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151. CMB 2005 (vol 48 pp. 607)

Park, Efton
 Toeplitz Algebras and Extensions of\\Irrational Rotation Algebras For a given irrational number $\theta$, we define Toeplitz operators with symbols in the irrational rotation algebra ${\mathcal A}_\theta$, and we show that the $C^*$-algebra $\mathcal T({\mathcal A}_\theta)$ generated by these Toeplitz operators is an extension of ${\mathcal A}_\theta$ by the algebra of compact operators. We then use these extensions to explicitly exhibit generators of the group $KK^1({\mathcal A}_\theta,\mathbb C)$. We also prove an index theorem for $\mathcal T({\mathcal A}_\theta)$ that generalizes the standard index theorem for Toeplitz operators on the circle. Keywords:Toeplitz operators, irrational rotation algebras, index theoryCategories:47B35, 46L80

152. CMB 2005 (vol 48 pp. 481)

Azagra, D.; Fabian, M.; Jiménez-Sevilla, M.
 Exact Filling of Figures with the Derivatives of Smooth Mappings Between Banach Spaces We establish sufficient conditions on the shape of a set $A$ included in the space $\mathcal L _s^n(X,Y)$ of the $n$-linear symmetric mappings between Banach spaces $X$ and $Y$, to ensure the existence of a $C^n$\nobreakdash-smooth mapping $f\colon X \rightarrow Y$, with bounded support, and such that $f^{(n)}(X)=A$, provided that $X$ admits a $C^{n}$-smooth bump with bounded $n$-th derivative and $\dens X=\dens \mathcal L ^n(X,Y)$. For instance, when $X$ is infinite-dimensional, every bounded connected and open set $U$ containing the origin is the range of the $n$-th derivative of such a mapping. The same holds true for the closure of $U$, provided that every point in the boundary of $U$ is the end point of a path within $U$. In the finite-dimensional case, more restrictive conditions are required. We also study the Fr\'echet smooth case for mappings from $\mathbb R^n$ to a separable infinite-dimensional Banach space and the G\^ateaux smooth case for mappings defined on a separable infinite-dimensional Banach space and with values in a separable Banach space. Category:46B20

153. CMB 2005 (vol 48 pp. 455)

Rychtář, Jan
 On GÃ¢teaux Differentiability of Convex Functions in WCG Spaces It is shown, using the Borwein--Preiss variational principle that for every continuous convex function $f$ on a weakly compactly generated space $X$, every $x_0\in X$ and every weakly compact convex symmetric set $K$ such that $\cspan K=X$, there is a point of G\^ateaux differentiability of $f$ in $x_0+K$. This extends a Klee's result for separable spaces. Keywords:GÃ¢teaux smoothness, Borwein--Preiss variational principle,, weakly compactly generated spacesCategory:46B20

154. CMB 2005 (vol 48 pp. 340)

Andruchow, Esteban
 Short Geodesics of Unitaries in the $L^2$ Metric Let $\M$ be a type II$_1$ von Neumann algebra, $\tau$ a trace in $\M$, and $\l2$ the GNS Hilbert space of $\tau$. We regard the unitary group $U_\M$ as a subset of $\l2$ and characterize the shortest smooth curves joining two fixed unitaries in the $L^2$ metric. As a consequence of this we obtain that $U_\M$, though a complete (metric) topological group, is not an embedded riemannian submanifold of $\l2$ Keywords:unitary group, short geodesics, infinite dimensional riemannian manifolds.Categories:46L51, 58B10, 58B25

155. CMB 2005 (vol 48 pp. 251)

Murphy, G. J.
 The Index Theory Associated to a Non-Finite Trace on a $C^\ast$-Algebra The index theory considered in this paper, a generalisation of the classical Fredholm index theory, is obtained in terms of a non-finite trace on a unital $C^\ast$-algebra. We relate it to the index theory of M.~Breuer, which is developed in a von~Neumann algebra setting, by means of a representation theorem. We show how our new index theory can be used to obtain an index theorem for Toeplitz operators on the compact group $\mathbf{U}(2)$, where the classical index theory does not give any interesting result. Categories:46L, 47B35, 47L80

156. CMB 2005 (vol 48 pp. 283)

Thibault, Lionel; Zagrodny, Dariusz
 Enlarged Inclusion of Subdifferentials This paper studies the integration of inclusion of subdifferentials. Under various verifiable conditions, we obtain that if two proper lower semicontinuous functions $f$ and $g$ have the subdifferential of $f$ included in the $\gamma$-enlargement of the subdifferential of $g$, then the difference of those functions is $\gamma$-Lipschitz over their effective domain. Keywords:subdifferential,, directionally regular function,, approximate convex function,, subdifferentially and directionally stable functionCategories:49J52, 46N10, 58C20

157. CMB 2005 (vol 48 pp. 161)

Betancor, Jorge J.
 Hankel Convolution Operators on Spaces of Entire Functions of Finite Order In this paper we study Hankel transforms and Hankel convolution operators on spaces of entire functions of finite order and their duals. Keywords:Hankel transform, convolution, entire functions, finite orderCategory:46F12

158. CMB 2005 (vol 48 pp. 69)

Fabian, M.; Montesinos, V.; Zizler, V.
 Biorthogonal Systems in Weakly LindelÃ¶f Spaces We study countable splitting of Markushevich bases in weakly Lindel\"of Banach spaces in connection with the geometry of these spaces. Keywords:Weak compactness, projectional resolutions,, Markushevich bases, Eberlein compacts, Va\v sÃ¡k spacesCategories:46B03, 46B20., 46B26

159. CMB 2005 (vol 48 pp. 97)

Katavolos, Aristides; Paulsen, Vern I.
 On the Ranges of Bimodule Projections We develop a symbol calculus for normal bimodule maps over a masa that is the natural analogue of the Schur product theory. Using this calculus we are easily able to give a complete description of the ranges of contractive normal bimodule idempotents that avoids the theory of J*-algebras. We prove that if $P$ is a normal bimodule idempotent and $\|P\| < 2/\sqrt{3}$ then $P$ is a contraction. We finish with some attempts at extending the symbol calculus to non-normal maps. Categories:46L15, 47L25

160. CMB 2005 (vol 48 pp. 50)

Elliott, George A.; Gong, Guihua; Li, Liangqing
 Injectivity of the Connecting Maps in AH Inductive Limit Systems Let $A$ be the inductive limit of a system $$A_{1}\xrightarrow{\phi_{1,2}}A_{2} \xrightarrow{\phi_{2,3}} A_{3}\longrightarrow \cd$$ with $A_n = \bigoplus_{i=1}^{t_n} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$, where $~X_{n,i}$ is a finite simplicial complex, and $P_{n,i}$ is a projection in $M_{[n,i]}(C(X_{n,i}))$. In this paper, we will prove that $A$ can be written as another inductive limit $$B_1\xrightarrow{\psi_{1,2}} B_2 \xrightarrow{\psi_{2,3}} B_3\longrightarrow \cd$$ with $B_n = \bigoplus_{i=1}^{s_n} Q_{n,i}M_{\{n,i\}}(C(Y_{n,i}))Q_{n,i}$, where $Y_{n,i}$ is a finite simplicial complex, and $Q_{n,i}$ is a projection in $M_{\{n,i\}}(C(Y_{n,i}))$, with the extra condition that all the maps $\psi_{n,n+1}$ are \emph{injective}. (The result is trivial if one allows the spaces $Y_{n,i}$ to be arbitrary compact metrizable spaces.) This result is important for the classification of simple AH algebras (see \cite{G5,G6,EGL}. The special case that the spaces $X_{n,i}$ are graphs is due to the third named author \cite{Li1}. Categories:46L05, 46L35, 19K14

161. 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 regionsCategories:46E35, 26D10

162. 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^*$-algebras, summing operators, diagonal operators,, Radon-Nikodym propertyCategories:46L50, 47D15 163. CMB 2004 (vol 47 pp. 553) Kerr, David  A Geometric Approach to Voiculescu-Brown Entropy A basic problem in dynamics is to identify systems with positive entropy, i.e., systems which are chaotic.'' While there is a vast collection of results addressing this issue in topological dynamics, the phenomenon of positive entropy remains by and large a mystery within the broader noncommutative domain of$C^*$-algebraic dynamics. To shed some light on the noncommutative situation we propose a geometric perspective inspired by work of Glasner and Weiss on topological entropy. This is a written version of the author's talk at the Winter 2002 Meeting of the Canadian Mathematical Society in Ottawa, Ontario. Categories:46L55, 37B40 164. CMB 2004 (vol 47 pp. 481) Bekjan, Turdebek N.  A New Characterization of Hardy Martingale Cotype Space We give a new characterization of Hardy martingale cotype property of complex quasi-Banach space by using the existence of a kind of plurisubharmonic functions. We also characterize the best constants of Hardy martingale inequalities with values in the complex quasi-Banach space. Keywords:Hardy martingale, Hardy martingale cotype,, plurisubharmonic functionCategories:46B20, 52A07, 60G44 165. CMB 2004 (vol 47 pp. 445) Pirkovskii, A. Yu.  Biprojectivity and Biflatness for Convolution Algebras of Nuclear Operators For a locally compact group$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$, such as compactness, finiteness, discreteness, and amenability. Categories:46M10, 46H25, 43A20, 16E65 166. CMB 2004 (vol 47 pp. 206) Hurri-Syrjänen, Ritva  The PoincarÃ© Inequality and Reverse Doubling Weights We show that Poincar\'e inequalities with reverse doubling weights hold in a large class of irregular domains whenever the weights satisfy certain conditions. Examples of these domains are John domains. Keywords:reverse doubling weights, PoincarÃ© inequality, John domainsCategory:46E35 167. CMB 2004 (vol 47 pp. 108) Śliwa, Wiesław  On Universal Schauder Bases in Non-Archimedean FrÃ©chet Spaces It is known that any non-archimedean Fr\'echet space of countable type is isomorphic to a subspace of$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)$. Keywords:universal bases, complemented subspaces with basesCategories:46S10, 46A35 168. CMB 2004 (vol 47 pp. 49) Lindström, Mikael; Makhmutov, Shamil; Taskinen, Jari  The Essential Norm of a Bloch-to-$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

169. 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

170. 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

171. 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

172. 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

173. 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

174. 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

175. 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
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