Expand all Collapse all | Results 51 - 75 of 188 |
51. CMB 2011 (vol 55 pp. 260)
A Note on the Antipode for Algebraic Quantum Groups Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra.
In this note, we show that this formula can be proved for any regular multiplier Hopf
algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a
finite-dimensional Hopf algebra, but also that of any
Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that
the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the
theory of regular multiplier Hopf algebras with integrals.
We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.
Keywords:multiplier Hopf algebras, algebraic quantum groups, the antipode Categories:16W30, 46L65 |
52. CMB 2011 (vol 55 pp. 410)
A Ramsey Theorem with an Application to Sequences in Banach Spaces The notion of a maximally conditional sequence is introduced for sequences in a Banach space. It is then proved using
Ramsey theory that every basic sequence in a Banach space has a subsequence which is either an unconditional
basic sequence or a maximally conditional sequence. An apparently novel, purely combinatorial lemma in the spirit of
Galvin's theorem is used in the proof. An alternative proof
of the dichotomy result for sequences in Banach spaces is
also sketched,
using the Galvin-Prikry theorem.
Keywords:Banach spaces, Ramsey theory Categories:46B15, 05D10 |
53. CMB 2011 (vol 54 pp. 577)
Erratum: The Duality Problem For The Class of AM-Compact Operators On Banach Lattices It is proved that if a positive operator
$S: E \rightarrow F$ is AM-compact whenever its adjoint
$S': F' \rightarrow E'$ is AM-compact, then either the
norm of F is order continuous or $E'$ is discrete.
This note corrects an error in the proof of Theorem 2.3 of
B. Aqzzouz, R. Nouira, and L. Zraoula, The duality problem for
the class of AM-compact operators on Banach lattices. Canad. Math. Bull.
51(2008).
Categories:46A40, 46B40, 46B42 |
54. 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 |
55. CMB 2011 (vol 55 pp. 73)
Classification of Inductive Limits of Outer Actions of ${\mathbb R}$ on Approximate Circle Algebras In this paper we present a classification,
up to equivariant isomorphism, of $C^*$-dynamical systems $(A,{\mathbb R},\alpha )$
arising as inductive limits of directed systems
$\{ (A_n,{\mathbb R},\alpha_n),\varphi_{nm}\}$, where each $A_n$
is a finite direct sum of matrix algebras over the continuous
functions on the unit circle, and the $\alpha_n$s are outer actions
generated by rotation of the spectrum.
Keywords:classification, $C^*$-dynamical system Categories:46L57, 46L35 |
56. CMB 2011 (vol 54 pp. 726)
Auerbach Bases and Minimal Volume Sufficient Enlargements Let $B_Y$ denote the unit ball of a
normed linear space $Y$. A symmetric, bounded, closed, convex set
$A$ in a finite dimensional normed linear space $X$ is called a
sufficient enlargement for $X$ if, for an arbitrary
isometric embedding of $X$ into a Banach space $Y$, there exists a
linear projection $P\colon Y\to X$ such that $P(B_Y)\subset A$. Each
finite dimensional normed space has a minimal-volume sufficient
enlargement that is a parallelepiped; some spaces have ``exotic''
minimal-volume sufficient enlargements. The main result of the
paper is a characterization of spaces having ``exotic''
minimal-volume sufficient enlargements in terms of Auerbach
bases.
Keywords:Banach space, Auerbach basis, sufficient enlargement Categories:46B07, 52A21, 46B15 |
57. CMB 2011 (vol 54 pp. 411)
Operator Algebras with Unique Preduals We show that every free semigroup algebra has a (strongly) unique
Banach space predual. We also provide a new simpler proof that a
weak-$*$ closed unital operator algebra containing a weak-$*$
dense subalgebra of compact operators has a unique Banach space
predual.
Keywords:unique predual, free semigroup algebra, CSL algebra Categories:47L50, 46B04, 47L35 |
58. CMB 2011 (vol 54 pp. 593)
Stability of Real $C^*$-Algebras We will give a characterization of stable real $C^*$-algebras
analogous to the one given for complex $C^*$-algebras by Hjelmborg
and RÃ¸rdam. Using this result, we will prove
that any real $C^*$-algebra satisfying the corona factorization
property is stable if and only if its complexification is stable.
Real $C^*$-algebras satisfying the corona factorization property
include AF-algebras and purely infinite $C^*$-algebras. We will also
provide an example of a simple unstable $C^*$-algebra, the
complexification of which is stable.
Keywords:stability, real C*-algebras Category:46L05 |
59. CMB 2011 (vol 54 pp. 680)
$2$-Local Isometries on Spaces of Lipschitz Functions Let $(X,d)$ be a metric space, and let $\mathop{\textrm{Lip}}(X)$ denote the Banach
space of all scalar-valued bounded Lipschitz functions $f$ on $X$
endowed with one of the natural norms
$
\| f\| =\max \{\| f\| _\infty ,L(f)\}$ or $\|f\| =\|
f\| _\infty +L(f),
$
where $L(f)$ is the
Lipschitz constant of $f.$ It is said that the isometry
group of $\mathop{\textrm{Lip}}(X)$ is canonical if every
surjective linear isometry of
$\mathop{\textrm{Lip}}(X) $ is induced by a surjective isometry of $X$.
In this paper
we prove that if $X$ is bounded separable and the isometry group of
$\mathop{\textrm{Lip}}(X)$ is canonical, then every $2$-local isometry
of $\mathop{\textrm{Lip}}(X)$ is
a surjective linear isometry. Furthermore, we give a complete
description of all $2$-local isometries of $\mathop{\textrm{Lip}}(X)$ when $X$ is
bounded.
Keywords:isometry, local isometry, Lipschitz function Categories:46B04, 46J10, 46E15 |
60. CMB 2011 (vol 54 pp. 338)
SzegÃ¶'s Theorem and Uniform Algebras We study SzegÃ¶'s theorem for a uniform algebra.
In particular, we do it for the disc algebra or the bidisc algebra.
Keywords:SzegÃ¶'s theorem, uniform algebras, disc algebra, weighted Bergman space Categories:32A35, 46J15, 60G25 |
61. CMB 2011 (vol 54 pp. 347)
The Haar System in the Preduals of Hyperfinite Factors We shall present examples of Schauder bases in the preduals to the
hyperfinite factors of types~$\hbox{II}_1$, $\hbox{II}_\infty$,
$\hbox{III}_\lambda$, $0 < \lambda \leq 1$. In the semifinite
(respectively, purely infinite) setting, these systems form Schauder bases
in any associated separable symmetric space of measurable operators
(respectively, in any non-commutative $L^p$-space).
Category:46L52 |
62. CMB 2011 (vol 54 pp. 302)
Structure of the Set of Norm-attaining Functionals on Strictly Convex Spaces Let $X$ be a separable non-reflexive Banach space. We show that there
is no Borel class which contains the set of norm-attaining functionals
for every strictly convex renorming of $X$.
Keywords:separable non-reflexive space, set of norm-attaining functionals, strictly convex norm, Borel class Categories:46B20, 54H05, 46B10 |
63. CMB 2010 (vol 54 pp. 82)
Lefschetz Numbers for $C^*$-Algebras
Using Poincar\'e duality, we obtain a formula of Lefschetz type
that computes the Lefschetz number of an endomorphism of a separable
nuclear $C^*$-algebra satisfying Poincar\'e duality and the Kunneth
theorem. (The Lefschetz number of an endomorphism is the graded trace
of the induced map on $\textrm{K}$-theory tensored with $\mathbb{C}$, as in the
classical case.) We then examine endomorphisms of Cuntz--Krieger
algebras $O_A$. An endomorphism has an invariant, which is a
permutation of an infinite set, and the contracting and expanding
behavior of this permutation describes the Lefschetz number of the
endomorphism. Using this description, we derive a closed polynomial
formula for the Lefschetz number depending on the matrix $A$ and the
presentation of the endomorphism.
Categories:19K35, 46L80 |
64. CMB 2010 (vol 54 pp. 141)
Linear Maps on $C^*$-Algebras Preserving the Set of Operators that are Invertible in $\mathcal{A}/\mathcal{I}$ |
Linear Maps on $C^*$-Algebras Preserving the Set of Operators that are Invertible in $\mathcal{A}/\mathcal{I}$
For $C^*$-algebras $\mathcal{A}$ of real rank zero, we describe
linear maps $\phi$ on $\mathcal{A}$ that are surjective up to ideals
$\mathcal{I}$, and $\pi(A)$ is invertible in $\mathcal{A}/\mathcal{I}$ if and only if
$\pi(\phi(A))$ is invertible in $\mathcal{A}/\mathcal{I}$, where $A\in\mathcal{A}$ and
$\pi:\mathcal{A}\to\mathcal{A}/\mathcal{I}$ is the quotient map. We also consider similar
linear maps preserving zero products on the Calkin algebra.
Keywords:preservers, Jordan automorphisms, invertible operators, zero products Categories:47B48, 47A10, 46H10 |
65. CMB 2010 (vol 54 pp. 68)
Non-splitting in Kirchberg's Ideal-related $KK$-Theory
A. Bonkat obtained a universal coefficient theorem in the setting of Kirchberg's
ideal-related $KK$-theory in the fundamental case of a
$C^*$-algebra with one
specified ideal. The universal coefficient sequence was shown to split, unnaturally, under certain
conditions. Employing certain $K$-theoretical information derivable
from the given operator algebras using a method introduced here, we shall
demonstrate that Bonkat's UCT does not split in general. Related
methods lead to information on the complexity of the $K$-theory which
must be used to
classify $*$-isomorphisms for purely infinite $C^*$-algebras with
one non-trivial ideal.
Keywords:KK-theory, UCT Category:46L35 |
66. CMB 2010 (vol 53 pp. 587)
Hulls of Ring Extensions We investigate the behavior of the quasi-Baer and the
right FI-extending right ring hulls under various ring extensions
including group ring extensions, full and triangular matrix ring
extensions, and infinite matrix ring extensions. As a consequence,
we show that for semiprime rings $R$ and $S$, if $R$ and $S$ are
Morita equivalent, then so are the quasi-Baer right ring hulls
$\widehat{Q}_{\mathfrak{qB}}(R)$ and $\widehat{Q}_{\mathfrak{qB}}(S)$ of
$R$ and $S$, respectively. As an application, we prove that if
unital $C^*$-algebras $A$ and $B$ are Morita equivalent as rings,
then the bounded central closure of $A$ and that of $B$ are
strongly Morita equivalent as $C^*$-algebras. Our results show
that the quasi-Baer property is always preserved by infinite
matrix rings, unlike the Baer property. Moreover, we give an
affirmative answer to an open question of Goel and Jain for the
commutative group ring $A[G]$ of a torsion-free Abelian group $G$
over a commutative semiprime quasi-continuous ring $A$. Examples
that illustrate and delimit the results of this paper are provided.
Keywords:(FI-)extending, Morita equivalent, ring of quotients, essential overring, (quasi-)Baer ring, ring hull, u.p.-monoid, $C^*$-algebra Categories:16N60, 16D90, 16S99, 16S50, 46L05 |
67. CMB 2010 (vol 53 pp. 690)
On the Maximal Operator Ideal Associated with a Tensor Norm Defined by Interpolation Spaces
The classical approach to studying operator ideals using tensor
norms mainly focuses on those tensor norms and operator ideals
defined by means of $\ell_p$ spaces. In a previous paper,
an interpolation space, defined via the real method
and using
$\ell_p$ spaces, was used to define a tensor
norm, and the associated minimal operator ideals were characterized.
In this paper, the next natural step is taken, that is, the
corresponding maximal operator
ideals are characterized. As an application, necessary and sufficient
conditions for the coincidence of
the maximal and minimal ideals are given.
Finally, the previous results are used in order to find some new
metric properties of the mentioned tensor norm.
Keywords:maximal operator ideals, ultraproducts of spaces, interpolation spaces Categories:46M05, 46M35, 46A32 |
68. CMB 2010 (vol 53 pp. 550)
Representing a Product System Representation as a Contractive Semigroup and Applications to Regular Isometric Dilations |
Representing a Product System Representation as a Contractive Semigroup and Applications to Regular Isometric Dilations
In this paper we propose a new technical tool for analyzing
representations of Hilbert $C^*$-product systems. Using this tool,
we give a new proof that every doubly commuting representation
over $\mathbb{N}^k$ has a regular isometric dilation, and we also
prove sufficient conditions for the existence of a regular
isometric dilation of representations over more general
subsemigroups of $\mathbb R_{+}^k$.
Categories:47A20, 46L08 |
69. 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 |
70. 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 |
71. 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 |
72. 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 |
73. 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 |
74. 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 |
75. 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 |