101. CMB 2009 (vol 53 pp. 118)
 Lewis, Paul

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 

102. CMB 2009 (vol 53 pp. 278)
 Galego, Elói M.

CantorBernstein 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 SchroederBernstein (or
CantorBernstein) 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, $\Phi=(pu)(sv)(q+u)(r+v)$ is different from zero and $\Phi$
divides $p+q$ and $r+s$. These sextuples are called CantorBernstein
sextuples for Banach spaces. The simplest case $(1, 0, 0, 1, 1, 1)$
indicates the wellknown PeÅczyÅski's decomposition method in
Banach space. On the other hand, by interchanging some Banach spaces
in the above decomposition scheme, refinements of
the SchroederBernstein problem become evident.
Keywords:Pel czyÅski's decomposition method, SchroederBernstein problem Categories:46B03, 46B20 

103. CMB 2009 (vol 53 pp. 239)
104. CMB 2009 (vol 53 pp. 64)
 Dodos, Pandelis

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 weaklynull 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 

105. CMB 2009 (vol 53 pp. 37)
 Choi, ManDuen; Latrémolière, Frédéric

$C^*$CrossedProducts by an OrderTwo Automorphism
We describe the representation theory of $C^*$crossedproducts 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 crossedproduct: 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 Ktheory of several crossedproducts of the free group on two generators.
Categories:46L55, 46L80 

106. CMB 2009 (vol 52 pp. 598)
 Moreno, M. A.; Nicola, J.; Pardo, E.; Thomas, H.

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
2squashed 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 

107. CMB 2009 (vol 52 pp. 424)
 Martini, Horst; Spirova, Margarita

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 sidelengths 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 

108. CMB 2009 (vol 52 pp. 213)
 Ghenciu, Ioana; Lewis, Paul

DunfordPettis 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 DunfordPettis 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 DunfordPettis sets and strong
DunfordPettis sets to extend each of the preceding theorems. The space
$L_{w^*}(X^* , Y)$ of $w^*w$ continuous operators is also studied.
Keywords:DunfordPettis property, DunfordPettis set, basic sequence, complemented spaces of operators Categories:46B20, 46B28 

109. CMB 2009 (vol 52 pp. 28)
110. CMB 2009 (vol 52 pp. 39)
 Cimpri\v{c}, Jakob

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
GelfandNaimark representation theorem for commutative $C^\ast$algebras.
A noncommutative version of GelfandNaimark 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 

111. CMB 2008 (vol 51 pp. 618)
 Valmorin, V.

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 

112. CMB 2008 (vol 51 pp. 604)
 {\'S}liwa, Wies{\l}aw

The Invariant Subspace Problem for NonArchimedean Banach Spaces
It is proved that every infinitedimensional
nonarchimedean Banach space of countable type admits a linear
continuous operator without a nontrivial closed invariant
subspace. This solves a problem stated by A.~C.~M. van Rooij and
W.~H. Schikhof in 1992.
Keywords:invariant subspaces, nonarchimedean Banach spaces Categories:47S10, 46S10, 47A15 

113. CMB 2008 (vol 51 pp. 545)
 Ionescu, Marius; Watatani, Yasuo

$C^{\ast}$Algebras Associated with MauldinWilliams Graphs
A MauldinWilliams graph $\mathcal{M}$ is a generalization of an
iterated function system by a directed graph. Its invariant set $K$
plays the role of the selfsimilar set. We associate a $C^{*}$algebra
$\mathcal{O}_{\mathcal{M}}(K)$ with a MauldinWilliams 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 

114. CMB 2008 (vol 51 pp. 321)
 Asaeda, Marta

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 

115. CMB 2008 (vol 51 pp. 378)
 Izuchi, Kou Hei

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 nonvanishing if and only if $f$ is cyclic.
Keywords:weighted $L^p$ spaces of entire functions, cyclic vectors Categories:47A16, 46J15, 46H25 

116. CMB 2008 (vol 51 pp. 236)
117. CMB 2008 (vol 51 pp. 205)
 Duda, Jakub

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 pointwiseLipschitz.
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, RadonNikodÃ½m property, differentiability of Lipschitz functions, pointwiseLipschitz functions, cone mononotone functions Categories:46G05, 46T20 

118. CMB 2008 (vol 51 pp. 26)
 Belinschi, S. T.; Bercovici, H.

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 halfline.
Categories:46L53, 60E07, 60E10 

119. CMB 2008 (vol 51 pp. 67)
 Kalton, Nigel; Sukochev, Fyodor

RearrangementInvariant 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 $x\pprec y$
(HardyLittlewoodPolya 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
symmetricallynormed ideals of compact operators), answering
questions raised by Guido and Isola.
Categories:46L52, 47B10, 46E30 

120. CMB 2008 (vol 51 pp. 15)
121. CMB 2007 (vol 50 pp. 519)
 Henson, C. Ward; Raynaud, Yves; Rizzo, Andrew

On Axiomatizability of NonCommutative $L_p$Spaces
It is shown that Schatten $p$classes
of operators between Hilbert spaces of different (infinite)
dimensions have ultrapowers which are (completely) isometric to
noncommutative $L_p$spaces. On the other hand, these Schatten
classes are not themselves isomorphic to noncommutative $L_p$
spaces. As a consequence, the class of noncommutative $L_p$spaces
is not axiomatizable in the firstorder 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 noncommutative $L_p$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 

122. CMB 2007 (vol 50 pp. 619)
 Tcaciuc, Adi

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
RyllNardzewski.
Categories:46B20, 46B40, 46B03 

123. CMB 2007 (vol 50 pp. 610)
 Rychtář, Jan; Spurný, Jiří

On Weak$^*$ KadecKlee Norms
We present partial positive results supporting a conjecture that
admitting an equivalent Lipschitz (or uniformly) weak$^*$ KadecKlee norm is
a three space property.
Keywords:weak$^*$ KadecKlee norms, threespace problem Categories:46B03, 46B2 

124. CMB 2007 (vol 50 pp. 460)
 Spielberg, Jack

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 

125. CMB 2007 (vol 50 pp. 227)