176. CMB 2003 (vol 46 pp. 538)
177. CMB 2003 (vol 46 pp. 509)
 Benson, David J.; Kumjian, Alex; Phillips, N. Christopher

Symmetries of Kirchberg Algebras
Let $G_0$ and $G_1$ be countable abelian groups. Let $\gamma_i$ be an
automorphism of $G_i$ of order two. Then there exists a unital
Kirchberg algebra $A$ satisfying the Universal Coefficient Theorem and
with $[1_A] = 0$ in $K_0 (A)$, and an automorphism $\alpha \in
\Aut(A)$ of order two, such that $K_0 (A) \cong G_0$, such that $K_1
(A) \cong G_1$, and such that $\alpha_* \colon K_i (A) \to K_i (A)$ is
$\gamma_i$. As a consequence, we prove that every
$\mathbb{Z}_2$graded countable module over the representation ring $R
(\mathbb{Z}_2)$ of $\mathbb{Z}_2$ is isomorphic to the equivariant
$K$theory $K^{\mathbb{Z}_2} (A)$ for some action of $\mathbb{Z}_2$ on
a unital Kirchberg algebra~$A$.
Along the way, we prove that every not necessarily finitely generated
$\mathbb{Z} [\mathbb{Z}_2]$module which is free as a
$\mathbb{Z}$module has a direct sum decomposition with only three
kinds of summands, namely $\mathbb{Z} [\mathbb{Z}_2]$ itself and
$\mathbb{Z}$ on which the nontrivial element of $\mathbb{Z}_2$ acts
either trivially or by multiplication by $1$.
Categories:20C10, 46L55, 19K99, 19L47, 46L40, 46L80 

178. CMB 2003 (vol 46 pp. 481)
 Bachir, M.; Lancien, G.

On the Composition of Differentiable Functions
We prove that a Banach space $X$ has the Schur property if and only if every
$X$valued weakly differentiable function is Fr\'echet differentiable. We
give a general result on the Fr\'echet differentiability of $f\circ T$, where
$f$ is a Lipschitz function and $T$ is a compact linear operator. Finally
we study, using in particular a smooth variational principle, the
differentiability of the semi norm $\Vert \ \Vert_{\lip}$ on various spaces
of Lipschitz functions.
Categories:58C20, 46B20 

179. CMB 2003 (vol 46 pp. 365)
180. CMB 2003 (vol 46 pp. 457)
 Toms, Andrew

Strongly Perforated $K_{0}$Groups of Simple $C^{*}$Algebras
In the sequel we construct simple, unital, separable, stable, amenable
$C^{*}$algebras for which the ordered $K_{0}$group is strongly
perforated and group isomorphic to $Z$. The particular order structures
to be constructed will be described in detail below, and all
known results of this type will be generalised.
Categories:46, 19 

181. CMB 2003 (vol 46 pp. 441)
182. CMB 2003 (vol 46 pp. 419)
183. CMB 2003 (vol 46 pp. 388)
 Lin, Huaxin

Tracially Quasidiagonal Extensions
It is known that a unital simple $C^*$algebra $A$ with tracial
topological rank zero has real rank zero. We show in this note that,
in general, there are unital $C^*$algebras with tracial topological
rank zero that have real rank other than zero.
Let $0\to J\to E\to A\to 0$ be a short exact sequence of
$C^*$algebras. Suppose that $J$ and $A$ have tracial topological
rank zero. It is known that $E$ has tracial topological rank zero
as a $C^*$algebra if and only if $E$ is tracially quasidiagonal
as an extension. We present an example of a tracially
quasidiagonal extension which is not quasidiagonal.
Keywords:tracially quasidiagonal extensions, tracial rank Categories:46L05, 46L80 

184. CMB 2003 (vol 46 pp. 164)
 Dean, Andrew J.

Classification of $\AF$ Flows
An $\AF$ flow is a oneparameter automorphism group of an $\AF$
$C^*$algebra $A$ such that there exists an increasing sequence of
invariant finite dimensional sub$C^*$algebras whose union is dense in
$A$. In this paper, a classification of $C^*$dynamical systems of
this form up to equivariant isomorphism is presented. Two pictures
of the actions are given, one in terms of a modified Bratteli
diagram/pathspace construction, and one in terms of a modified
$K_0$ functor.
Categories:46L57, 46L35 

185. 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 

186. CMB 2003 (vol 46 pp. 161)
187. 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 

188. 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 

189. 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 

190. 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 

191. 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 

192. 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 

193. 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 

194. CMB 2002 (vol 45 pp. 60)
195. CMB 2002 (vol 45 pp. 46)
196. 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 

197. 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 

198. 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 

199. 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 

200. CMB 2001 (vol 44 pp. 105)