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76. CJM 2006 (vol 58 pp. 548)

 Hausdorff and Quasi-Hausdorff Matrices on Spaces of Analytic Functions We consider Hausdorff and quasi-Hausdorff matrices as operators on classical spaces of analytic functions such as the Hardy and the Bergman spaces, the Dirichlet space, the Bloch spaces and $\BMOA$. When the generating sequence of the matrix is the moment sequence of a measure $\mu$, we find the conditions on $\mu$ which are equivalent to the boundedness of the matrix on the various spaces. Categories:47B38, 46E15, 40G05, 42A20

77. CJM 2006 (vol 58 pp. 492)

Chua, Seng-Kee
 Extension Theorems on Weighted Sobolev Spaces and Some Applications We extend the extension theorems to weighted Sobolev spaces $L^p_{w,k}(\mathcal D)$ on $(\varepsilon,\delta)$ domains with doubling weight $w$ that satisfies a Poincar\'e inequality and such that $w^{-1/p}$ is locally $L^{p'}$. We also make use of the main theorem to improve weighted Sobolev interpolation inequalities. Keywords:PoincarÃ© inequalities, $A_p$ weights, doubling weights, $(\ep,\delta)$ domain, $(\ep,\infty)$ domainCategory:46E35

78. CJM 2006 (vol 58 pp. 39)

Exel, R.; Vershik, A.
 $C^*$-Algebras of Irreversible Dynamical Systems We show that certain $C^*$-algebras which have been studied by, among others, Arzumanian, Vershik, Deaconu, and Renault, in connection with a measure-preserving transformation of a measure space or a covering map of a compact space, are special cases of the endomorphism crossed-product construction recently introduced by the first named author. As a consequence these algebras are given presentations in terms of generators and relations. These results come as a consequence of a general theorem on faithfulness of representations which are covariant with respect to certain circle actions. For the case of topologically free covering maps we prove a stronger result on faithfulness of representations which needs no covariance. We also give a necessary and sufficient condition for simplicity. Categories:46L55, 37A55

79. CJM 2005 (vol 57 pp. 1249)

Lindström, Mikael; Saksman, Eero; Tylli, Hans-Olav
 Strictly Singular and Cosingular Multiplications Let $L(X)$ be the space of bounded linear operators on the Banach space $X$. We study the strict singularity andcosingularity of the two-sided multiplication operators $S \mapsto ASB$ on $L(X)$, where $A,B \in L(X)$ are fixed bounded operators and $X$ is a classical Banach space. Let $1 Categories:47B47, 46B28 80. CJM 2005 (vol 57 pp. 897) Berezhnoĭ, Evgenii I.; Maligranda, Lech  Representation of Banach Ideal Spaces and Factorization of Operators Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calder{\'o}n--Lozanovski\u\i\ construction. Factorization theorems for operators in spaces more general than the Lebesgue$L^{p}$spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de~Francia theorem on factorization of the Muckenhoupt$A_{p}$weights to reflexive Orlicz spaces. However, it turns out that for the scales far from$L^{p}$-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calder{\'o}n--Lozanovski\u\i\ construction are involved in the proofs. Keywords:Banach ideal spaces, weighted spaces, weight functions,, CalderÃ³n--Lozanovski\u\i\ spaces, Orlicz spaces, representation of, spaces, uniqueness problem, positive linear operators, positive sublinear, operators, Schur test, factorization of operators, fCategories:46E30, 46B42, 46B70 81. CJM 2005 (vol 57 pp. 1056) Ozawa, Narutaka; Rieffel, Marc A.  Hyperbolic Group$C^*$-Algebras and Free-Product$C^*$-Algebras as Compact Quantum Metric Spaces Let$\ell$be a length function on a group$G$, and let$M_{\ell}$denote the operator of pointwise multiplication by$\ell$on$\bell^2(G)$. Following Connes,$M_{\ell}$can be used as a Dirac'' operator for$C_r^*(G)$. It defines a Lipschitz seminorm on$C_r^*(G)$, which defines a metric on the state space of$C_r^*(G)$. We show that if$G$is a hyperbolic group and if$\ell$is a word-length function on$G$, then the topology from this metric coincides with the weak-$*$topology (our definition of a compact quantum metric space''). We show that a convenient framework is that of filtered$C^*$-algebras which satisfy a suitable Haagerup-type'' condition. We also use this framework to prove an analogous fact for certain reduced free products of$C^*$-algebras. Categories:46L87, 20F67, 46L09 82. CJM 2005 (vol 57 pp. 983) an Huef, Astrid; Raeburn, Iain; Williams, Dana P.  A Symmetric Imprimitivity Theorem for Commuting Proper Actions We prove a symmetric imprimitivity theorem for commuting proper actions of locally compact groups$H$and$K$on a$C^*$-algebra. Categories:46L05, 46L08, 46L55 83. CJM 2005 (vol 57 pp. 673) Androulakis, G.; Odell, E.; Schlumprecht, Th.; Tomczak-Jaegermann, N.  On the Structure of the Spreading Models of a Banach Space We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space$X$. In particular we give an example of a reflexive$X$so that all spreading models of$X$contain$\ell_1$but none of them is isomorphic to$\ell_1$. We also prove that for any countable set$C$of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of$C$. In certain cases this ensures that$X$admits, for each$\alpha < \omega_1$, a spreading model$(\tilde x_i^{(\alpha)})_i$such that if$\alpha < \beta$then$(\tilde x_i^{(\alpha)})_i$is dominated by (and not equivalent to)$(\tilde x_i^{(\beta)})_i$. Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map. Category:46B03 84. CJM 2005 (vol 57 pp. 351) Lin, Huaxin  Extensions by Simple$C^*$-Algebras: Quasidiagonal Extensions Let$A$be an amenable separable$C^*$-algebra and$B$be a non-unital but$\sigma$-unital simple$C^*$-algebra with continuous scale. We show that two essential extensions$\tau_1$and$\tau_2$of$A$by$B$are approximately unitarily equivalent if and only if $$[\tau_1]=[\tau_2] \text{ in } KL(A, M(B)/B).$$ If$A$is assumed to satisfy the Universal Coefficient Theorem, there is a bijection from approximate unitary equivalence classes of the above mentioned extensions to$KL(A, M(B)/B)$. Using$KL(A, M(B)/B)$, we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be approximately trivial. We also study the approximately trivial extensions. Keywords:Extensions, Simple$C^*$-algebrasCategories:46L05, 46L35 85. CJM 2005 (vol 57 pp. 17) Bédos, Erik; Conti, Roberto; Tuset, Lars  On Amenability and Co-Amenability of Algebraic Quantum Groups and Their Corepresentations We introduce and study several notions of amenability for unitary corepresentations and$*$-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background for this study, we investigate the associated tensor C$^{*}$-categories. Keywords:quantum group, amenabilityCategories:46L05, 46L65, 22D10, 22D25, 43A07, 43A65, 58B32 86. CJM 2005 (vol 57 pp. 61) Binding, Paul; Strauss, Vladimir  On Operators with Spectral Square but without Resolvent Points Decompositions of spectral type are obtained for closed Hilbert space operators with empty resolvent set, but whose square has closure which is spectral. Krein space situations are also discussed. Keywords:unbounded operators, closed operators,, spectral resolution, indefinite metricCategories:47A05, 47A15, 47B40, 47B50, 46C20 87. CJM 2004 (vol 56 pp. 1237) Kishimoto, Akitaka  Central Sequence Algebras of a Purely Infinite Simple$C^{*}$-algebra We are concerned with a unital separable nuclear purely infinite simple$C^{*}$-algebra\$A$satisfying UCT with a Rohlin flow, as a continuation of~\cite{Kismh}. Our first result (which is independent of the Rohlin flow) is to characterize when two {\em central} projections in$A$are equivalent by a {\em central} partial isometry. Our second result shows that the K-theory of the central sequence algebra$A'\cap A^\omega$(for an$\omega\in \beta\N\setminus\N$) and its {\em fixed point} algebra under the flow are the same (incorporating the previous result). We will also complete and supplement the characterization result of the Rohlin property for flows stated in~ \cite{Kismh}. Category:46L40 88. CJM 2004 (vol 56 pp. 1121) Chaumat, Jacques; Chollet, Anne-Marie  Division par un polynÃ´me hyperbolique On se donne un intervalle ouvert non vide$\omega$de$\mathbb R$, un ouvert connexe non vide$\Omega$de$\mathbb R_s$et un polyn\^ome unitaire $P_m(z, \lambda) = z^m + a_1(\lambda)z^{m-1} = +\dots + a_{m-1}(\lambda) z + a_m(\lambda),$ de degr\'e$m>0$, d\'ependant du param\etre$\lambda \in \Omega$. Un tel polyn\^ome est dit$\omega$-hyperbolique si, pour tout$\lambda \in \Omega$, ses racines sont r\'eelles et appartiennent \a$\omega$. On suppose que les fonctions$a_k, \, k=1, \dots, m$, appartiennent \a une classe ultradiff\'erentiable$C_M(\Omega)$. On sint\'eresse au probl\eme suivant. Soit$f$appartient \a$C_M(\Omega)$, existe-t-il des fonctions$Q_f$et$R_{f,k},\, k=0, \dots, m-1$, appartenant respectivement \a$C_M(\omega \times \Omega)$et \a$C_M(\Omega)$, telles que l'on ait, pour$(x,\lambda) \in \omega \times \Omega$, $f(x) = P_m(x,\lambda) Q_f (x,\lambda) + \sum^{m-1}_{k=0} x^k R_{f,k}(\lambda)~?$ On donne ici une r\'eponse positive d\es que le polyn\^ome est$\omega$-hyperbolique, que la class untradiff\'eren\-tiable soit quasi-analytique ou non ; on obtient alors, des exemples d'id\'eaux ferm\'es dans$C_M(\mathbb R^n)$. On compl\ete ce travail par une g\'en\'eralisation d'un r\'esultat de C.~L. Childress dans le cadre quasi-analytique et quelques remarques. Categories:26E10, 46E25, 46J20 89. CJM 2004 (vol 56 pp. 983) Junge, Marius  Fubini's Theorem for Ultraproducts \\of Noncommutative$L_p$-Spaces Let$(\M_i)_{i\in I}$,$(\N_j)_{j\in J}$be families of von Neumann algebras and$\U$,$\U'$be ultrafilters in$I$,$J$, respectively. Let$1\le p<\infty$and$\nen$. Let$x_1$,\dots,$x_n$in$\prod L_p(\M_i)$and$y_1$,\dots,$y_n$in$\prod L_p(\N_j)$be bounded families. We show the following equality $$\lim_{i,\U} \lim_{j,\U'} \Big\| \summ_{k=1}^n x_k(i)\otimes y_k(j)\Big\|_{L_p(\M_i\otimes \N_j)} = \lim_{j,\U'} \lim_{i,\U} \Big\| \summ_{k=1}^n x_k(i)\otimes y_k(j)\Big\|_{L_p(\M_i\otimes \N_j)} .$$ For$p=1$this Fubini type result is related to the local reflexivity of duals of$C^*$-algebras. This fails for$p=\infty$. Keywords:noncommutative$L_p$-spaces, ultraproductsCategories:46L52, 46B08, 46L07 90. CJM 2004 (vol 56 pp. 926) Hadfield, Tom  K-Homology of the Rotation Algebras$A_{\theta}$We study the K-homology of the rotation algebras$A_{\theta}$using the six-term cyclic sequence for the K-homology of a crossed product by${\bf Z}$. In the case that$\theta$is irrational, we use Pimsner and Voiculescu's work on AF-embeddings of the$A_{\theta}$to search for the missing generator of the even K-homology. Categories:58B34, 19K33, 46L 91. CJM 2004 (vol 56 pp. 843) Ruan, Zhong-Jin  Type Decomposition and the Rectangular AFD Property for$W^*$-TRO's We study the type decomposition and the rectangular AFD property for$W^*$-TRO's. Like von Neumann algebras, every$W^*$-TRO can be uniquely decomposed into the direct sum of$W^*$-TRO's of type$I$, type$II$, and type$III$. We may further consider$W^*$-TRO's of type$I_{m, n}$with cardinal numbers$m$and$n$, and consider$W^*$-TRO's of type$II_{\lambda, \mu}$with$\lambda, \mu = 1$or$\infty$. It is shown that every separable stable$W^*$-TRO (which includes type$I_{\infty,\infty}$, type$II_{\infty, \infty}$and type$III$) is TRO-isomorphic to a von Neumann algebra. We also introduce the rectangular version of the approximately finite dimensional property for$W^*$-TRO's. One of our major results is to show that a separable$W^*$-TRO is injective if and only if it is rectangularly approximately finite dimensional. As a consequence of this result, we show that a dual operator space is injective if and only if its operator predual is a rigid rectangular${\OL}_{1, 1^+}$space (equivalently, a rectangular Categories:46L07, 46L08, 46L89 92. CJM 2004 (vol 56 pp. 699) Gaspari, Thierry  Bump Functions with HÃ¶lder Derivatives We study the range of the gradients of a$C^{1,\al}$-smooth bump function defined on a Banach space. We find that this set must satisfy two geometrical conditions: It can not be too flat and it satisfies a strong compactness condition with respect to an appropriate distance. These notions are defined precisely below. With these results we illustrate the differences with the case of$C^1$-smooth bump functions. Finally, we give a sufficient condition on a subset of$X^{\ast}$so that it is the set of the gradients of a$C^{1,1}$-smooth bump function. In particular, if$X$is an infinite dimensional Banach space with a$C^{1,1}$-smooth bump function, then any convex open bounded subset of$X^{\ast}$containing$0$is the set of the gradients of a$C^{1,1}$-smooth bump function. Keywords:Banach space, bump function, range of the derivativeCategories:46T20, 26E15, 26B05 93. CJM 2004 (vol 56 pp. 472) Fonf, Vladimir P.; Veselý, Libor  Infinite-Dimensional Polyhedrality This paper deals with generalizations of the notion of a polytope to infinite dimensions. The most general definition is the following: a bounded closed convex subset of a Banach space is called a \emph{polytope} if each of its finite-dimensional affine sections is a (standard) polytope. We study the relationships between eight known definitions of infinite-dimensional polyhedrality. We provide a complete isometric classification of them, which gives solutions to several open problems. An almost complete isomorphic classification is given as well (only one implication remains open). Categories:46B20, 46B03, 46B04, 52B99 94. CJM 2004 (vol 56 pp. 225) Blower, Gordon; Ransford, Thomas  Complex Uniform Convexity and Riesz Measure The norm on a Banach space gives rise to a subharmonic function on the complex plane for which the distributional Laplacian gives a Riesz measure. This measure is calculated explicitly here for Lebesgue$L^p$spaces and the von~Neumann-Schatten trace ideals. Banach spaces that are$q$-uniformly$\PL$-convex in the sense of Davis, Garling and Tomczak-Jaegermann are characterized in terms of the mass distribution of this measure. This gives a new proof that the trace ideals$c^p$are$2$-uniformly$\PL$-convex for$1\leq p\leq 2$. Keywords:subharmonic functions, Banach spaces, Schatten trace idealsCategories:46B20, 46L52 95. CJM 2004 (vol 56 pp. 3) Amini, Massoud  Locally Compact Pro-$C^*$-Algebras Let$X$be a locally compact non-compact Hausdorff topological space. Consider the algebras$C(X)$,$C_b(X)$,$C_0(X)$, and$C_{00}(X)$of respectively arbitrary, bounded, vanishing at infinity, and compactly supported continuous functions on$X$. Of these, the second and third are$C^*$-algebras, the fourth is a normed algebra, whereas the first is only a topological algebra (it is indeed a pro-$C^\ast$-algebra). The interesting fact about these algebras is that if one of them is given, the others can be obtained using functional analysis tools. For instance, given the$C^\ast$-algebra$C_0(X)$, one can get the other three algebras by$C_{00}(X)=K\bigl(C_0(X)\bigr)$,$C_b(X)=M\bigl(C_0(X)\bigr)$,$C(X)=\Gamma\bigl( K(C_0(X))\bigr)$, where the right hand sides are the Pedersen ideal, the multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of$C_0(X)$, respectively. In this article we consider the possibility of these transitions for general$C^\ast$-algebras. The difficult part is to start with a pro-$C^\ast$-algebra$A$and to construct a$C^\ast$-algebra$A_0$such that$A=\Gamma\bigl(K(A_0)\bigr)$. The pro-$C^\ast$-algebras for which this is possible are called {\it locally compact\/} and we have characterized them using a concept similar to that of an approximate identity. Keywords:pro-$C^\ast$-algebras, projective limit, multipliers of Pedersen's idealCategories:46L05, 46M40 96. CJM 2003 (vol 55 pp. 1302) Katsura, Takeshi  The Ideal Structures of Crossed Products of Cuntz Algebras by Quasi-Free Actions of Abelian Groups We completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A.~Kishimoto's result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra and$K$-groups of our algebras. Categories:46L05, 46L55, 46L45 97. CJM 2003 (vol 55 pp. 969) Glöckner, Helge  Lie Groups of Measurable Mappings We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space$(X,\Sigma,\mu)$and (possibly infinite-dimensional) Lie group$G$, we construct a Lie group$L^\infty (X,G)$, which is a Fr\'echet-Lie group if$G$is so. We also show that the weak direct product$\prod^*_{i\in I} G_i$of an arbitrary family$(G_i)_{i\in I}$of Lie groups can be made a Lie group, modelled on the locally convex direct sum$\bigoplus_{i\in I} L(G_i)$. Categories:22E65, 46E40, 46E30, 22E67, 46T20, 46T25 98. CJM 2003 (vol 55 pp. 204) Yan, Yaqiang  On the Nonsquare Constants of Orlicz Spaces with Orlicz Norm Let$l^{\Phi}$and$L^\Phi (\Omega)$be the Orlicz sequence space and function space generated by$N$-function$\Phi(u)$with Orlicz norm. We give equivalent expressions for the nonsquare constants$C_J (l^\Phi)$,$C_J \bigl( L^\Phi (\Omega) \bigr)$in sense of James and$C_S (l^\Phi)$,$C_S \bigl( L^\Phi(\Omega) \bigr)$in sense of Sch\"affer. We are devoted to get practical computational formulas giving estimates of these constants and to obtain their exact value in a class of spaces$l^{\Phi}$and$L^\Phi (\Omega)$. Keywords:James nonsquare constant, SchÃ¤ffer nonsquare constant, Orlicz sequence space, Orlicz function spaceCategory:46E30 99. CJM 2002 (vol 54 pp. 1165) Blasco, Oscar; Arregui, José Luis  Multipliers on Vector Valued Bergman Spaces Let$X$be a complex Banach space and let$B_p(X)$denote the vector-valued Bergman space on the unit disc for$1\le p<\infty$. A sequence$(T_n)_n$of bounded operators between two Banach spaces$X$and$Y$defines a multiplier between$B_p(X)$and$B_q(Y)$(resp.\$B_p(X)$and$\ell_q(Y)$) if for any function$f(z) = \sum_{n=0}^\infty x_n z^n$in$B_p(X)$we have that$g(z) = \sum_{n=0}^\infty T_n (x_n) z^n$belongs to$B_q(Y)$(resp.\$\bigl( T_n (x_n) \bigr)_n \in \ell_q(Y)$). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces$X$and$Y$. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in$B_p(X)$are introduced. Categories:42A45, 46E40 100. CJM 2002 (vol 54 pp. 1280) Skrzypczak, Leszek  Besov Spaces and Hausdorff Dimension For Some Carnot-CarathÃ©odory Metric Spaces We regard a system of left invariant vector fields$\mathcal{X}=\{X_1,\dots,X_k\}$satisfying the H\"ormander condition and the related Carnot-Carath\'eodory metric on a unimodular Lie group$G$. We define Besov spaces corresponding to the sub-Laplacian$\Delta=\sum X_i^2\$ both with positive and negative smoothness. The atomic decomposition of the spaces is given. In consequence we get the distributional characterization of the Hausdorff dimension of Borel subsets with the Haar measure zero. Keywords:Besov spaces, sub-elliptic operators, Carnot-CarathÃ©odory metric, Hausdorff dimensionCategories:46E35, 43A15, 28A78
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