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126. 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

127. 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 functionCategories:46B20, 52A07, 60G44

128. 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

129. CMB 2004 (vol 47 pp. 206)

Hurri-Syrjänen, Ritva
 The PoincarÃ© 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, PoincarÃ© inequality, John domainsCategory:46E35

130. CMB 2004 (vol 47 pp. 108)

Śliwa, Wiesław
 On Universal Schauder Bases in Non-Archimedean FrÃ©chet 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 basesCategories:46S10, 46A35

131. 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$spaces,, composition operator, compact operator, essential normCategories:47B38, 47B10, 46E40, 46E15 132. CMB 2003 (vol 46 pp. 481) Bachir, M.; Lancien, G.  On the Composition of Differentiable Functions We prove that a Banach space$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}$on various spaces of Lipschitz functions. Categories:58C20, 46B20 133. CMB 2003 (vol 46 pp. 632) Runde, Volker  The Operator Amenability of Uniform Algebras We prove a quantized version of a theorem by M.~V.~She\u{\i}nberg: A uniform algebra equipped with its canonical, {\it i.e.}, minimal, operator space structure is operator amenable if and only if it is a commutative$C^\ast$-algebra. Keywords:uniform algebras, amenable Banach algebras, operator amenability, minimal, operator spaceCategories:46H20, 46H25, 46J10, 46J40, 47L25 134. CMB 2003 (vol 46 pp. 588) Monteiro, Martha Salerno  Weakly Stable Relations and Inductive Limits of$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$. Category:46L05 135. CMB 2003 (vol 46 pp. 575) Marshall, M.  Optimization of Polynomial Functions This paper develops a refinement of Lasserre's algorithm for optimizing a polynomial on a basic closed semialgebraic set via semidefinite programming and addresses an open question concerning the duality gap. It is shown that, under certain natural stability assumptions, the problem of optimization on a basic closed set reduces to the compact case. Categories:14P10, 46L05, 90C22 136. CMB 2003 (vol 46 pp. 538) Borwein, Jonathan; Fitzpatrick, Simon; Girgensohn, Roland  Subdifferentials Whose Graphs Are Not Norm$\times$Weak* Closed In this note we give examples of convex functions whose subdifferentials have unpleasant properties. Particularly, we exhibit a proper lower semicontinuous convex function on a separable Hilbert space such that the graph of its subdifferential is not closed in the product of the norm and bounded weak topologies. We also exhibit a set whose sequential normal cone is not norm closed. Categories:46N10, 47H05 137. CMB 2003 (vol 46 pp. 509) Benson, David J.; Kumjian, Alex; Phillips, N. Christopher  Symmetries of Kirchberg Algebras Let$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$. Categories:20C10, 46L55, 19K99, 19L47, 46L40, 46L80 138. CMB 2003 (vol 46 pp. 365) Kishimoto, Akitaka; Ozawa, Narutaka; Sakai, Shôichirô  Homogeneity of the Pure State Space of a Separable$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^*$-algebras. The first result of this kind was shown by Powers for the UHF algbras some 30 years ago. Categories:46L40, 46L30 139. CMB 2003 (vol 46 pp. 457) Toms, Andrew  Strongly Perforated$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$. The particular order structures to be constructed will be described in detail below, and all known results of this type will be generalised. Categories:46, 19 140. CMB 2003 (vol 46 pp. 441) Stacey, P. J.  An Inductive Limit Model for the$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$. Categories:46L35, 46L80 141. CMB 2003 (vol 46 pp. 419) Masuda, Toshihiko  On Non-Strongly Free Automorphisms of Subfactors of Type III$_0$We determine when an automorphism of a subfactor of type III$_0$with finite index is non-strongly free in the sense of C.~Winsl\o w in terms of the modular endomorphisms introduced by M.~Izumi. Category:46L37 142. CMB 2003 (vol 46 pp. 388) Lin, Huaxin  Tracially Quasidiagonal Extensions It is known that a unital simple$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 rankCategories:46L05, 46L80 143. 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 convexityCategories:46B07, 46B09, 46B20, 52A21 144. CMB 2003 (vol 46 pp. 164) Dean, Andrew J.  Classification of$\AF$Flows An$\AF$flow is a one-parameter automorphism group of an$\AFC^*$-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 145. 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 146. 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. Category:46L55 147. 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. Category:46L37 148. 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 149. 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 envelopeCategories:46E10, 30A78, 30A76 150. 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
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