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76. CJM 2007 (vol 59 pp. 614)

Labuschagne, C. C. A.
 Preduals and Nuclear Operators Associated with Bounded, $p$-Convex, $p$-Concave and Positive $p$-Summing Operators We use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of $p$-convex, $p$-concave and positive $p$-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators. Keywords:$p$-convex operator, $p$-concave operator, $p$-summing operator, Banach space, Banach lattice, nuclear operator, sequence spaceCategories:46B28, 47B10, 46B42, 46B45

77. CJM 2007 (vol 59 pp. 343)

Lin, Huaxin
 Weak Semiprojectivity in Purely Infinite Simple $C^*$-Algebras Let $A$ be a separable amenable purely infinite simple \CA which satisfies the Universal Coefficient Theorem. We prove that $A$ is weakly semiprojective if and only if $K_i(A)$ is a countable direct sum of finitely generated groups ($i=0,1$). Therefore, if $A$ is such a \CA, for any $\ep>0$ and any finite subset ${\mathcal F}\subset A$ there exist $\dt>0$ and a finite subset ${\mathcal G}\subset A$ satisfying the following: for any contractive positive linear map $L: A\to B$ (for any \CA $B$) with $\|L(ab)-L(a)L(b)\|<\dt$ for $a, b\in {\mathcal G}$ there exists a homomorphism $h\from A\to B$ such that $\|h(a)-L(a)\|<\ep$ for $a\in {\mathcal F}$. Keywords:weakly semiprojective, purely infinite simple $C^*$-algebrasCategories:46L05, 46L80

78. CJM 2007 (vol 59 pp. 276)

Bernardis, A. L.; Martín-Reyes, F. J.; Salvador, P. Ortega
 Weighted Inequalities for Hardy--Steklov Operators We characterize the pairs of weights $(v,w)$ for which the operator $Tf(x)=g(x)\int_{s(x)}^{h(x)}f$ with $s$ and $h$ increasing and continuous functions is of strong type $(p,q)$ or weak type $(p,q)$ with respect to the pair $(v,w)$ in the case $0 Keywords:Hardy--Steklov operator, weights, inequalitiesCategories:26D15, 46E30, 42B25 79. CJM 2007 (vol 59 pp. 63) Ferenczi, Valentin; Galego, Elói Medina  Some Results on the Schroeder--Bernstein Property for Separable Banach Spaces We construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder--Bernstein Index of any of these spaces is$2^{\aleph_0}$. Our construction is based on a Banach space introduced by W. T. Gowers and B. Maurey in 1997. We also use classical descriptive set theory methods, as in some work of the first author and C. Rosendal, to improve some results of P. G. Casazza and of N. J. Kalton on the Schroeder--Bernstein Property for spaces with an unconditional finite-dimensional Schauder decomposition. Keywords:complemented subspaces,, Schroeder-Bernstein propertyCategories:46B03, 46B20 80. CJM 2007 (vol 59 pp. 3) Biller, Harald  Holomorphic Generation of Continuous Inverse Algebras We study complex commutative Banach algebras (and, more generally, continuous inverse algebras) in which the holomorphic functions of a fixed$n$-tuple of elements are dense. In particular, we characterize the compact subsets of~$\C^n$which appear as joint spectra of such$n$-tuples. The characterization is compared with several established notions of holomorphic convexity by means of approximation conditions. Keywords:holomorphic functional calculus, commutative continuous inverse algebra, holomorphic convexity, Stein compacta, meromorphic convexity, holomorphic approximationCategories:46H30, 32A38, 32E30, 41A20, 46J15 81. CJM 2006 (vol 58 pp. 1144) Hamana, Masamichi  Partial$*$-Automorphisms, Normalizers, and Submodules in Monotone Complete$C^*$-Algebras For monotone complete$C^*$-algebras$A\subset B$with$A$contained in$B$as a monotone closed$C^*$-subalgebra, the relation$X = AsA$gives a bijection between the set of all monotone closed linear subspaces$X$of$B$such that$AX + XA \subset X$and$XX^* + X^*X \subset A$and a set of certain partial isometries$s$in the normalizer" of$A$in$B$, and similarly for the map$s \mapsto \Ad s$between the latter set and a set of certain partial$*$-automorphisms" of$A$. We introduce natural inverse semigroup structures in the set of such$X$'s and the set of partial$*$-automorphisms of$A$, modulo a certain relation, so that the composition of these maps induces an inverse semigroup homomorphism between them. For a large enough$B$the homomorphism becomes surjective and all the partial$*$-automorphisms of$A$are realized via partial isometries in$B$. In particular, the inverse semigroup associated with a type${\rm II}_1$von Neumann factor, modulo the outer automorphism group, can be viewed as the fundamental group of the factor. We also consider the$C^*$-algebra version of these results. Categories:46L05, 46L08, 46L40, 20M18 82. CJM 2006 (vol 58 pp. 1268) Sims, Aidan  Gauge-Invariant Ideals in the$C^*$-Algebras of Finitely Aligned Higher-Rank Graphs We produce a complete description of the lattice of gauge-invariant ideals in$C^*(\Lambda)for a finitely alignedk$-graph$\Lambda$. We provide a condition on$\Lambda$under which every ideal is gauge-invariant. We give conditions on$\Lambda$under which$C^*(\Lambda)$satisfies the hypotheses of the Kirchberg--Phillips classification theorem. Keywords:Graphs as categories, graph algebra,$C^*$-algebraCategory:46L05 83. 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 84. CJM 2006 (vol 58 pp. 859) Read, C. J.  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 85. 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 86. 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$. Examples are discussed. Categories:60B15, 43A77, 35H10, 46F25, 60J45, 60J60 87. CJM 2006 (vol 58 pp. 548) Galanopoulos, P.; Papadimitrakis, M.  Hausdorff and Quasi-Hausdorff Matrices on Spaces of Analytic Functions We consider Hausdorff and quasi-Hausdorff matrices as operators on classical spaces of analytic functions such as the Hardy and the Bergman spaces, the Dirichlet space, the Bloch spaces and$\BMOA$. When the generating sequence of the matrix is the moment sequence of a measure$\mu$, we find the conditions on$\mu$which are equivalent to the boundedness of the matrix on the various spaces. Categories:47B38, 46E15, 40G05, 42A20 88. CJM 2006 (vol 58 pp. 492) Chua, Seng-Kee  Extension Theorems on Weighted Sobolev Spaces and Some Applications We extend the extension theorems to weighted Sobolev spaces$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 89. 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 90. 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

91. 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

92. 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

93. 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

94. 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

95. 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

96. CJM 2005 (vol 57 pp. 61)

 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

97. 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

98. 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

99. 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

100. 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
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