101. 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 

102. 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 

103. 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 

104. 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 

105. 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 

106. CMB 2007 (vol 50 pp. 227)
107. 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 nonexact $C^*$algebras and
property T groups, we show that one of his examples of noninvertible
$C^*$extensions is not semiinvertible. 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 

108. CMB 2007 (vol 50 pp. 172)
109. 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 complexvalued functions on the
spectrum. These spaces are defined in terms of solutions to the nonlinear
CauchyRiemann 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 

110. CMB 2007 (vol 50 pp. 149)
 Śliwa, Wiesław

On Quotients of NonArchimedean KÃ¶the Spaces
We show that there exists a nonarchimedean
Fr\'echetMontel space $W$ with a basis and with a continuous norm
such that any nonarchimedean Fr\'echet space of countable type is isomorphic
to a quotient of $W$. We also prove that any nonarchimedean nuclear
Fr\'echet space is isomorphic to a quotient of some nonarchimedean nuclear
Fr\'echet space with a basis and with a continuous norm.
Keywords:Nonarchimedean KÃ¶the spaces, nuclear FrÃ©chet spaces, pseudobases Categories:46S10, 46A45 

111. CMB 2007 (vol 50 pp. 138)
112. CMB 2007 (vol 50 pp. 85)
 Han, Deguang

Classification of Finite GroupFrames and SuperFrames
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, groupframes, frame representations, disjoint frames Categories:42C15, 46C05, 47B10 

113. CMB 2006 (vol 49 pp. 536)
114. 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 

115. CMB 2006 (vol 49 pp. 414)
116. 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 

117. CMB 2006 (vol 49 pp. 213)
118. CMB 2006 (vol 49 pp. 313)
119. 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 RogersShephard 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 crosspolytope. 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 BanachMazur distance to the
regular hexagon.
Keywords:convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, BanachMazur compactum, (modified) BanachMazur distance, volume ratio Categories:52A40, 46B20 

120. 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 

121. 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 

122. CMB 2005 (vol 48 pp. 481)
 Azagra, D.; Fabian, M.; JiménezSevilla, 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 $C^n$\nobreakdashsmooth
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
infinitedimensional, 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 finitedimensional case, more
restrictive conditions are required. We also study the Fr\'echet
smooth case for mappings from $\mathbb R^n$ to a separable
infinitedimensional Banach space and the G\^ateaux smooth case
for mappings defined on a separable infinitedimensional Banach
space and with values in a separable Banach space.
Category:46B20 

123. 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 

124. CMB 2005 (vol 48 pp. 455)
 Rychtář, Jan

On GÃ¢teaux Differentiability of Convex Functions in WCG Spaces
It is shown, using the BorweinPreiss 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:GÃ¢teaux smoothness, BorweinPreiss variational principle,, weakly compactly generated spaces Category:46B20 

125. 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 
