Expand all Collapse all | Results 76 - 100 of 184 |
76. CMB 2009 (vol 52 pp. 424)
Covering Discs in Minkowski Planes We investigate the following version of the circle covering
problem in strictly convex (normed or) Minkowski planes: to cover
a circle of largest possible diameter by $k$ unit circles. In
particular, we study the cases $k=3$, $k=4$, and $k=7$. For $k=3$
and $k=4$, the diameters under consideration are described in
terms of side-lengths and circumradii of certain inscribed regular
triangles or quadrangles. This yields also simple explanations of
geometric meanings that the corresponding homothety ratios have.
It turns out that basic notions from Minkowski geometry play an
essential role in our proofs, namely Minkowskian bisectors,
$d$-segments, and the monotonicity lemma.
Keywords:affine regular polygon, bisector, circle covering problem, circumradius, $d$-segment, Minkowski plane, (strictly convex) normed plane Categories:46B20, 52A21, 52C15 |
77. CMB 2009 (vol 52 pp. 213)
Dunford--Pettis Properties and Spaces of Operators J. Elton used an application of Ramsey theory to show that
if $X$ is an infinite dimensional Banach space,
then $c_0$ embeds in $X$, $\ell_1$ embeds in $X$, or there
is a subspace of $X$ that fails to have the Dunford--Pettis property.
Bessaga and Pelczynski showed that if $c_0$ embeds in $X^*$,
then $\ell_\infty$ embeds in $X^*$. Emmanuele and John showed
that if $c_0$ embeds in $K(X,Y)$, then $K(X,Y)$ is not
complemented in $L(X,Y)$. Classical results from Schauder basis theory
are used in a study of Dunford--Pettis sets and strong
Dunford--Pettis sets to extend each of the preceding theorems. The space
$L_{w^*}(X^* , Y)$ of $w^*-w$ continuous operators is also studied.
Keywords:Dunford--Pettis property, Dunford--Pettis set, basic sequence, complemented spaces of operators Categories:46B20, 46B28 |
78. CMB 2009 (vol 52 pp. 39)
A Representation Theorem for Archimedean Quadratic Modules on $*$-Rings We present a new approach to noncommutative real algebraic geometry
based on the representation theory of $C^\ast$-algebras.
An important result in commutative real algebraic geometry is
Jacobi's representation theorem for archimedean quadratic modules
on commutative rings.
We show that this theorem is a consequence of the
Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras.
A noncommutative version of Gelfand--Naimark theory was studied by
I. Fujimoto. We use his results to generalize
Jacobi's theorem to associative rings with involution.
Keywords:Ordered rings with involution, $C^\ast$-algebras and their representations, noncommutative convexity theory, real algebraic geometry Categories:16W80, 46L05, 46L89, 14P99 |
79. CMB 2009 (vol 52 pp. 28)
Right and Left Weak Approximation Properties in Banach Spaces New necessary and sufficient conditions are established for Banach
spaces to have the approximation property; these conditions are
easier to check than the known ones. A shorter proof of a result
of Grothendieck is presented, and some properties of a weak
version of the approximation property are addressed.
Keywords:approximation property, quasi approximation property, weak approximation property Categories:46B28, 46B10 |
80. CMB 2008 (vol 51 pp. 604)
The Invariant Subspace Problem for Non-Archimedean Banach Spaces It is proved that every infinite-dimensional
non-archimedean Banach space of countable type admits a linear
continuous operator without a non-trivial closed invariant
subspace. This solves a problem stated by A.~C.~M. van Rooij and
W.~H. Schikhof in 1992.
Keywords:invariant subspaces, non-archimedean Banach spaces Categories:47S10, 46S10, 47A15 |
81. CMB 2008 (vol 51 pp. 618)
Vanishing Theorems in Colombeau Algebras of Generalized Functions Using a canonical linear embedding of the algebra
${\mathcal G}^{\infty}(\Omega)$ of Colombeau generalized functions in the space of
$\overline{\C}$-valued $\C$-linear maps on the space
${\mathcal D}(\Omega)$ of smooth functions with compact support, we give vanishing
conditions for functions and linear integral operators of class
${\mathcal G}^\infty$. These results are then applied to the zeros of holomorphic
generalized functions in dimension greater than one.
Keywords:Colombeau generalized functions, linear integral operators, generalized holomorphic functions Categories:32A60, 45P05, 46F30 |
82. CMB 2008 (vol 51 pp. 545)
$C^{\ast}$-Algebras Associated with Mauldin--Williams Graphs A Mauldin--Williams graph $\mathcal{M}$ is a generalization of an
iterated function system by a directed graph. Its invariant set $K$
plays the role of the self-similar set. We associate a $C^{*}$-algebra
$\mathcal{O}_{\mathcal{M}}(K)$ with a Mauldin--Williams graph $\mathcal{M}$
and the invariant set $K$, laying emphasis on the singular points.
We assume that the underlying graph $G$ has no sinks and no sources.
If $\mathcal{M}$ satisfies the open set condition in $K$, and $G$
is irreducible and is not a cyclic permutation, then the associated
$C^{*}$-algebra $\mathcal{O}_{\mathcal{M}}(K)$ is simple and purely
infinite. We calculate the $K$-groups for some examples including the
inflation rule of the Penrose tilings.
Categories:46L35, 46L08, 46L80, 37B10 |
83. CMB 2008 (vol 51 pp. 378)
Cyclic Vectors in Some Weighted $L^p$ Spaces of Entire Functions In this paper,
we generalize a result recently obtained by the author.
We characterize the cyclic vectors in $\Lp$.
Let $f\in\Lp$ and $f\poly$ be contained in the space.
We show that $f$ is non-vanishing if and only if $f$ is cyclic.
Keywords:weighted $L^p$ spaces of entire functions, cyclic vectors Categories:47A16, 46J15, 46H25 |
84. CMB 2008 (vol 51 pp. 321)
Quantum Multiple Construction of Subfactors We construct the quantum $s$-tuple subfactors for an AFD II$_{1}$
subfactor with finite index and depth, for an arbitrary natural number
$s$. This is a generalization of the quantum multiple subfactors by
Erlijman and Wenzl, which in turn generalized the quantum double
construction of a subfactor for the case that the original subfactor
gives rise to a braided tensor category. In this paper we give a
multiple construction for a subfactor with a weaker condition than
braidedness of the bimodule system.
Categories:46L37, 81T05 |
85. CMB 2008 (vol 51 pp. 236)
86. CMB 2008 (vol 51 pp. 205)
On GÃ¢teaux Differentiability of Pointwise Lipschitz Mappings We prove that for every function $f\from X\to Y$,
where $X$ is a separable Banach space and $Y$ is a Banach space
with RNP, there exists a set $A\in\tilde\mcA$ such that $f$ is
G\^ateaux differentiable at all $x\in S(f)\setminus A$, where
$S(f)$ is the set of points where $f$ is pointwise-Lipschitz.
This improves a result of Bongiorno. As a corollary,
we obtain that every $K$-monotone function on a separable Banach space
is Hadamard differentiable outside of a set belonging to $\tilde\mcC$;
this improves a result due to Borwein and Wang.
Another corollary is that if $X$ is Asplund, $f\from X\to\R$ cone monotone,
$g\from X\to\R$ continuous convex, then there exists a point in $X$, where $f$ is Hadamard
differentiable and $g$ is Fr\'echet differentiable.
Keywords:GÃ¢teaux differentiable function, Radon-NikodÃ½m property, differentiability of Lipschitz functions, pointwise-Lipschitz functions, cone mononotone functions Categories:46G05, 46T20 |
87. CMB 2008 (vol 51 pp. 15)
The Duality Problem for the Class of AM-Compact Operators on Banach Lattices We prove the converse of a
theorem of Zaanen about the duality problem of
positive AM-compact operators.
Keywords:AM-compact operator, order continuous norm, discrete vector lattice Categories:46A40, 46B40, 46B42 |
88. CMB 2008 (vol 51 pp. 67)
Rearrangement-Invariant Functionals with Applications to Traces on Symmetrically Normed Ideals We present a construction of singular rearrangement
invariant functionals on Marcinkiewicz function/operator spaces.
The functionals constructed differ from all previous examples in
the literature in that they fail to be symmetric. In other words,
the functional $\phi$ fails the condition that if $x\pprec y$
(Hardy-Littlewood-Polya submajorization) and $0\leq x,y$, then
$0\le \phi(x)\le \phi(y).$ We apply our results to singular traces
on symmetric operator spaces (in particular on
symmetrically-normed ideals of compact operators), answering
questions raised by Guido and Isola.
Categories:46L52, 47B10, 46E30 |
89. CMB 2008 (vol 51 pp. 26)
Hin\v cin's Theorem for Multiplicative Free Convolution Hin\v cin proved that any limit law, associated with a triangular
array of infinitesimal random variables, is infinitely divisible.
The analogous result for additive free convolution was proved earlier by
Bercovici and Pata.
In this paper we will prove corresponding results for the multiplicative
free convolution of measures definded on the unit circle and on the
positive half-line.
Categories:46L53, 60E07, 60E10 |
90. CMB 2007 (vol 50 pp. 519)
On Axiomatizability of Non-Commutative $L_p$-Spaces It is shown that Schatten $p$-classes
of operators between Hilbert spaces of different (infinite)
dimensions have ultrapowers which are (completely) isometric to
non-commutative $L_p$-spaces. On the other hand, these Schatten
classes are not themselves isomorphic to non-commutative $L_p$
spaces. As a consequence, the class of non-commutative $L_p$-spaces
is not axiomatizable in the first-order language developed by
Henson and Iovino for normed space structures, neither in the
signature of Banach spaces, nor in that of operator spaces. Other
examples of the same phenomenon are presented that belong to the
class of corners of non-commutative $L_p$-spaces. For $p=1$ this
last class, which is the same as the class of preduals of ternary
rings of operators, is itself axiomatizable in the signature of
operator spaces.
Categories:46L52, 03C65, 46B20, 46L07, 46M07 |
91. CMB 2007 (vol 50 pp. 610)
On Weak$^*$ Kadec--Klee Norms We present partial positive results supporting a conjecture that
admitting an equivalent Lipschitz (or uniformly) weak$^*$ Kadec--Klee norm is
a three space property.
Keywords:weak$^*$ Kadec--Klee norms, three-space problem Categories:46B03, 46B2 |
92. CMB 2007 (vol 50 pp. 619)
On the Existence of Asymptotic-$l_p$ Structures in Banach Spaces It is shown that if a Banach space is saturated with infinite
dimensional subspaces in which all ``special" $n$-tuples of
vectors are equivalent with constants independent of $n$-tuples and
of $n$, then the space contains asymptotic-$l_p$ subspaces
for some $1 \leq p \leq \infty$.
This extends a result by Figiel, Frankiewicz, Komorowski and
Ryll-Nardzewski.
Categories:46B20, 46B40, 46B03 |
93. CMB 2007 (vol 50 pp. 460)
Weak Semiprojectivity for Purely Infinite $C^*$-Algebras We prove that a separable, nuclear, purely infinite, simple
$C^*$-algebra satisfying the universal coefficient theorem
is weakly semiprojective if and only if
its $K$-groups are direct sums of cyclic groups.
Keywords:Kirchberg algebra, weak semiprojectivity, graph $C^*$-algebra Categories:46L05, 46L80, 22A22 |
94. CMB 2007 (vol 50 pp. 172)
An Infinite Dimensional Vector Space of Universal Functions for $H^\infty$ of the Ball We show that there exists a closed infinite dimensional subspace
of $H^\infty(B^n)$ such that every function of norm one is
universal for some sequence of automorphisms of $B^n$.
Categories:47B38, 47B33, 46J10 |
95. 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 |
96. 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 |
97. CMB 2007 (vol 50 pp. 3)
Higher Dimensional Spaces of Functions on the Spectrum of a Uniform Algebra In this paper we introduce a nested family of spaces of continuous functions defined
on the spectrum of a uniform algebra. The smallest space in the family is the
uniform algebra itself. In the ``finite dimensional'' case, from some point on the
spaces will be the space of all continuous complex-valued functions on the
spectrum. These spaces are defined in terms of solutions to the nonlinear
Cauchy--Riemann equations as introduced by the author in 1976, so they are not
generally linear spaces of functions. However, these spaces do shed light on the
higher dimensional properties of a uniform algebra. In particular, these spaces are
directly related to the generalized Shilov boundary of the uniform algebra (as
defined by the author and, independently, by Sibony in the early 1970s).
Categories:32A99, 46J10 |
98. CMB 2007 (vol 50 pp. 149)
On Quotients of Non-Archimedean KÃ¶the Spaces We show that there exists a non-archimedean
Fr\'echet-Montel space $W$ with a basis and with a continuous norm
such that any non-archimedean Fr\'echet space of countable type is isomorphic
to a quotient of $W$. We also prove that any non-archimedean nuclear
Fr\'echet space is isomorphic to a quotient of some non-archimedean nuclear
Fr\'echet space with a basis and with a continuous norm.
Keywords:Non-archimedean KÃ¶the spaces, nuclear FrÃ©chet spaces, pseudo-bases Categories:46S10, 46A45 |
99. CMB 2007 (vol 50 pp. 138)
On the Structure of the Set of Symmetric Sequences in Orlicz Sequence Spaces We study the structure of the sets of symmetric sequences and
spreading models of an Orlicz sequence space equipped with partial
order with respect to domination of bases. In the cases that these
sets are ``small'', some descriptions of the structure of these posets
are obtained.
Categories:46B20, 46B45, 46B07 |
100. CMB 2007 (vol 50 pp. 85)
Classification of Finite Group-Frames and Super-Frames Given a finite group $G$, we examine the classification of all
frame representations of $G$ and the classification of all
$G$-frames, \emph{i.e.,} frames induced by group representations of $G$.
We show that the exact number of equivalence classes of $G$-frames
and the exact number of frame representations can be explicitly
calculated. We also discuss how to calculate the largest number
$L$ such that there exists an $L$-tuple of strongly disjoint
$G$-frames.
Keywords:frames, group-frames, frame representations, disjoint frames Categories:42C15, 46C05, 47B10 |