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76. CJM 2006 (vol 58 pp. 768)

Hu, Zhiguo; Neufang, Matthias
 Decomposability of von Neumann Algebras and the Mazur Property of Higher Level The decomposability number of a von Neumann algebra $\m$ (denoted by $\dec(\m)$) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in $\m$. In this paper, we explore the close connection between $\dec(\m)$ and the cardinal level of the Mazur property for the predual $\m_*$ of $\m$, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group $G$ as the group algebra $\lone$, the Fourier algebra $A(G)$, the measure algebra $M(G)$, the algebra $\luc^*$, etc. We show that for any of these von Neumann algebras, say $\m$, the cardinal number $\dec(\m)$ and a certain cardinal level of the Mazur property of $\m_*$ are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of $G$: the compact covering number $\kg$ of $G$ and the least cardinality $\bg$ of an open basis at the identity of $G$. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra $\ag^{**}$. Keywords:Mazur property, predual of a von Neumann algebra, locally compact group and its cardinal invariants, group algebra, Fourier algebra, topological centreCategories:22D05, 43A20, 43A30, 03E55, 46L10

77. CJM 2006 (vol 58 pp. 859)

 Nonstandard Ideals from Nonstandard Dual Pairs for $L^1(\omega)$ and $l^1(\omega)$ The Banach convolution algebras $l^1(\omega)$ and their continuous counterparts $L^1(\bR^+,\omega)$ are much studied, because (when the submultiplicative weight function $\omega$ is radical) they are pretty much the prototypic examples of commutative radical Banach algebras. In cases of nice'' weights $\omega$, the only closed ideals they have are the obvious, or standard'', ideals. But in the general case, a brilliant but very difficult paper of Marc Thomas shows that nonstandard ideals exist in $l^1(\omega)$. His proof was successfully exported to the continuous case $L^1(\bR^+,\omega)$ by Dales and McClure, but remained difficult. In this paper we first present a small improvement: a new and easier proof of the existence of nonstandard ideals in $l^1(\omega)$ and $L^1(\bR^+,\omega)$. The new proof is based on the idea of a nonstandard dual pair'' which we introduce. We are then able to make a much larger improvement: we find nonstandard ideals in $L^1(\bR^+,\omega)$ containing functions whose supports extend all the way down to zero in $\bR^+$, thereby solving what has become a notorious problem in the area. Keywords:Banach algebra, radical, ideal, standard ideal, semigroupCategories:46J45, 46J20, 47A15

78. CJM 2006 (vol 58 pp. 820)

Moreno, J. P.; Papini, P. L.; Phelps, R. R.
 Diametrically Maximal and Constant Width Sets in Banach Spaces We characterize diametrically maximal and constant width sets in $C(K)$, where $K$ is any compact Hausdorff space. These results are applied to prove that the sum of two diametrically maximal sets needs not be diametrically maximal, thus solving a question raised in a paper by Groemer. A~characterization of diametrically maximal sets in $\ell_1^3$ is also given, providing a negative answer to Groemer's problem in finite dimensional spaces. We characterize constant width sets in $c_0(I)$, for every $I$, and then we establish the connections between the Jung constant of a Banach space and the existence of constant width sets with empty interior. Porosity properties of families of sets of constant width and rotundity properties of diametrically maximal sets are also investigated. Finally, we present some results concerning non-reflexive and Hilbert spaces. Categories:52A05, 46B20

79. CJM 2006 (vol 58 pp. 691)

Bendikov, A.; Saloff-Coste, L.
 Hypoelliptic Bi-Invariant Laplacians on Infinite Dimensional Compact Groups On a compact connected group $G$, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup $(\mu_t)_{t>0}$. Using appropriate notions of distribution and smooth function spaces, we prove that $L$ is hypoelliptic if and only if $(\mu_t)_{t>0}$ is absolutely continuous with respect to Haar measure and admits a continuous density $x\mapsto \mu_t(x)$, $t>0$, such that $\lim_{t\ra 0} t\log \mu_t(e)=0$. In particular, this condition holds if and only if any Borel measure $u$ which is solution of $Lu=0$ in an open set $\Omega$ can be represented by a continuous function in $\Omega$. Examples are discussed. Categories:60B15, 43A77, 35H10, 46F25, 60J45, 60J60

80. 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

81. 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

82. 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

83. 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 $1 Categories:47B47, 46B28 84. CJM 2005 (vol 57 pp. 1056) Ozawa, Narutaka; Rieffel, Marc A.  Hyperbolic Group$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^*$-algebras. Categories:46L87, 20F67, 46L09 85. CJM 2005 (vol 57 pp. 983) an Huef, Astrid; Raeburn, Iain; Williams, Dana P.  A Symmetric Imprimitivity Theorem for Commuting Proper Actions We prove a symmetric imprimitivity theorem for commuting proper actions of locally compact groups$H$and$K$on a$C^*$-algebra. Categories:46L05, 46L08, 46L55 86. CJM 2005 (vol 57 pp. 897) Berezhnoĭ, Evgenii I.; Maligranda, Lech  Representation of Banach Ideal Spaces and Factorization of Operators Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calder{\'o}n--Lozanovski\u\i\ construction. Factorization theorems for operators in spaces more general than the Lebesgue$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}$-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calder{\'o}n--Lozanovski\u\i\ construction are involved in the proofs. Keywords:Banach ideal spaces, weighted spaces, weight functions,, CalderÃ³n--Lozanovski\u\i\ spaces, Orlicz spaces, representation of, spaces, uniqueness problem, positive linear operators, positive sublinear, operators, Schur test, factorization of operators, fCategories:46E30, 46B42, 46B70 87. CJM 2005 (vol 57 pp. 673) Androulakis, G.; Odell, E.; Schlumprecht, Th.; Tomczak-Jaegermann, N.  On the Structure of the Spreading Models of a Banach Space We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space$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$. Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map. Category:46B03 88. CJM 2005 (vol 57 pp. 351) Lin, Huaxin  Extensions by Simple$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 89. 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 90. 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 91. 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$) and its {\em fixed point} algebra under the flow are the same (incorporating the previous result). We will also complete and supplement the characterization result of the Rohlin property for flows stated in~ \cite{Kismh}. Category:46L40 92. CJM 2004 (vol 56 pp. 1121) Chaumat, Jacques; Chollet, Anne-Marie  Division par un polynÃ´me hyperbolique On se donne un intervalle ouvert non vide$\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 93. 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 94. 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 95. 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 96. 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 97. 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 98. 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 99. 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 100. 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
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