176. CMB 2003 (vol 46 pp. 242)
 Litvak, A. E.; Milman, V. D.

Euclidean Sections of Direct Sums of Normed Spaces
We study the dimension of ``random'' Euclidean sections of direct sums of
normed spaces. We compare the obtained results with results from \cite{LMS},
to show that for the direct sums the standard randomness with respect to the
Haar measure on Grassmanian coincides with a much ``weaker'' randomness of
``diagonal'' subspaces (Corollary~\ref{sle} and explanation after). We also
add some relative information on ``phase transition''.
Keywords:Dvoretzky theorem, ``random'' Euclidean section, phase transition in asymptotic convexity Categories:46B07, 46B09, 46B20, 52A21 

177. CMB 2003 (vol 46 pp. 98)
 Larsen, Nadia S.

Crossed Products by Semigroups of Endomorphisms and Groups of Partial Automorphisms
We consider a class $(A, S, \alpha)$ of dynamical systems,
where $S$ is an Ore semigroup and $\alpha$ is an action such that
each $\alpha_s$ is injective and extendible ({\it i.e.} it extends to a
nonunital endomorphism of the multiplier algebra), and has range an
ideal of $A$. We show that there is a partial action on the fixedpoint
algebra under the canonical coaction of the enveloping group $G$ of $S$
constructed in \cite[Proposition~6.1]{LR2}. It turns out that the full
crossed product by this coaction is isomorphic to $A\rtimes_\alpha S$.
If the coaction is moreover normal, then the isomorphism can be extended
to include the reduced crossed product. We look then at invariant ideals
and finally, at examples of systems where our results apply.
Category:46L55 

178. CMB 2003 (vol 46 pp. 80)
 Erlijman, Juliana

MultiSided Braid Type Subfactors, II
We show that the multisided inclusion $R^{\otimes l} \subset R$ of
braidtype subfactors of the hyperfinite II$_1$ factor $R$, introduced
in {\it Multisided braid type subfactors} [E3], contains a sequence
of intermediate subfactors: $R^{\otimes l} \subset R^{\otimes l1}
\subset \cdots \subset R^{\otimes 2} \subset R$. That is, every
$t$sided subfactor is an intermediate subfactor for the inclusion
$R^{\otimes l} \subset R$, for $2\leq t\leq l$. Moreover, we also
show that if $t>m$ then $R^{\otimes t} \subset R^{\otimes m}$ is
conjugate to $R^{\otimes tm+1} \subset R$. Thus, if the braid
representation considered is associated to one of the classical Lie
algebras then the asymptotic inclusions for the JonesWenzl subfactors
are intermediate subfactors.
Category:46L37 

179. CMB 2002 (vol 45 pp. 321)
 Brenken, Berndt

$C^{\ast}$Algebras of Infinite Graphs and CuntzKrieger Algebras
The CuntzKrieger algebra $\mathcal{O}_B$ is defined for an
arbitrary, possibly infinite and infinite valued, matrix $B$. A graph
$C^{\ast}$algebra $G^{\ast} (E)$ is introduced for an arbitrary
directed graph $E$, and is shown to coincide with a previously defined
graph algebra $C^{\ast} (E)$ if each source of $E$ emits only finitely
many edges. Each graph algebra $G^{\ast} (E)$ is isomorphic to the
CuntzKrieger algebra $\mathcal{O}_B$ where $B$ is the vertex matrix
of~$E$.
Categories:46LXX, 05C50 

180. CMB 2002 (vol 45 pp. 309)
 Xia, Jingbo

Joint Mean Oscillation and Local Ideals in the Toeplitz Algebra II: Local Commutivity and Essential Commutant
A wellknown theorem of Sarason [11] asserts that if $[T_f,T_h]$ is
compact for every $h \in H^\infty$, then $f \in H^\infty + C(T)$.
Using local analysis in the full Toeplitz algebra $\calT = \calT
(L^\infty)$, we show that the membership $f \in H^\infty + C(T)$ can
be inferred from the compactness of a much smaller collection of
commutators $[T_f,T_h]$. Using this strengthened result and a theorem
of Davidson [2], we construct a proper $C^\ast$subalgebra $\calT
(\calL)$ of $\calT$ which has the same essential commutant as that of
$\calT$. Thus the image of $\calT (\calL)$ in the Calkin algebra does
not satisfy the double commutant relation [12], [1]. We will also
show that no {\it separable} subalgebra $\calS$ of $\calT$ is capable
of conferring the membership $f \in H^\infty + C(T)$ through the
compactness of the commutators $\{[T_f,S] : S \in \calS\}$.
Categories:46H10, 47B35, 47C05 

181. CMB 2002 (vol 45 pp. 265)
 Nawrocki, Marek

On the Smirnov Class Defined by the Maximal Function
H.~O.~Kim has shown that contrary to the case of
$H^p$space, the Smirnov class $M$ defined by the radial maximal
function is essentially smaller than the classical Smirnov class
of the disk. In the paper we show that these two classes have the
same corresponding locally convex structure, {\it i.e.} they have the
same dual spaces and the same Fr\'echet envelopes. We describe a
general form of a continuous linear functional on $M$ and
multiplier from $M$ into $H^p$, $0 < p \leq \infty$.
Keywords:Smirnov class, maximal radial function, multipliers, dual space, FrÃ©chet envelope Categories:46E10, 30A78, 30A76 

182. CMB 2002 (vol 45 pp. 232)
 Ji, Min; Shen, Zhongmin

On Strongly Convex Indicatrices in Minkowski Geometry
The geometry of indicatrices is the foundation of Minkowski geometry.
A strongly convex indicatrix in a vector space is a strongly convex
hypersurface. It admits a Riemannian metric and has a distinguished
invariant(Cartan) torsion. We prove the existence of nontrivial
strongly convex indicatrices with vanishing mean torsion and discuss
the relationship between the mean torsion and the Riemannian curvature
tensor for indicatrices of Randers type.
Categories:46B20, 53C21, 53A55, 52A20, 53B40, 53A35 

183. CMB 2002 (vol 45 pp. 60)
184. CMB 2002 (vol 45 pp. 46)
185. CMB 2002 (vol 45 pp. 3)
 Azagra, D.; Dobrowolski, T.

RealAnalytic Negligibility of Points and Subspaces in Banach Spaces, with Applications
We prove that every infinitedimensional Banach space $X$ having a
(not necessarily equivalent) realanalytic norm is realanalytic
diffeomorphic to $X \setminus \{0\}$. More generally, if $X$ is an
infinitedimensional Banach space and $F$ is a closed subspace of $X$
such that there is a realanalytic seminorm on $X$ whose set of zeros
is $F$, and $X/F$ is infinitedimensional, then $X$ and $X \setminus
F$ are realanalytic diffeomorphic. As an application we show the
existence of realanalytic free actions of the circle and the
$n$torus on certain Banach spaces.
Categories:46B20, 58B99 

186. CMB 2001 (vol 44 pp. 504)
 Zhang, Yong

Weak Amenability of a Class of Banach Algebras
We show that, if a Banach algebra $\A$ is a left ideal in its second
dual algebra and has a left bounded approximate identity, then the
weak amenability of $\A$ implies the ($2m+1$)weak amenability of $\A$
for all $m\geq 1$.
Keywords:$n$weak amenability, left ideals, left bounded approximate identity Categories:46H20, 46H10, 46H25 

187. CMB 2001 (vol 44 pp. 335)
 Stacey, P. J.

Inductive Limit Toral Automorphisms of Irrational Rotation Algebras
Irrational rotation $C^*$algebras have an inductive limit
decomposition in terms of matrix algebras over the space of continuous
functions on the circle and this decomposition can be chosen to be
invariant under the flip automorphism. It is shown that the flip is
essentially the only toral automorphism with this property.
Categories:46L40, 46L35 

188. CMB 2001 (vol 44 pp. 370)
 Weston, Anthony

On Locating Isometric $\ell_{1}^{(n)}$
Motivated by a question of Per Enflo, we develop a hypercube criterion
for locating linear isometric copies of $\lone$ in an arbitrary real
normed space $X$.
The said criterion involves finding $2^{n}$ points in $X$ that satisfy
one metric equality. This contrasts nicely to the standard classical
criterion wherein one seeks $n$ points that satisfy $2^{n1}$ metric
equalities.
Keywords:normed spaces, hypercubes Categories:46B04, 05C10, 05B99 

189. CMB 2001 (vol 44 pp. 355)
 Weaver, Nik

Hilbert Bimodules with Involution
We examine Hilbert bimodules which possess a (generally unbounded)
involution. Topics considered include a linking algebra
representation, duality, locality, and the role of these bimodules
in noncommutative differential geometry
Categories:46L08, 46L57, 46L87 

190. CMB 2001 (vol 44 pp. 105)
191. CMB 2000 (vol 43 pp. 418)
 Gong, Guihua; Jiang, Xinhui; Su, Hongbing

Obstructions to $\mathcal{Z}$Stability for Unital Simple $C^*$Algebras
Let $\cZ$ be the unital simple nuclear infinite dimensional
$C^*$algebra which has the same Elliott invariant as $\bbC$,
introduced in \cite{JS}. A $C^*$algebra is called $\cZ$stable
if $A \cong A \otimes \cZ$. In this note we give some necessary
conditions for a unital simple $C^*$algebra to be $\cZ$stable.
Keywords:simple $C^*$algebra, $\mathcal{Z}$stability, weak (un)perforation in $K_0$ group, property $\Gamma$, finiteness Category:46L05 

192. CMB 2000 (vol 43 pp. 257)
193. CMB 2000 (vol 43 pp. 320)
194. CMB 2000 (vol 43 pp. 368)
 Litvak, A. E.

KahaneKhinchin's Inequality for QuasiNorms
We extend the recent results of R.~Lata{\l}a and O.~Gu\'edon about
equivalence of $L_q$norms of logconcave random variables
(KahaneKhinchin's inequality) to the quasiconvex case. We
construct examples of quasiconvex bodies $K_n \subset \R$ which
demonstrate that this equivalence fails for uniformly distributed
vector on $K_n$ (recall that the uniformly distributed vector on a
convex body is logconcave). Our examples also show the lack of the
exponential decay of the ``tail" volume (for convex bodies such
decay was proved by M.~Gromov and V.~Milman).
Categories:46B09, 52A30, 60B11 

195. CMB 2000 (vol 43 pp. 193)
 Magajna, Bojan

C$^*$Convexity and the Numerical Range
If $A$ is a prime C$^*$algebra, $a \in A$ and $\lambda$ is in the
numerical range $W(a)$ of $a$, then for each $\varepsilon > 0$ there
exists an element $h \in A$ such that $\norm{h} = 1$ and $\norm{h^*
(a\lambda)h} < \varepsilon$. If $\lambda$ is an extreme point of
$W(a)$, the same conclusion holds without the assumption that $A$ is
prime. Given any element $a$ in a von Neumann algebra (or in a
general C$^*$algebra) $A$, all normal elements in the weak* closure
(the norm closure, respectively) of the C$^*$convex hull of $a$ are
characterized.
Categories:47A12, 46L05, 46L10 

196. CMB 2000 (vol 43 pp. 138)
 Boyd, C.

Exponential Laws for the Nachbin Ported Topology
We show that for $U$ and $V$ balanced open subsets of (Qno) Fr\'echet
spaces $E$ and $F$ that we have the topological identity
$$
\bigl( {\cal H}(U\times V), \tau_\omega \bigr) = \biggl( {\cal H}
\Bigl( U; \bigl( {\cal H}(V), \tau_\omega \bigr) \Bigr), \tau_\omega
\biggr).
$$
Analogous results for the compact open topology have long been
established. We also give an example to show that the (Qno)
hypothesis on both $E$ and $F$ is necessary.
Categories:46G20, 18D15, 46M05 

197. CMB 2000 (vol 43 pp. 208)
 Matoušková, Eva

Extensions of Continuous and Lipschitz Functions
We show a result slightly more general than the following. Let $K$
be a compact Hausdorff space, $F$ a closed subset of $K$, and $d$ a
lower semicontinuous metric on $K$. Then each continuous function
$f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on
$K$ which is Lipschitz in $d$. The extension has the same supremum
norm and the same Lipschitz constant.
As a corollary we get that a Banach space $X$ is reflexive if and only
if each bounded, weakly continuous and norm Lipschitz function
defined on a weakly closed subset of $X$ admits a weakly continuous,
norm Lipschitz extension defined on the entire space $X$.
Keywords:extension, continous, Lipschitz, Banach space Categories:54C20, 46B10 

198. CMB 2000 (vol 43 pp. 69)
199. CMB 1999 (vol 42 pp. 344)
200. CMB 1999 (vol 42 pp. 321)
 Kikuchi, Masato

Averaging Operators and Martingale Inequalities in Rearrangement Invariant Function Spaces
We shall study some connection between averaging operators and
martingale inequalities in rearrangement invariant function spaces.
In Section~2 the equivalence between Shimogaki's theorem and some
martingale inequalities will be established, and in Section~3 the
equivalence between Boyd's theorem and martingale inequalities with
change of probability measure will be established.
Keywords:martingale inequalities, rearrangement invariant function spaces Categories:60G44, 60G46, 46E30 
