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101. CMB 2009 (vol 53 pp. 37)

Choi, Man-Duen; Latrémolière, Frédéric
$C^*$-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

102. CMB 2009 (vol 53 pp. 133)

Moritoh, Shinya; Tomoeda, Kyoko
A Further Decay Estimate for the Dziubański-Hernández Wavelets
We give a further decay estimate for the Dziubański-Hernández 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

103. 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

104. CMB 2009 (vol 53 pp. 278)

Galego, Elói M.
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 Pełczyński'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 czyński's decomposition method, Schroeder-Bernstein problem
Categories:46B03, 46B20

105. CMB 2009 (vol 53 pp. 239)

Dong, Z.
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

106. 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 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

107. CMB 2009 (vol 53 pp. 51)

Cobos, Fernando; Fernández-Cabrera, Luz M.
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

108. 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 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

109. 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 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

110. CMB 2009 (vol 52 pp. 213)

Ghenciu, Ioana; Lewis, Paul
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

111. 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 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

112. CMB 2009 (vol 52 pp. 28)

Choi, Changsun; Kim, Ju Myung; Lee, Keun Young
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

113. CMB 2008 (vol 51 pp. 545)

Ionescu, Marius; Watatani, Yasuo
$C^{\ast}$-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

114. 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

115. CMB 2008 (vol 51 pp. 604)

{\'S}liwa, Wies{\l}aw
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

116. 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 non-vanishing if and only if $f$ is cyclic.

Keywords:weighted $L^p$ spaces of entire functions, cyclic vectors
Categories:47A16, 46J15, 46H25

117. 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

118. CMB 2008 (vol 51 pp. 236)

Konovalov, Victor N.; Kopotun, Kirill A.

119. CMB 2008 (vol 51 pp. 205)

Duda, Jakub
On Gâteaux 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:Gâteaux differentiable function, Radon-Nikodým property, differentiability of Lipschitz functions, pointwise-Lipschitz functions, cone mononotone functions
Categories:46G05, 46T20

120. 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 half-line.

Categories:46L53, 60E07, 60E10

121. CMB 2008 (vol 51 pp. 67)

Kalton, Nigel; Sukochev, Fyodor
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

122. CMB 2008 (vol 51 pp. 15)

Aqzzouz, Belmesnaoui; Nouira, Redouane; Zraoula, Larbi
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

123. CMB 2007 (vol 50 pp. 519)

Henson, C. Ward; Raynaud, Yves; Rizzo, Andrew
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

124. 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 Ryll-Nardzewski.

Categories:46B20, 46B40, 46B03

125. CMB 2007 (vol 50 pp. 610)

Rychtář, Jan; Spurný, Jiří
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
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