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126. CMB 2004 (vol 47 pp. 49)
The Essential Norm of a Blochto$Q_p$ Composition Operator The $Q_p$ spaces coincide with the Bloch space for $p>1$ and are
subspaces of $\BMOA$ for $0

127. CMB 2003 (vol 46 pp. 509)
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 
128. CMB 2003 (vol 46 pp. 632)
The Operator Amenability of Uniform Algebras We prove a quantized version of a theorem by M.~V.~She\u{\i}nberg:
A uniform algebra equipped with its canonical, {\it i.e.}, minimal,
operator space structure is operator amenable if and only if it is
a commutative $C^\ast$algebra.
Keywords:uniform algebras, amenable Banach algebras, operator amenability, minimal, operator space Categories:46H20, 46H25, 46J10, 46J40, 47L25 
129. CMB 2003 (vol 46 pp. 588)
Weakly Stable Relations and Inductive Limits of $C^\ast$algebras We show that if $\mathcal{A}$ is a class of $C^\ast$algebras for which
the set of formal relations $\mathcal{R}$ is weakly stable, then $\mathcal{R}$
is weakly stable for the class $\mathcal{B}$ that contains $\mathcal{A}$ and
all the inductive limits that can be constructed with the $C^\ast$algebras in
$\mathcal{A}$.
A set of formal relations $\mathcal{R}$ is said to be {\it weakly stable\/} for
a class $\mathcal{C}$ of $C^\ast$algebras if, in any $C^\ast$algebra $A\in
\mathcal{C}$, close to an approximate representation of the set $\mathcal{R}$
in $A$ there is an exact representation of $\mathcal{R}$ in $A$.
Category:46L05 
130. CMB 2003 (vol 46 pp. 575)
Optimization of Polynomial Functions This paper develops a refinement of Lasserre's algorithm for
optimizing a polynomial on a basic closed semialgebraic set via
semidefinite programming and addresses an open question concerning the
duality gap. It is shown that, under certain natural stability
assumptions, the problem of optimization on a basic closed set reduces
to the compact case.
Categories:14P10, 46L05, 90C22 
131. CMB 2003 (vol 46 pp. 538)
Subdifferentials Whose Graphs Are Not Norm$\times$Weak* Closed In this note we give examples of convex functions whose
subdifferentials have unpleasant properties. Particularly, we
exhibit a proper lower semicontinuous convex function on a
separable Hilbert space such that the graph of its subdifferential
is not closed in the product of the norm and bounded weak
topologies. We also exhibit a set whose sequential normal cone is
not norm closed.
Categories:46N10, 47H05 
132. CMB 2003 (vol 46 pp. 481)
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 
133. CMB 2003 (vol 46 pp. 365)
Homogeneity of the Pure State Space of a Separable $C^*$Algebra We prove that the pure state space is homogeneous under the action of
the automorphism group (or the subgroup of asymptotically inner
automorphisms) for all the separable simple $C^*$algebras. The
first result of this kind was shown by Powers for the UHF algbras
some 30 years ago.
Categories:46L40, 46L30 
134. CMB 2003 (vol 46 pp. 457)
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 
135. CMB 2003 (vol 46 pp. 441)
An Inductive Limit Model for the $K$Theory of the GeneratorInterchanging Antiautomorphism of an Irrational Rotation Algebra 
An Inductive Limit Model for the $K$Theory of the GeneratorInterchanging Antiautomorphism of an Irrational Rotation Algebra Let $A_\theta$ be the universal $C^*$algebra generated by two
unitaries $U$, $V$ satisfying $VU=e^{2\pi i\theta} UV$ and let $\Phi$
be the antiautomorphism of $A_\theta$ interchanging $U$ and $V$. The
$K$theory of $R_\theta=\{a\in A_\theta:\Phi(a)=a^*\}$ is computed. When
$\theta$ is irrational, an inductive limit of algebras of the form
$M_q(C(\mathbb{T})) \oplus M_{q'} (\mathbb{R}) \oplus M_q(\mathbb{R})$
is constructed which has complexification $A_\theta$ and the same
$K$theory as $R_\theta$.
Categories:46L35, 46L80 
136. CMB 2003 (vol 46 pp. 419)
On NonStrongly Free Automorphisms of Subfactors of Type III$_0$ We determine when an automorphism of a subfactor of type III$_0$
with finite index is nonstrongly free in the sense of C.~Winsl\o w
in terms of the modular endomorphisms introduced by M.~Izumi.
Category:46L37 
137. CMB 2003 (vol 46 pp. 388)
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 
138. CMB 2003 (vol 46 pp. 242)
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 
139. CMB 2003 (vol 46 pp. 164)
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 
140. CMB 2003 (vol 46 pp. 161)
Answer to a Question of S.~Rolewicz We exhibit examples of $F$spaces with trivial dual which are
isomorphic to its quotient by a line, thus solving a problem in
Rolewicz's {\it Metric Linear Spaces}.
Categories:46M99, 46M15, 46A16, 46B20 
141. CMB 2003 (vol 46 pp. 98)
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 
142. CMB 2003 (vol 46 pp. 80)
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 
143. CMB 2002 (vol 45 pp. 321)
$C^{\ast}$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 
144. CMB 2002 (vol 45 pp. 265)
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 
145. CMB 2002 (vol 45 pp. 309)
Joint Mean Oscillation and Local Ideals in the Toeplitz Algebra II: Local Commutivity and Essential Commutant 
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 
146. CMB 2002 (vol 45 pp. 232)
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 
147. CMB 2002 (vol 45 pp. 3)
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 
148. CMB 2002 (vol 45 pp. 60)
Uniform Embeddings into Hilbert Space and a Question of Gromov Gromov introduced the concept of uniform embedding into Hilbert space
and asked if every separable metric space admits a uniform embedding
into Hilbert space. In this paper, we study uniform embedding into
Hilbert space and answer Gromov's question negatively.
Category:46C05 
149. CMB 2002 (vol 45 pp. 46)
Local $\VMO$ and Weak Convergence in $\hone$ A local version of $\VMO$ is defined, and the local Hardy space
$\hone$ is shown to be its dual. An application to weak$*$
convergence in $\hone$ is proved.
Categories:42B30, 46E99 
150. CMB 2001 (vol 44 pp. 504)
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 