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151. CMB 2004 (vol 47 pp. 481)

Bekjan, Turdebek N.
A New Characterization of Hardy Martingale Cotype Space
We give a new characterization of Hardy martingale cotype property of complex quasi-Banach space by using the existence of a kind of plurisubharmonic functions. We also characterize the best constants of Hardy martingale inequalities with values in the complex quasi-Banach space.

Keywords:Hardy martingale, Hardy martingale cotype,, plurisubharmonic function
Categories:46B20, 52A07, 60G44

152. CMB 2004 (vol 47 pp. 615)

Randrianantoanina, Narcisse
$C^*$-Algebras and Factorization Through Diagonal Operators
Let $\cal A$ be a $C^*$-algebra and $E$ be a Banach space with the Radon-Nikodym property. We prove that if $j$ is an embedding of $E$ into an injective Banach space then for every absolutely summing operator $T:\mathcal{A}\longrightarrow E$, the composition $j \circ T$ factors through a diagonal operator from $l^{2}$ into $l^{1}$. In particular, $T$ factors through a Banach space with the Schur property. Similarly, we prove that for $2
Keywords:$C^*$-algebras, summing operators, diagonal operators,, Radon-Nikodym property
Categories:46L50, 47D15

153. CMB 2004 (vol 47 pp. 553)

Kerr, David
A Geometric Approach to Voiculescu-Brown Entropy
A basic problem in dynamics is to identify systems with positive entropy, i.e., systems which are ``chaotic.'' While there is a vast collection of results addressing this issue in topological dynamics, the phenomenon of positive entropy remains by and large a mystery within the broader noncommutative domain of $C^*$-algebraic dynamics. To shed some light on the noncommutative situation we propose a geometric perspective inspired by work of Glasner and Weiss on topological entropy. This is a written version of the author's talk at the Winter 2002 Meeting of the Canadian Mathematical Society in Ottawa, Ontario.

Categories:46L55, 37B40

154. CMB 2004 (vol 47 pp. 540)

Jain, Pankaj; Jain, Pawan K.; Gupta, Babita
Compactness of Hardy-Type Operators over Star-Shaped Regions in $\mathbb{R}^N$
We study a compactness property of the operators between weighted Lebesgue spaces that average a function over certain domains involving a star-shaped region. The cases covered are (i) when the average is taken over a difference of two dilations of a star-shaped region in $\RR^N$, and (ii) when the average is taken over all dilations of star-shaped regions in $\RR^N$. These cases include, respectively, the average over annuli and the average over balls centered at origin.

Keywords:Hardy operator, Hardy-Steklov operator, compactness, boundedness, star-shaped regions
Categories:46E35, 26D10

155. CMB 2004 (vol 47 pp. 445)

Pirkovskii, A. Yu.
Biprojectivity and Biflatness for Convolution Algebras of Nuclear Operators
For a locally compact group $G$, the convolution product on the space $\nN(L^p(G))$ of nuclear operators was defined by Neufang \cite{Neuf_PhD}. We study homological properties of the convolution algebra $\nN(L^p(G))$ and relate them to some properties of the group $G$, such as compactness, finiteness, discreteness, and amenability.

Categories:46M10, 46H25, 43A20, 16E65

156. CMB 2004 (vol 47 pp. 206)

Hurri-Syrjänen, Ritva
The Poincaré Inequality and Reverse Doubling Weights
We show that Poincar\'e inequalities with reverse doubling weights hold in a large class of irregular domains whenever the weights satisfy certain conditions. Examples of these domains are John domains.

Keywords:reverse doubling weights, Poincaré inequality, John domains

157. CMB 2004 (vol 47 pp. 49)

Lindström, Mikael; Makhmutov, Shamil; Taskinen, Jari
The Essential Norm of a Bloch-to-$Q_p$ Composition Operator
The $Q_p$ spaces coincide with the Bloch space for $p>1$ and are subspaces of $\BMOA$ for $0
Keywords:Bloch space, little Bloch space, $\BMOA$, $\VMOA$, $Q_p$ spaces,, composition operator, compact operator, essential norm
Categories:47B38, 47B10, 46E40, 46E15

158. CMB 2004 (vol 47 pp. 108)

Śliwa, Wiesław
On Universal Schauder Bases in Non-Archimedean Fréchet Spaces
It is known that any non-archimedean Fr\'echet space of countable type is isomorphic to a subspace of $c_0^{\mathbb{N}}$. In this paper we prove that there exists a non-archimedean Fr\'echet space $U$ with a basis $(u_n)$ such that any basis $(x_n)$ in a non-archimedean Fr\'echet space $X$ is equivalent to a subbasis $(u_{k_n})$ of $(u_n)$. Then any non-archimedean Fr\'echet space with a basis is isomorphic to a complemented subspace of $U$. In contrast to this, we show that a non-archimedean Fr\'echet space $X$ with a basis $(x_n)$ is isomorphic to a complemented subspace of $c_0^{\mathbb{N}}$ if and only if $X$ is isomorphic to one of the following spaces: $c_0$, $c_0 \times \mathbb{K}^{\mathbb{N}}$, $\mathbb{K}^{\mathbb{N}}$, $c_0^{\mathbb{N}}$. Finally, we prove that there is no nuclear non-archimedean Fr\'echet space $H$ with a basis $(h_n)$ such that any basis $(y_n)$ in a nuclear non-archimedean Fr\'echet space $Y$ is equivalent to a subbasis $(h_{k_n})$ of $(h_n)$.

Keywords:universal bases, complemented subspaces with bases
Categories:46S10, 46A35

159. 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

160. 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

161. 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 $A$.


162. 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

163. 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

164. 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

165. 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

166. 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

167. 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

168. 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.


169. 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

170. 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

171. 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

172. 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

173. 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.


174. 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.


175. 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
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