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126. 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

127. 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

128. 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

129. 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

130. 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

131. 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

132. 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

133. 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

134. 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

135. 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

136. 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

137. CMB 2006 (vol 49 pp. 389)

Hiai, Fumio; Petz, Dénes; Ueda, Yoshimichi
A Free Logarithmic Sobolev Inequality on the Circle
Free analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.

Categories:46L54, 60E15, 94A17

138. 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

139. CMB 2006 (vol 49 pp. 185)

Averkov, Gennadiy
On the Inequality for Volume and Minkowskian Thickness
Given a centrally symmetric convex body $B$ in $\E^d,$ we denote by $\M^d(B)$ the Minkowski space ({\em i.e.,} finite dimensional Banach space) with unit ball $B.$ Let $K$ be an arbitrary convex body in $\M^d(B).$ The relationship between volume $V(K)$ and the Minkowskian thickness ($=$ minimal width) $\thns_B(K)$ of $K$ can naturally be given by the sharp geometric inequality $V(K) \ge \alpha(B) \cdot \thns_B(K)^d,$ where $\alpha(B)>0.$ As a simple corollary of the Rogers--Shephard inequality we obtain that $\binom{2d}{d}{}^{-1} \le \alpha(B)/V(B) \le 2^{-d}$ with equality on the left attained if and only if $B$ is the difference body of a simplex and on the right if $B$ is a cross-polytope. The main result of this paper is that for $d=2$ the equality on the right implies that $B$ is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach--Mazur distance to the regular hexagon.

Keywords:convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, Banach-Mazur compactum, (modified) Banach-Mazur distance, volume ratio
Categories:52A40, 46B20

140. CMB 2006 (vol 49 pp. 313)

Wagner, Roy
On the Relation Between the Gaussian Orthogonal Ensemble and Reflections, or a Self-Adjoint Version of the Marcus--Pisier Inequality
We prove a self-adjoint analogue of the Marcus--Pisier inequality, comparing the expected value of convex functionals on randomreflection matrices and on elements of the Gaussian orthogonal (or unitary) ensemble.

Categories:15A52, 46B09, 46L54

141. CMB 2006 (vol 49 pp. 213)

Dean, Andrew J.
On Inductive Limit Type Actions of the Euclidean Motion Group on Stable UHF Algebras
An invariant is presented which classifies, up to equivariant isomorphism, $C^*$-dynamical systems arising as limits from inductive systems of elementary $C^*$-algebras on which the Euclidean motion group acts by way of unitary representations that decompose into finite direct sums of irreducibles.

Keywords:classification, $C^*$-dynamical system
Categories:46L57, 46L35

142. CMB 2006 (vol 49 pp. 117)

Levene, R. H.
A Double Triangle Operator Algebra From $SL_2(\R)$
We consider the w$^*$-closed operator algebra $\cA_+$ generated by the image of the semigroup $SL_2(\R_+)$ under a unitary representation $\rho$ of $SL_2(\R)$ on the Hilbert~space $L_2(\R)$. We show that $\cA_+$ is a reflexive operator algebra and $\cA_+=\Alg\cD$ where $\cD$ is a double triangle subspace lattice. Surprisingly, $\cA_+$ is also generated as a w$^*$-closed algebra by the image under $\rho$ of a strict subsemigroup of $SL_2(\R_+)$.

Categories:46K50, 47L55

143. CMB 2006 (vol 49 pp. 82)

Gogatishvili, Amiran; Pick, Luboš
Embeddings and Duality Theorem for Weak Classical Lorentz Spaces
We characterize the weight functions $u,v,w$ on $(0,\infty)$ such that $$ \left(\int_0^\infty f^{*}(t)^ qw(t)\,dt\right)^{1/q} \leq C \sup_{t\in(0,\infty)}f^{**}_u(t)v(t), $$ where $$ f^{**}_u(t):=\left(\int_{0}^{t}u(s)\,ds\right)^{-1} \int_{0}^{t}f^*(s)u(s)\,ds. $$ As an application we present a~new simple characterization of the associate space to the space $\Gamma^ \infty(v)$, determined by the norm $$ \|f\|_{\Gamma^ \infty(v)}=\sup_{t\in(0,\infty)}f^{**}(t)v(t), $$ where $$ f^{**}(t):=\frac1t\int_{0}^{t}f^*(s)\,ds. $$

Keywords:Discretizing sequence, antidiscretization, classical Lorentz spaces, weak Lorentz spaces, embeddings, duality, Hardy's inequality
Categories:26D10, 46E20

144. CMB 2005 (vol 48 pp. 607)

Park, Efton
Toeplitz Algebras and Extensions of\\Irrational Rotation Algebras
For a given irrational number $\theta$, we define Toeplitz operators with symbols in the irrational rotation algebra ${\mathcal A}_\theta$, and we show that the $C^*$-algebra $\mathcal T({\mathcal A}_\theta)$ generated by these Toeplitz operators is an extension of ${\mathcal A}_\theta$ by the algebra of compact operators. We then use these extensions to explicitly exhibit generators of the group $KK^1({\mathcal A}_\theta,\mathbb C)$. We also prove an index theorem for $\mathcal T({\mathcal A}_\theta)$ that generalizes the standard index theorem for Toeplitz operators on the circle.

Keywords:Toeplitz operators, irrational rotation algebras, index theory
Categories:47B35, 46L80

145. CMB 2005 (vol 48 pp. 481)

Azagra, D.; Fabian, M.; Jiménez-Sevilla, M.
Exact Filling of Figures with the Derivatives of Smooth Mappings Between Banach Spaces
We establish sufficient conditions on the shape of a set $A$ included in the space $\mathcal L _s^n(X,Y)$ of the $n$-linear symmetric mappings between Banach spaces $X$ and $Y$, to ensure the existence of a $C^n$\nobreakdash-smooth mapping $f\colon X \rightarrow Y$, with bounded support, and such that $f^{(n)}(X)=A$, provided that $X$ admits a $C^{n}$-smooth bump with bounded $n$-th derivative and $\dens X=\dens \mathcal L ^n(X,Y)$. For instance, when $X$ is infinite-dimensional, every bounded connected and open set $U$ containing the origin is the range of the $n$-th derivative of such a mapping. The same holds true for the closure of $U$, provided that every point in the boundary of $U$ is the end point of a path within $U$. In the finite-dimensional case, more restrictive conditions are required. We also study the Fr\'echet smooth case for mappings from $\mathbb R^n$ to a separable infinite-dimensional Banach space and the G\^ateaux smooth case for mappings defined on a separable infinite-dimensional Banach space and with values in a separable Banach space.


146. CMB 2005 (vol 48 pp. 455)

Rychtář, Jan
On Gâteaux Differentiability of Convex Functions in WCG Spaces
It is shown, using the Borwein--Preiss variational principle that for every continuous convex function $f$ on a weakly compactly generated space $X$, every $x_0\in X$ and every weakly compact convex symmetric set $K$ such that $\cspan K=X$, there is a point of G\^ateaux differentiability of $f$ in $x_0+K$. This extends a Klee's result for separable spaces.

Keywords:Gâteaux smoothness, Borwein--Preiss variational principle,, weakly compactly generated spaces

147. CMB 2005 (vol 48 pp. 340)

Andruchow, Esteban
Short Geodesics of Unitaries in the $L^2$ Metric
Let $\M$ be a type II$_1$ von Neumann algebra, $\tau$ a trace in $\M$, and $\l2$ the GNS Hilbert space of $\tau$. We regard the unitary group $U_\M$ as a subset of $\l2$ and characterize the shortest smooth curves joining two fixed unitaries in the $L^2$ metric. As a consequence of this we obtain that $U_\M$, though a complete (metric) topological group, is not an embedded riemannian submanifold of $\l2$

Keywords:unitary group, short geodesics, infinite dimensional riemannian manifolds.
Categories:46L51, 58B10, 58B25

148. CMB 2005 (vol 48 pp. 251)

Murphy, G. J.
The Index Theory Associated to a Non-Finite Trace on a $C^\ast$-Algebra
The index theory considered in this paper, a generalisation of the classical Fredholm index theory, is obtained in terms of a non-finite trace on a unital $C^\ast$-algebra. We relate it to the index theory of M.~Breuer, which is developed in a von~Neumann algebra setting, by means of a representation theorem. We show how our new index theory can be used to obtain an index theorem for Toeplitz operators on the compact group $\mathbf{U}(2)$, where the classical index theory does not give any interesting result.

Categories:46L, 47B35, 47L80

149. CMB 2005 (vol 48 pp. 283)

Thibault, Lionel; Zagrodny, Dariusz
Enlarged Inclusion of Subdifferentials
This paper studies the integration of inclusion of subdifferentials. Under various verifiable conditions, we obtain that if two proper lower semicontinuous functions $f$ and $g$ have the subdifferential of $f$ included in the $\gamma$-enlargement of the subdifferential of $g$, then the difference of those functions is $ \gamma$-Lipschitz over their effective domain.

Keywords:subdifferential,, directionally regular function,, approximate convex function,, subdifferentially and directionally stable function
Categories:49J52, 46N10, 58C20

150. CMB 2005 (vol 48 pp. 161)

Betancor, Jorge J.
Hankel Convolution Operators on Spaces of Entire Functions of Finite Order
In this paper we study Hankel transforms and Hankel convolution operators on spaces of entire functions of finite order and their duals.

Keywords:Hankel transform, convolution, entire functions, finite order
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