Expand all Collapse all | Results 76 - 100 of 195 |
76. CMB 2010 (vol 53 pp. 447)
Injective Convolution Operators on l^{∞}(Γ) are Surjective Let $\Gamma$ be a discrete group and let $f \in \ell^{1}(\Gamma)$. We observe that if the natural convolution operator $\rho_f: \ell^{\infty}(\Gamma)\to \ell^{\infty}(\Gamma)$ is injective, then $f$ is invertible in $\ell^{1}(\Gamma)$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt by appealing to the direct finiteness of the algebra $\ell^{1}(\Gamma)$. We give simple examples to show that in general one cannot replace $\ell^{\infty}$ with $\ell^{p}$, $1\leq p< \infty$, nor with $L^{\infty}(G)$ for nondiscrete $G$. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on $\Gamma$, and give some partial results.
Categories:43A20, 46L05, 43A22 |
77. CMB 2010 (vol 53 pp. 256)
Equivalent Definitions of Infinite Positive Elements in Simple C^{*}-algebras We prove the equivalence of three definitions given by different comparison relations for infiniteness of positive elements in simple $C^*$-algebras.
Keywords:Infinite positive element, Comparison relation Category:46L99 |
78. CMB 2010 (vol 53 pp. 466)
Separating Maps between Spaces of Vector-Valued Absolutely Continuous Functions In this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and biseparating in the finite-dimensional case. The infinite-dimensional case is also studied.
Keywords:separating maps, disjointness preserving, vector-valued absolutely continuous functions, automatic continuity Categories:47B38, 46E15, 46E40, 46H40, 47B33 |
79. CMB 2009 (vol 53 pp. 51)
On the Relationship Between Interpolation of Banach Algebras and Interpolation of Bilinear Operators |
On the Relationship Between Interpolation of Banach Algebras and Interpolation of Bilinear Operators We show that if the general real method $(\cdot ,\cdot )_\Gamma$
preserves the Banach-algebra structure, then a bilinear
interpolation theorem holds for $(\cdot ,\cdot )_\Gamma$.
Keywords:real interpolation, bilinear operators, Banach algebras Categories:46B70, 46M35, 46H05 |
80. CMB 2009 (vol 53 pp. 37)
$C^*$-Crossed-Products by an Order-Two Automorphism We describe the representation theory of $C^*$-crossed-products of a unital $C^*$-algebra A by the cyclic group of order~2. We prove that there are two main types of irreducible representations for the crossed-product: those whose restriction to A is irreducible and those who are the sum of two unitarily unequivalent representations of~A. We characterize each class in term of the restriction of the representations to the fixed point $C^*$-subalgebra of~A. We apply our results to compute the K-theory of several crossed-products of the free group on two generators.
Categories:46L55, 46L80 |
81. CMB 2009 (vol 53 pp. 133)
A Further Decay Estimate for the DziubaÅski-HernÃ¡ndez Wavelets We give a further decay estimate for the DziubaÅski-HernÃ¡ndez wavelets that are band-limited and have subexponential decay. This is done by constructing an appropriate bell function and using the Paley-Wiener theorem for ultradifferentiable functions.
Keywords:wavelets, ultradifferentiable functions Categories:42C40, 46E10 |
82. CMB 2009 (vol 53 pp. 118)
The Uncomplemented Spaces $W(X,Y)$ and $K(X,Y)$ Classical results of Kalton and techniques of Feder are used to study the complementation of the space $W(X, Y)$ of weakly compact operators and the space $K(X,Y)$ of compact operators in the space $L(X,Y)$ of all bounded linear maps from X to Y.
Keywords:spaces of operators, complemented subspace, weakly compact operator, basic sequence Categories:46B28, 46B15, 46B20 |
83. CMB 2009 (vol 53 pp. 278)
Cantor-Bernstein Sextuples for Banach Spaces Let $X$ and $Y$ be Banach spaces isomorphic
to complemented subspaces of each other with supplements $A$ and
$B$. In 1996, W. T. Gowers solved the Schroeder--Bernstein (or
Cantor--Bernstein) problem for Banach spaces by showing that $X$ is not
necessarily isomorphic to $Y$. In this paper, we obtain a necessary
and sufficient condition on the sextuples $(p, q, r, s, u, v)$ in
$\mathbb N$
with $p+q \geq 1$, $r+s \geq 1$ and $u, v \in \mathbb N^*$, to provide that
$X$ is isomorphic to $Y$, whenever these spaces satisfy the following
decomposition scheme
$$
A^u \sim X^p \oplus Y^q, \quad
B^v \sim X^r \oplus Y^s.
$$
Namely, $\Phi=(p-u)(s-v)-(q+u)(r+v)$ is different from zero and $\Phi$
divides $p+q$ and $r+s$. These sextuples are called Cantor--Bernstein
sextuples for Banach spaces. The simplest case $(1, 0, 0, 1, 1, 1)$
indicates the well-known PeÅczyÅski's decomposition method in
Banach space. On the other hand, by interchanging some Banach spaces
in the above decomposition scheme, refinements of
the Schroeder--Bernstein problem become evident.
Keywords:Pel czyÅski's decomposition method, Schroeder-Bernstein problem Categories:46B03, 46B20 |
84. CMB 2009 (vol 53 pp. 239)
A Note on the Exactness of Operator Spaces In this paper, we give two characterizations of the exactness of operator spaces.
Keywords:operator space, exactness Category:46L07 |
85. CMB 2009 (vol 53 pp. 64)
On Antichains of Spreading Models of Banach Spaces We show that for every separable Banach space $X$,
either $\mathrm{SP_w}(X)$ (the set of all spreading models
of $X$ generated by weakly-null sequences in $X$, modulo
equivalence) is countable, or $\mathrm{SP_w}(X)$ contains an
antichain of the size of the continuum. This answers
a question of S.~J. Dilworth, E. Odell, and B. Sari.
Categories:46B20, 03E15 |
86. CMB 2009 (vol 52 pp. 598)
Numerical Semigroups That Are Not Intersections of $d$-Squashed Semigroups We say that a numerical semigroup is \emph{$d$-squashed} if it can
be written in the form
$$ S=\frac 1 N \langle a_1,\dots,a_d \rangle \cap \mathbb{Z}$$
for $N,a_1,\dots,a_d$ positive integers with
$\gcd(a_1,\dots, a_d)=1$.
Rosales and Urbano have shown that a numerical semigroup is
2-squashed if and only if it is proportionally modular.
Recent works by Rosales \emph{et al.} give a concrete example of a
numerical semigroup that cannot be written as an intersection of
$2$-squashed semigroups. We will show the existence of infinitely
many numerical semigroups that cannot be written as an
intersection of $2$-squashed semigroups. We also will prove the
same result for $3$-squashed semigroups. We conjecture that there
are numerical semigroups that cannot be written as the
intersection of $d$-squashed semigroups for any fixed $d$, and we
prove some partial results towards this conjecture.
Keywords:numerical semigroup, squashed semigroup, proportionally modular semigroup Categories:20M14, 06F05, 46L80 |
87. 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 |
88. 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 |
89. 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 |
90. 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 |
91. 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 |
92. 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 |
93. 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 |
94. 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 |
95. 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 |
96. CMB 2008 (vol 51 pp. 236)
97. 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 |
98. 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 |
99. 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 |
100. 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 |