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101. 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

102. 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

103. 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

104. 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

105. 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

106. 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

107. CMB 2008 (vol 51 pp. 236)

Konovalov, Victor N.; Kopotun, Kirill A.

108. 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

109. 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

110. 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

111. 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

112. 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

113. 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

114. 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

115. 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

116. CMB 2007 (vol 50 pp. 227)

Kucerovsky, D.; Ng, P. W.
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

117. CMB 2007 (vol 50 pp. 268)

Manuilov, V.; Thomsen, K.
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

118. CMB 2007 (vol 50 pp. 172)

Aron, Richard; Gorkin, Pamela
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

119. CMB 2007 (vol 50 pp. 3)

Basener, Richard F.
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

120. CMB 2007 (vol 50 pp. 149)

Śliwa, Wiesław
On Quotients of Non-Archimedean Köthe 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 Köthe spaces, nuclear Fréchet spaces, pseudo-bases
Categories:46S10, 46A45

121. CMB 2007 (vol 50 pp. 138)

Sari, Bünyamin
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

122. CMB 2007 (vol 50 pp. 85)

Han, Deguang
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

123. CMB 2006 (vol 49 pp. 536)

Dostál, Petr; Lukeš, Jaroslav; Spurný, Jiří
Measure Convex and Measure Extremal Sets
We prove that convex sets are measure convex and extremal sets are measure extremal provided they are of low Borel complexity. We also present examples showing that the positive results cannot be strengthened.

Keywords:measure convex set, measure extremal set, face
Categories:46A55, 52A07

124. CMB 2006 (vol 49 pp. 371)

Floricel, Remus
Inner $E_0$-Semigroups on Infinite Factors
This paper is concerned with the structure of inner $E_0$-semigroups. We show that any inner $E_0$-semigroup acting on an infinite factor $M$ is completely determined by a continuous tensor product system of Hilbert spaces in $M$ and that the product system associated with an inner $E_0$-semigroup is a complete cocycle conjugacy invariant.

Keywords:von Neumann algebras, semigroups of endomorphisms, product systems, cocycle conjugacy
Categories:46L40, 46L55

125. CMB 2006 (vol 49 pp. 414)

Jiang, Liya; Jia, Houyu; Xu, Han
Commutators Estimates on Triebel--Lizorkin Spaces
In this paper, we consider the behavior of the commutators of convolution operators on the Triebel--Lizorkin spaces $\dot{F}^{s, q} _p$.

Keywords:commutators, Triebel--Lizorkin spaces, paraproduct
Categories:42B, 46F
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