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51. CJM 2010 (vol 62 pp. 889)

Xia, Jingbo
 Singular Integral Operators and Essential Commutativity on the Sphere Let ${\mathcal T}$ be the $C^\ast$-algebra generated by the Toeplitz operators $\{T_\varphi : \varphi \in L^\infty (S,d\sigma )\}$ on the Hardy space $H^2(S)$ of the unit sphere in $\mathbf{C}^n$. It is well known that ${\mathcal T}$ is contained in the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$. We show that the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$ is strictly larger than ${\mathcal T}$. Categories:32A55, 46L05, 47L80

52. CJM 2010 (vol 62 pp. 595)

Martínez, J. F.; Moltó, A.; Orihuela, J.; Troyanski, S.
 On Locally Uniformly Rotund Renormings in C(K) Spaces A characterization of the Banach spaces of type $C(K)$ that admit an equivalent locally uniformly rotund norm is obtained, and a method to apply it to concrete spaces is developed. As an application the existence of such renorming is deduced when $K$ is a Namioka--Phelps compact or for some particular class of Rosenthal compacta, results which were originally proved with \emph{ad hoc} methods. Categories:46B03, 46B20

53. CJM 2009 (vol 62 pp. 242)

Azagra, Daniel; Fry, Robb
 A Second Order Smooth Variational Principle on Riemannian Manifolds We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature. Keywords:smooth variational principle, Riemannian manifoldCategories:58E30, 49J52, 46T05, 47J30, 58B20

54. CJM 2009 (vol 62 pp. 646)

Rupp, R.; Sasane, A.
 Reducibility in AR(K), CR(K), and A(K) Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let $A_{\mathbb R}(K)$ denote the real Banach algebra of all real symmetric continuous functions on $K$ that are analytic in the interior $K^\circ$ of $K$, endowed with the supremum norm. We characterize all unimodular pairs $(f,g)$ in $A_{\mathbb R}(K)^2$ which are reducible. In addition, for an arbitrary compact $K$ in $\mathbb C$, we give a new proof (not relying on Banach algebra theory or elementary stable rank techniques) of the fact that the Bass stable rank of $A(K)$ is $1$. Finally, we also characterize all compact real symmetric sets $K$ such that $A_{\mathbb R}(K)$, respectively $C_{\mathbb R}(K)$, has Bass stable rank $1$. Keywords:real Banach algebras, Bass stable rank, topological stable rank, reducibilityCategories:46J15, 19B10, 30H05, 93D15

55. CJM 2009 (vol 62 pp. 305)

Hua, He; Yunbai, Dong; Xianzhou, Guo
 Approximation and Similarity Classification of Stably Finitely Strongly Irreducible Decomposable Operators Let $\mathcal H$ be a complex separable Hilbert space and ${\mathcal L}({\mathcal H})$ denote the collection of bounded linear operators on ${\mathcal H}$. In this paper, we show that for any operator $A\in{\mathcal L}({\mathcal H})$, there exists a stably finitely (SI) decomposable operator $A_\epsilon$, such that $\|A-A_{\epsilon}\|<\epsilon$ and ${\mathcal{\mathcal A}'(A_{\epsilon})}/\operatorname{rad} {{\mathcal A}'(A_{\epsilon})}$ is commutative, where $\operatorname{rad}{{\mathcal A}'(A_{\epsilon})}$ is the Jacobson radical of ${{\mathcal A}'(A_{\epsilon})}$. Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of Cowen-Douglas operators given by C. L. Jiang. Keywords:$K_{0}$-group, strongly irreducible decomposition, CowenâDouglas operators, commutant algebra, similarity classificationCategories:47A05, 47A55, 46H20

56. CJM 2009 (vol 61 pp. 1262)

Dong, Z.
 On the Local Lifting Properties of Operator Spaces In this paper, we mainly study operator spaces which have the locally lifting property (LLP). The dual of any ternary ring of operators is shown to satisfy the strongly local reflexivity, and this is used to prove that strongly local reflexivity holds also for operator spaces which have the LLP. Several homological characterizations of the LLP and weak expectation property are given. We also prove that for any operator space $V$, $V^{**}$ has the LLP if and only if $V$ has the LLP and $V^{*}$ is exact. Keywords:operator space, locally lifting property, strongly locally reflexiveCategory:46L07

57. CJM 2009 (vol 61 pp. 1239)

Davidson, Kenneth R.; Yang, Dilian
 Periodicity in Rank 2 Graph Algebras Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of $\mathrm{C}^*(\mathbb{F}^+_{\theta})$. The periodic $\mathrm{C}^*$-algebras are characterized, and it is shown that $\mathrm{C}^*(\mathbb{F}^+_{\theta}) \simeq \mathrm{C}(\mathbb{T})\otimes\mathfrak{A}$ where $\mathfrak{A}$ is a simple $\mathrm{C}^*$-algebra. Keywords:higher rank graph, aperiodicity condition, simple $\mathrm{C}^*$-algebra, expectationCategories:47L55, 47L30, 47L75, 46L05

58. CJM 2009 (vol 61 pp. 503)

Baranov, Anton; Woracek, Harald
 Subspaces of de~Branges Spaces Generated by Majorants For a given de~Branges space $\mc H(E)$ we investigate de~Branges subspaces defined in terms of majorants on the real axis. If $\omega$ is a nonnegative function on $\mathbb R$, we consider the subspace $\mc R_\omega(E)=\clos_{\mc H(E)} \big\{F\in\mc H(E): \text{ there exists } C>0: |E^{-1} F|\leq C\omega \mbox{ on }{\mathbb R}\big\} .$ We show that $\mc R_\omega(E)$ is a de~Branges subspace and describe all subspaces of this form. Moreover, we give a criterion for the existence of positive minimal majorants. Keywords:de~Branges subspace, majorant, Beurling-Malliavin TheoremCategories:46E20, 30D15, 46E22

59. CJM 2009 (vol 61 pp. 241)

Azamov, N. A.; Carey, A. L.; Dodds, P. G.; Sukochev, F. A.
 Operator Integrals, Spectral Shift, and Spectral Flow We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general \vNa s. For semifinite \vNa s we give applications to the Fr\'echet differentiation of operator functions that sharpen existing results, and establish the Birman--Solomyak representation of the spectral shift function of M.\,G.\,Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow. Categories:47A56, 47B49, 47A55, 46L51

60. CJM 2009 (vol 61 pp. 282)

Bouya, Brahim
 Closed Ideals in Some Algebras of Analytic Functions We obtain a complete description of closed ideals of the algebra $\cD\cap \cL$, $0<\alpha\leq\frac{1}{2}$, where $\cD$ is the Dirichlet space and $\cL$ is the algebra of analytic functions satisfying the Lipschitz condition of order $\alpha$. Categories:46E20, 30H05, 47A15

61. CJM 2009 (vol 61 pp. 50)

Chen, Huaihui; Gauthier, Paul
 Composition operators on $\mu$-Bloch spaces Given a positive continuous function $\mu$ on the interval $0 Categories:47B33, 32A70, 46E15 62. CJM 2009 (vol 61 pp. 124) Dijkstra, Jan J.; Mill, Jan van  Characterizing Complete Erd\H os Space The space now known as {\em complete Erd\H os space\/}$\cerdos$was introduced by Paul Erd\H os in 1940 as the closed subspace of the Hilbert space$\ell^2$consisting of all vectors such that every coordinate is in the convergent sequence$\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G. Oversteegen we present simple and useful topological characterizations of$\cerdos$. As an application we determine the class of factors of$\cerdos$. In another application we determine precisely which of the spaces that can be constructed in the Banach spaces$\ell^p$according to the Erd\H os method' are homeomorphic to$\cerdos$. A novel application states that if$I$is a Polishable$F_\sigma$-ideal on$\omega$, then$I$with the Polish topology is homeomorphic to either$\Z$, the Cantor set$2^\omega$,$\Z\times2^\omega$, or$\cerdos$. This last result answers a question that was asked by Stevo Todor{\v{c}}evi{\'c}. Keywords:Complete Erd\H os space, Lelek fan, almost zero-dimensional, nowhere zero-dimensional, Polishable ideals, submeasures on$\omega$,$\R$-trees, line-free groups in Banach spacesCategories:28C10, 46B20, 54F65 63. CJM 2008 (vol 60 pp. 1108) Lopez-Abad, J.; Manoussakis, A.  A Classification of Tsirelson Type Spaces We give a complete classification of mixed Tsirelson spaces$T[(\mathcal F_i,\theta_i)_{i=1}^{r}]$for finitely many pairs of given compact and hereditary families$\mathcal F_i$of finite sets of integers and$0<\theta_i<1$in terms of the Cantor--Bendixson indices of the families$\mathcal F_i$, and$\theta_i$($1\le i\le r$). We prove that there are unique countable ordinal$\alpha$and$0<\theta<1$such that every block sequence of$T[(\mathcal F_i,\theta_i)_{i=1}^{r}]$has a subsequence equivalent to a subsequence of the natural basis of the$T(\mathcal S_{\omega^\alpha},\theta)$. Finally, we give a complete criterion of comparison in between two of these mixed Tsirelson spaces. Categories:46B20, 05D10 64. CJM 2008 (vol 60 pp. 1010) Galé, José E.; Miana, Pedro J. $H^\infty$Functional Calculus and Mikhlin-Type Multiplier Conditions Let$T$be a sectorial operator. It is known that the existence of a bounded (suitably scaled)$H^\infty$calculus for$T$, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra$\Lambda_{\infty,1}^\alpha(\R^+)$. Such an algebra includes functions defined by Mikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for$T$. In this paper, we use fractional derivation to analyse in detail the relationship between$\Lambda_{\infty,1}^\alpha$and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence. Keywords:functional calculus, fractional calculus, Mikhlin multipliers, analytic semigroups, unbounded operators, quasimultipliersCategories:47A60, 47D03, 46J15, 26A33, 47L60, 47B48, 43A22 65. CJM 2008 (vol 60 pp. 975) Boca, Florin P.  An AF Algebra Associated with the Farey Tessellation We associate with the Farey tessellation of the upper half-plane an AF algebra$\AA$encoding the `cutting sequences'' that define vertical geodesics. The Effros--Shen AF algebras arise as quotients of$\AA$. Using the path algebra model for AF algebras we construct, for each$\tau \in \big(0,\frac{1}{4}\big]$, projections$(E_n)$in$\AA$such that$E_n E_{n\pm 1}E_n \leq \tau E_n$. Categories:46L05, 11A55, 11B57, 46L55, 37E05, 82B20 66. CJM 2008 (vol 60 pp. 520) Chen, Chang-Pao; Huang, Hao-Wei; Shen, Chun-Yen  Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences Let$A=(a_{j,k})_{j,k \ge 1}$be a non-negative matrix. In this paper, we characterize those$A$for which$\|A\|_{E, F}$are determined by their actions on decreasing sequences, where$E$and$F$are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces:$\ell_p$,$d(w,p)$, and$\ell_p(w)$. The results established here generalize ones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour. Keywords:norms of matrices, normed Riesz spaces, weighted mean matrices, NÃ¶rlund mean matrices, summability matrices, matrices with row decreasingCategories:15A60, 40G05, 47A30, 47B37, 46B42 67. CJM 2008 (vol 60 pp. 703) Toms, Andrew S.; Winter, Wilhelm $\mathcal{Z}$-Stable ASH Algebras The Jiang--Su algebra$\mathcal{Z}$has come to prominence in the classification program for nuclear$C^*$-algebras of late, due primarily to the fact that Elliott's classification conjecture in its strongest form predicts that all simple, separable, and nuclear$C^*$-algebras with unperforated$\mathrm{K}$-theory will absorb$\mathcal{Z}$tensorially, i.e., will be$\mathcal{Z}$-stable. There exist counterexamples which suggest that the conjecture will only hold for simple, nuclear, separable and$\mathcal{Z}$-stable$C^*$-algebras. We prove that virtually all classes of nuclear$C^*$-algebras for which the Elliott conjecture has been confirmed so far consist of$\mathcal{Z}$-stable$C^*$-algebras. This follows in large part from the following result, also proved herein: separable and approximately divisible$C^*$-algebras are$\mathcal{Z}$-stable. Keywords:nuclear$C^*$-algebras, K-theory, classificationCategories:46L85, 46L35 68. CJM 2008 (vol 60 pp. 189) Lin, Huaxin  Furstenberg Transformations and Approximate Conjugacy Let$\alpha$and$\beta$be two Furstenberg transformations on$2$-torus associated with irrational numbers$\theta_1,\theta_2,$integers$d_1, d_2$and Lipschitz functions$f_1$and$f_2$. It is shown that$\alpha$and$\beta$are approximately conjugate in a measure theoretical sense if (and only if)$\overline{\theta_1\pm \theta_2}=0$in$\R/\Z.$Closely related to the classification of simple amenable \CAs, it is shown that$\af$and$\bt$are approximately$K$-conjugate if (and only if)$\overline{\theta_1\pm \theta_2}=0$in$\R/\Z$and$|d_1|=|d_2|.$This is also shown to be equivalent to the condition that the associated crossed product \CAs are isomorphic. Keywords:Furstenberg transformations, approximate conjugacyCategories:37A55, 46L35 69. CJM 2007 (vol 59 pp. 1135) Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari  Sobolev Extensions of HÃ¶lder Continuous and Characteristic Functions on Metric Spaces We study when characteristic and H\"older continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and H\"older continuous functions into globally defined Sobolev functions. Keywords:characteristic function, Newtonian function, metric space, resolutivity, HÃ¶lder continuous, Perron solution,$p$-harmonic, Sobolev extension, Whitney coveringCategories:46E35, 31C45 70. CJM 2007 (vol 59 pp. 1029) Kalton, N. J.; Koldobsky, A.; Yaskin, V.; Yaskina, M.  The Geometry of$L_0$Suppose that we have the unit Euclidean ball in$\R^n$and construct new bodies using three operations --- linear transformations, closure in the radial metric, and multiplicative summation defined by$\|x\|_{K+_0L} = \sqrt{\|x\|_K\|x\|_L}.$We prove that in dimension$3$this procedure gives all origin-symmetric convex bodies, while this is no longer true in dimensions$4$and higher. We introduce the concept of embedding of a normed space in$L_0$that naturally extends the corresponding properties of$L_p$-spaces with$p\ne0$, and show that the procedure described above gives exactly the unit balls of subspaces of$L_0$in every dimension. We provide Fourier analytic and geometric characterizations of spaces embedding in$L_0$, and prove several facts confirming the place of$L_0$in the scale of$L_p$-spaces. Categories:52A20, 52A21, 46B20 71. CJM 2007 (vol 59 pp. 966) Forrest, Brian E.; Runde, Volker; Spronk, Nico  Operator Amenability of the Fourier Algebra in the$\cb$-Multiplier Norm Let$G$be a locally compact group, and let$A_{\cb}(G)$denote the closure of$A(G)$, the Fourier algebra of$G$, in the space of completely bounded multipliers of$A(G)$. If$G$is a weakly amenable, discrete group such that$\cstar(G)$is residually finite-dimensional, we show that$A_{\cb}(G)$is operator amenable. In particular,$A_{\cb}(\free_2)$is operator amenable even though$\free_2$, the free group in two generators, is not an amenable group. Moreover, we show that if$G$is a discrete group such that$A_{\cb}(G)$is operator amenable, a closed ideal of$A(G)$is weakly completely complemented in$A(G)$if and only if it has an approximate identity bounded in the$\cb$-multiplier norm. Keywords:$\cb$-multiplier norm, Fourier algebra, operator amenability, weak amenabilityCategories:43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25 72. CJM 2007 (vol 59 pp. 897) Bruneau, Laurent  The Ground State Problem for a Quantum Hamiltonian Model Describing Friction In this paper, we consider the quantum version of a Hamiltonian model describing friction. This model consists of a particle which interacts with a bosonic reservoir representing a homogeneous medium through which the particle moves. We show that if the particle is confined, then the Hamiltonian admits a ground state if and only if a suitable infrared condition is satisfied. The latter is violated in the case of linear friction, but satisfied when the friction force is proportional to a higher power of the particle speed. Categories:81Q10, 46N50 73. 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 74. 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

75. 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
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