every compact subset $K$ of the complete
regularization $\mathrm{Glimm}(A)$ of $\mathrm{Prim}(A)$ there exists
$\alpha \gt 0$ such that $K\subset \{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq
\alpha\}$. This extends
a result of J. Dauns
to all $\sigma$-unital $C^*$-algebras. However, there are a $C^*$-algebra $A$
and a compact subset of $\mathrm{Glimm}(A)$ that is not contained in any set of the form $\{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq
\alpha\}$, $a\in A$ and $\alpha \gt 0$.
Keywords:primitive ideal space, complete regularization Category:46L05 |
33. CMB 2012 (vol 57 pp. 42)
Covering the Unit Sphere of Certain Banach Spaces by Sequences of Slices and Balls e prove that, given any covering of any infinite-dimensional Hilbert space $H$ by countably many closed balls, some point exists in $H$ which belongs to infinitely many balls. We do that by characterizing isomorphically polyhedral separable Banach spaces as those whose unit sphere admits a point-finite covering by the union of countably many slices of the unit ball.
Keywords:point finite coverings, slices, polyhedral spaces, Hilbert spaces Categories:46B20, 46C05, 52C17 |
34. CMB 2012 (vol 57 pp. 166)
On Minimal and Maximal $p$-operator Space Structures We show that for $p$-operator spaces, there are natural notions of minimal and maximal
structures. These are useful for dealing with tensor products.
Keywords:$p$-operator space, min space, max space Categories:46L07, 47L25, 46G10 |
35. CMB 2012 (vol 57 pp. 3)
A Short Proof of Paouris' Inequality We give a short proof of a result of G.~Paouris on
the tail behaviour of the Euclidean norm $|X|$ of an isotropic
log-concave random vector $X\in\mathbb{R}^n,$
stating that for every $t\geq 1$,
\[\mathbb{P} \big( |X|\geq ct\sqrt n\big)\leq \exp(-t\sqrt n).\]
More precisely we show that for any log-concave random vector $X$
and any $p\geq 1$,
\[(\mathbb{E}|X|^p)^{1/p}\sim \mathbb{E} |X|+\sup_{z\in
S^{n-1}}(\mathbb{E} |\langle
z,X\rangle|^p)^{1/p}.\]
Keywords:log-concave random vectors, deviation inequalities Categories:46B06, 46B09, 52A23 |
36. CMB 2012 (vol 57 pp. 37)
Character Amenability of Lipschitz Algebras Let ${\mathcal X}$ be a locally compact metric space and let
${\mathcal A}$ be any of the Lipschitz algebras
${\operatorname{Lip}_{\alpha}{\mathcal X}}$, ${\operatorname{lip}_{\alpha}{\mathcal X}}$ or
${\operatorname{lip}_{\alpha}^0{\mathcal X}}$. In this paper, we show, as a
consequence of rather more general results on Banach algebras,
that ${\mathcal A}$ is $C$-character amenable if and only if
${\mathcal X}$ is uniformly discrete.
Keywords:character amenable, character contractible, Lipschitz algebras, spectrum Categories:43A07, 46H05, 46J10 |
37. 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 |
38. CMB 2012 (vol 56 pp. 870)
Note on Kasparov Product of $C^*$-algebra Extensions Using the Dadarlat isomorphism, we give a characterization for the
Kasparov product of $C^*$-algebra extensions. A certain relation
between $KK(A, \mathcal q(B))$ and $KK(A, \mathcal q(\mathcal k B))$ is also considered when
$B$ is not stable and it is proved that $KK(A, \mathcal q(B))$ and
$KK(A, \mathcal q(\mathcal k B))$ are not isomorphic in general.
Keywords:extension, Kasparov product, $KK$-group Category:46L80 |
39. CMB 2012 (vol 56 pp. 503)
Weak Sequential Completeness of $\mathcal K(X,Y)$ For Banach spaces $X$ and $Y$, we show that if $X^\ast$ and $Y$ are
weakly sequentially complete and every weakly compact operator from
$X$ to $Y$ is compact then the space of all compact operators from $X$
to $Y$ is weakly sequentially complete. The converse is also true if,
in addition, either $X^\ast$ or $Y$ has the bounded compact
approximation property.
Keywords:weak sequential completeness, reflexivity, compact operator space Categories:46B25, 46B28 |
40. CMB 2012 (vol 56 pp. 534)
A Cohomological Property of $\pi$-invariant Elements Let $A$ be a Banach algebra and $\pi \colon A \longrightarrow \mathscr L(H)$
be a continuous representation of $A$ on a separable Hilbert space $H$
with $\dim H =\frak m$. Let $\pi_{ij}$ be the coordinate functions of
$\pi$ with respect to an orthonormal basis and suppose that for each
$1\le j \le \frak m$, $C_j=\sum_{i=1}^{\frak m}
\|\pi_{ij}\|_{A^*}\lt \infty$ and $\sup_j C_j\lt \infty$. Under these
conditions, we call an element $\overline\Phi \in l^\infty (\frak m , A^{**})$
left $\pi$-invariant if $a\cdot \overline\Phi ={}^t\pi (a) \overline\Phi$ for all
$a\in A$. In this paper we prove a link between the existence
of left $\pi$-invariant elements and the vanishing of certain
Hochschild cohomology groups of $A$. Our results extend an earlier
result by Lau on $F$-algebras and recent results of Kaniuth-Lau-Pym
and the second named author in the special case that $\pi \colon A
\longrightarrow \mathbf C$ is a non-zero character on $A$.
Keywords:Banach algebras, $\pi$-invariance, derivations, representations Categories:46H15, 46H25, 13N15 |
41. 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 |
42. CMB 2011 (vol 56 pp. 337)
Certain Properties of $K_0$-monoids Preserved by Tracial Approximation We show that the following $K_0$-monoid properties of $C^*$-algebras
in the class $\Omega$ are inherited by simple unital $C^*$-algebras in
the class $TA\Omega$: (1) weak comparability, (2) strictly
unperforated, (3) strictly cancellative.
Keywords:$C^*$-algebra, tracial approximation, $K_0$-monoid Categories:46L05, 46L80, 46L35 |
43. CMB 2011 (vol 55 pp. 783)
Products and Direct Sums in Locally Convex Cones In this paper we define lower, upper, and symmetric completeness and
discuss closure of the sets in product and direct sums. In particular,
we introduce suitable bases for these topologies, which leads us to
investigate completeness of the direct sum and its components. Some
results obtained about $X$-topologies and polars of the neighborhoods.
Keywords:product and direct sum, duality, locally convex cone Categories:20K25, 46A30, 46A20 |
44. CMB 2011 (vol 56 pp. 400)
A Factorization Theorem for Multiplier Algebras of Reproducing Kernel Hilbert Spaces Let $(X,\mathcal B,\mu)$ be a $\sigma$-finite
measure space and let $H\subset L^2(X,\mu)$
be a separable reproducing kernel Hilbert
space on $X$. We show that the multiplier
algebra of $H$ has property $(A_1(1))$.
Keywords:reproducing kernel Hilbert space, Berezin transform, dual algebra Categories:46E22, 47B32, 47L45 |
45. CMB 2011 (vol 56 pp. 272)
On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate |
On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate In this note, we first give a characterization of super weakly
compact convex sets of a Banach space $X$:
a closed bounded convex set $K\subset X$ is
super weakly compact if and only if there exists a $w^*$ lower
semicontinuous seminorm $p$ with $p\geq\sigma_K\equiv\sup_{x\in
K}\langle\,\cdot\,,x\rangle$ such that $p^2$ is uniformly FrÃ©chet
differentiable on each bounded set of $X^*$. Then we present a
representation theorem for the dual of the semigroup $\textrm{swcc}(X)$
consisting of all the nonempty super weakly compact convex sets of the
space $X$.
Keywords:super weakly compact set, dual of normed semigroup, uniform FrÃ©chet differentiability, representation Categories:20M30, 46B10, 46B20, 46E15, 46J10, 49J50 |
46. CMB 2011 (vol 56 pp. 388)
Application of Measure of Noncompactness to Infinite Systems of Differential Equations In this paper we determine the Hausdorff measure of noncompactness on
the sequence space $n(\phi)$ of W. L. C. Sargent.
Further we apply
the technique of measures of noncompactness to the theory of infinite
systems of differential equations in the Banach sequence spaces
$n(\phi)$ and $m(\phi)$. Our aim is to present some existence results
for infinite systems of differential equations formulated with the help
of measures of noncompactness.
Keywords:sequence spaces, BK spaces, measure of noncompactness, infinite system of differential equations Categories:46B15, 46B45, 46B50, 34A34, 34G20 |
47. CMB 2011 (vol 56 pp. 434)
Some Remarks on the Algebraic Sum of Ideals and Riesz Subspaces Following ideas used by Drewnowski and Wilansky we prove that if $I$
is an infinite dimensional and
infinite codimensional closed ideal in a complete metrizable locally
solid Riesz space and $I$ does
not contain any order copy of $\mathbb R^{\mathbb N}$ then there exists a
closed, separable, discrete Riesz subspace
$G$ such that the topology induced on $G$ is Lebesgue, $I \cap G =
\{0\}$, and $I + G$ is not closed.
Keywords:locally solid Riesz space, Riesz subspace, ideal, minimal topological vector space, Lebesgue property Categories:46A40, 46B42, 46B45 |
48. CMB 2011 (vol 56 pp. 65)
The Uncomplemented Subspace $\mathbf K(X,Y) $ A vector measure result is used to study the complementation of the
space $K(X,Y)$ of compact operators in the spaces $W(X,Y)$ of weakly
compact operators, $CC(X,Y)$ of completely continuous operators, and
$U(X,Y)$ of unconditionally converging operators.
Results of Kalton and Emmanuele concerning the complementation of
$K(X,Y)$ in $L(X,Y)$ and in $W(X,Y)$ are generalized. The containment
of $c_0$ and $\ell_\infty$ in spaces of operators is also studied.
Keywords:compact operators, weakly compact operators, uncomplemented subspaces of operators Categories:46B20, 46B28 |
49. CMB 2011 (vol 55 pp. 673)
Multiplicity Free Jacquet Modules Let $F$ be a non-Archimedean local field or a finite field.
Let $n$ be a natural number and $k$ be $1$ or $2$.
Consider $G:=\operatorname{GL}_{n+k}(F)$ and let
$M:=\operatorname{GL}_n(F) \times \operatorname{GL}_k(F)\lt G$ be a maximal Levi subgroup.
Let $U\lt G$ be the corresponding unipotent subgroup and let $P=MU$ be the corresponding parabolic subgroup.
Let $J:=J_M^G: \mathcal{M}(G) \to \mathcal{M}(M)$ be the Jacquet functor, i.e., the functor of coinvariants with respect to $U$.
In this paper we prove that $J$ is a multiplicity free functor, i.e.,
$\dim \operatorname{Hom}_M(J(\pi),\rho)\leq 1$,
for any irreducible representations $\pi$ of $G$ and $\rho$ of $M$.
We adapt the classical method of Gelfand and Kazhdan, which proves the ``multiplicity free" property of certain representations to prove the ``multiplicity free" property of certain functors.
At the end we discuss whether other Jacquet functors are multiplicity free.
Keywords:multiplicity one, Gelfand pair, invariant distribution, finite group Categories:20G05, 20C30, 20C33, 46F10, 47A67 |
50. CMB 2011 (vol 55 pp. 821)
New Examples of Non-Archimedean Banach Spaces and Applications The study carried out in this paper about some new examples of
Banach spaces, consisting of certain valued fields extensions, is
a typical non-archimedean feature. We determine whether these
extensions are of countable type, have $t$-orthogonal bases, or are
reflexive.
As an application we construct, for a class of base fields, a norm
$\|\cdot\|$ on $c_0$, equivalent to the canonical supremum norm,
without non-zero vectors that are $\|\cdot\|$-orthogonal and such
that there is a multiplication on $c_0$ making $(c_0,\|\cdot\|)$
into a valued field.
Keywords:non-archimedean Banach spaces, valued field extensions, spaces of countable type, orthogonal bases Categories:46S10, 12J25 |