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151. CMB 2003 (vol 46 pp. 632)

Runde, Volker
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

152. CMB 2003 (vol 46 pp. 588)

Monteiro, Martha Salerno
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 $A$.


153. CMB 2003 (vol 46 pp. 575)

Marshall, M.
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

154. CMB 2003 (vol 46 pp. 538)

Borwein, Jonathan; Fitzpatrick, Simon; Girgensohn, Roland
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

155. 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

156. CMB 2003 (vol 46 pp. 365)

Kishimoto, Akitaka; Ozawa, Narutaka; Sakai, Shôichirô
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

157. 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

158. CMB 2003 (vol 46 pp. 441)

Stacey, P. J.
An Inductive Limit Model for the $K$-Theory of the Generator-Interchanging 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

159. CMB 2003 (vol 46 pp. 419)

Masuda, Toshihiko
On Non-Strongly Free Automorphisms of Subfactors of Type III$_0$
We determine when an automorphism of a subfactor of type III$_0$ with finite index is non-strongly free in the sense of C.~Winsl\o w in terms of the modular endomorphisms introduced by M.~Izumi.


160. 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

161. CMB 2003 (vol 46 pp. 164)

Dean, Andrew J.
Classification of $\AF$ Flows
An $\AF$ flow is a one-parameter 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/path-space construction, and one in terms of a modified $K_0$ functor.

Categories:46L57, 46L35

162. 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

163. CMB 2003 (vol 46 pp. 161)

Cabello Sánchez, Félix; Castillo, Jesús M. F.
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

164. CMB 2003 (vol 46 pp. 80)

Erlijman, Juliana
Multi-Sided Braid Type Subfactors, II
We show that the multi-sided inclusion $R^{\otimes l} \subset R$ of braid-type subfactors of the hyperfinite II$_1$ factor $R$, introduced in {\it Multi-sided braid type subfactors} [E3], contains a sequence of intermediate subfactors: $R^{\otimes l} \subset R^{\otimes l-1} \subset \cdots \subset R^{\otimes 2} \subset R$. 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 $R^{\otimes t-m+1} \subset R$. Thus, if the braid representation considered is associated to one of the classical Lie algebras then the asymptotic inclusions for the Jones-Wenzl subfactors are intermediate subfactors.


165. 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 non-unital endomorphism of the multiplier algebra), and has range an ideal of $A$. We show that there is a partial action on the fixed-point 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.


166. CMB 2002 (vol 45 pp. 321)

Brenken, Berndt
$C^{\ast}$-Algebras of Infinite Graphs and Cuntz-Krieger Algebras
The Cuntz-Krieger 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 Cuntz-Krieger algebra $\mathcal{O}_B$ where $B$ is the vertex matrix of~$E$.

Categories:46LXX, 05C50

167. 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, Fréchet envelope
Categories:46E10, 30A78, 30A76

168. 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 well-known 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

169. 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 non-trivial 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

170. CMB 2002 (vol 45 pp. 3)

Azagra, D.; Dobrowolski, T.
Real-Analytic Negligibility of Points and Subspaces in Banach Spaces, with Applications
We prove that every infinite-dimensional Banach space $X$ having a (not necessarily equivalent) real-analytic norm is real-analytic diffeomorphic to $X \setminus \{0\}$. More generally, if $X$ is an infinite-dimensional Banach space and $F$ is a closed subspace of $X$ such that there is a real-analytic seminorm on $X$ whose set of zeros is $F$, and $X/F$ is infinite-dimensional, then $X$ and $X \setminus F$ are real-analytic diffeomorphic. As an application we show the existence of real-analytic free actions of the circle and the $n$-torus on certain Banach spaces.

Categories:46B20, 58B99

171. CMB 2002 (vol 45 pp. 60)

Dranishnikov, A. N.; Gong, G.; Lafforgue, V.; Yu, G.
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.


172. CMB 2002 (vol 45 pp. 46)

Dafni, Galia
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

173. 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

174. 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

175. 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 $2^{n-1}$ metric equalities.

Keywords:normed spaces, hypercubes
Categories:46B04, 05C10, 05B99
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