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26. CJM 2011 (vol 63 pp. 1161)

Neuwirth, Stefan; Ricard, Éric
 Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group $\varGamma$ and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum $\varLambda\subseteq\varGamma$, the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than 1. Keywords:Fourier multiplier, Toeplitz Schur multiplier, lacunary set, unconditional approximation property, Hilbert transform, Riesz projectionCategories:47B49, 43A22, 43A46, 46B28

27. CJM 2011 (vol 64 pp. 573)

Nawata, Norio
 Fundamental Group of Simple $C^*$-algebras with Unique Trace III We introduce the fundamental group ${\mathcal F}(A)$ of a simple $\sigma$-unital $C^*$-algebra $A$ with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of Fundamental Group of Simple $C^*$-algebras with Unique Trace I and II'' by Nawata and Watatani. Our definition in this paper makes sense for stably projectionless $C^*$-algebras. We show that there exist separable stably projectionless $C^*$-algebras such that their fundamental groups are equal to $\mathbb{R}_+^\times$ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian. Keywords:fundamental group, Picard group, Hilbert module, countable basis, stably projectionless algebra, dimension functionCategories:46L05, 46L08, 46L35

28. CJM 2011 (vol 64 pp. 455)

Sherman, David
 On Cardinal Invariants and Generators for von Neumann Algebras We demonstrate how most common cardinal invariants associated with a von Neumann algebra $\mathcal M$ can be computed from the decomposability number, $\operatorname{dens}(\mathcal M)$, and the minimal cardinality of a generating set, $\operatorname{gen}(\mathcal M)$. Applications include the equivalence of the well-known generator problem, Is every separably-acting von Neumann algebra singly-generated?", with the formally stronger questions, Is every countably-generated von Neumann algebra singly-generated?" and Is the $\operatorname{gen}$ invariant monotone?" Modulo the generator problem, we determine the range of the invariant $\bigl( \operatorname{gen}(\mathcal M), \operatorname{dens}(\mathcal M) \bigr)$, which is mostly governed by the inequality $\operatorname{dens}(\mathcal M) \leq \mathfrak C^{\operatorname{gen}(\mathcal M)}$. Keywords:von Neumann algebra, cardinal invariant, generator problem, decomposability number, representation densityCategory:46L10

29. CJM 2011 (vol 63 pp. 798)

Daws, Matthew
 Representing Multipliers of the Fourier Algebra on Non-Commutative $L^p$ Spaces We show that the multiplier algebra of the Fourier algebra on a locally compact group $G$ can be isometrically represented on a direct sum on non-commutative $L^p$ spaces associated with the right von Neumann algebra of $G$. The resulting image is the idealiser of the image of the Fourier algebra. If these spaces are given their canonical operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the non-commutative $L^p$ spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca-Herz algebra built out of these non-commutative $L^p$ spaces, say $A_p(\widehat G)$. It is shown that $A_2(\widehat G)$ is isometric to $L^1(G)$, generalising the abelian situation. Keywords:multiplier, Fourier algebra, non-commutative $L^p$ space, complex interpolationCategories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52

30. CJM 2011 (vol 63 pp. 1188)

Śliwa, Wiesław; Ziemkowska, Agnieszka
 On Complemented Subspaces of Non-Archimedean Power Series Spaces The non-archimedean power series spaces, $A_1(a)$ and $A_\infty(b)$, are the best known and most important examples of non-archimedean nuclear FrÃ©chet spaces. We prove that the range of every continuous linear map from $A_p(a)$ to $A_q(b)$ has a Schauder basis if either $p=1$ or $p=\infty$ and the set $M_{b,a}$ of all bounded limit points of the double sequence $(b_i/a_j)_{i,j\in\mathbb{N}}$ is bounded. It follows that every complemented subspace of a power series space $A_p(a)$ has a Schauder basis if either $p=1$ or $p=\infty$ and the set $M_{a,a}$ is bounded. Keywords:non-archimedean KÃ¶the space, range of a continuous linear map, Schauder basisCategories:46S10, 47S10, 46A35

31. CJM 2011 (vol 63 pp. 551)

Hadwin, Don; Li, Qihui; Shen, Junhao
 Topological Free Entropy Dimensions in Nuclear C$^*$-algebras and in Full Free Products of Unital C$^*$-algebras In the paper, we introduce a new concept, topological orbit dimension of an $n$-tuple of elements in a unital C$^{\ast}$-algebra. Using this concept, we conclude that Voiculescu's topological free entropy dimension of every finite family of self-adjoint generators of a nuclear C$^{\ast}$-algebra is less than or equal to $1$. We also show that the Voiculescu's topological free entropy dimension is additive in the full free product of some unital C$^{\ast}$-algebras. We show that the unital full free product of Blackadar and Kirchberg's unital MF algebras is also an MF algebra. As an application, we obtain that $\mathop{\textrm{Ext}}(C_{r}^{\ast}(F_{2})\ast_{\mathbb{C}}C_{r}^{\ast}(F_{2}))$ is not a group. Keywords: topological free entropy dimension, unital C$^{*}$-algebraCategories:46L10, 46L54

32. CJM 2011 (vol 63 pp. 460)

Pavlíček, Libor
 Monotonically Controlled Mappings We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a RadÃ³-Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the FrÃ©chet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation. Keywords: monotone mapping, DM mapping, RadÃ³-Reichelderfer property, UDM mapping, differentiabilityCategories:26B05, 46G05

33. CJM 2011 (vol 63 pp. 381)

Ji, Kui ; Jiang, Chunlan
 A Complete Classification of AI Algebras with the Ideal Property Let $A$ be an AI algebra; that is, $A$ is the $\mbox{C}^{*}$-algebra inductive limit of a sequence $$A_{1}\stackrel{\phi_{1,2}}{\longrightarrow}A_{2}\stackrel{\phi_{2,3}}{\longrightarrow}A_{3} \longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots,$$ where $A_{n}=\bigoplus_{i=1}^{k_n}M_{[n,i]}(C(X^{i}_n))$, $X^{i}_n$ are $[0,1]$, $k_n$, and $[n,i]$ are positive integers. Suppose that $A$ has the ideal property: each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two-sided ideal. In this article, we give a complete classification of AI algebras with the ideal property. Keywords:AI algebras, K-group, tracial state, ideal property, classificationCategories:46L35, 19K14, 46L05, 46L08

34. CJM 2011 (vol 63 pp. 500)

Dadarlat, Marius; Elliott, George A.; Niu, Zhuang
 One-Parameter Continuous Fields of Kirchberg Algebras. II Parallel to the first two authors' earlier classification of separable, unita one-parameter, continuous fields of Kirchberg algebras with torsion free $\mathrm{K}$-groups supported in one dimension, one-parameter, separable, uni continuous fields of AF-algebras are classified by their ordered $\mathrm{K}_0$-sheaves. Effros-Handelman-Shen type theorems are pr separable unital one-parameter continuous fields of AF-algebras and Kirchberg algebras. Keywords:continuous fields of C$^*$-algebras, $\mathrm{K}_0$-presheaves, Effros--Handeen type theoremCategory:46L35

35. CJM 2010 (vol 63 pp. 436)

Mine, Kotaro; Sakai, Katsuro
 Simplicial Complexes and Open Subsets of Non-Separable LF-Spaces Let $F$ be a non-separable LF-space homeomorphic to the direct sum $\sum_{n\in\mathbb{N}} \ell_2(\tau_n)$, where $\aleph_0 < \tau_1 < \tau_2 < \cdots$. It is proved that every open subset $U$ of $F$ is homeomorphic to the product $|K| \times F$ for some locally finite-dimensional simplicial complex $K$ such that every vertex $v \in K^{(0)}$ has the star $\operatorname{St}(v,K)$ with $\operatorname{card} \operatorname{St}(v,K)^{(0)} < \tau = \sup\tau_n$ (and $\operatorname{card} K^{(0)} \le \tau$), and, conversely, if $K$ is such a simplicial complex, then the product $|K| \times F$ can be embedded in $F$ as an open set, where $|K|$ is the polyhedron of $K$ with the metric topology. Keywords:LF-space, open set, simplicial complex, metric topology, locally finite-dimensional, star, small box product, ANR, $\ell_2(\tau)$, $\ell_2(\tau)$-manifold, open embedding, $\sum_{i\in\mathbb{N}}\ell_2(\tau_i)$Categories:57N20, 46A13, 46T05, 57N17, 57Q05, 57Q40

36. CJM 2010 (vol 63 pp. 123)

Granirer, Edmond E.
 Strong and Extremely Strong Ditkin sets for the Banach Algebras $A_p^r(G)=A_p\cap L^r(G)$ Let $A_p(G)$ be the Figa-Talamanca, Herz Banach Algebra on $G$; thus $A_2(G)$ is the Fourier algebra. Strong Ditkin (SD) and Extremely Strong Ditkin (ESD) sets for the Banach algebras $A_p^r(G)$ are investigated for abelian and nonabelian locally compact groups $G$. It is shown that SD and ESD sets for $A_p(G)$ remain SD and ESD sets for $A_p^r(G)$, with strict inclusion for ESD sets. The case for the strict inclusion of SD sets is left open. A result on the weak sequential completeness of $A_2(F)$ for ESD sets $F$ is proved and used to show that Varopoulos, Helson, and Sidon sets are not ESD sets for $A_2(G)$, yet they are such for $A_2^r(G)$ for discrete groups $G$, for any $1\le r\le 2$. A result is given on the equivalence of the sequential and the net definitions of SD or ESD sets for $\sigma$-compact groups. The above results are new even if $G$ is abelian. Keywords:Fourier algebra, Figa-Talamanca-Herz algebra, locally compact group, Ditkin sets, Helson sets, Sidon sets, weak sequential completenessCategories:43A15, 43A10, 46J10, 43A45

37. CJM 2010 (vol 63 pp. 222)

Wang, Jiun-Chau
 Limit Theorems for Additive Conditionally Free Convolution In this paper we determine the limiting distributional behavior for sums of infinitesimal conditionally free random variables. We show that the weak convergence of classical convolution and that of conditionally free convolution are equivalent for measures in an infinitesimal triangular array, where the measures may have unbounded support. Moreover, we use these limit theorems to study the conditionally free infinite divisibility. These results are obtained by complex analytic methods without reference to the combinatorics of c-free convolution. Keywords:additive c-free convolution, limit theorems, infinitesimal arraysCategories:46L53, 60F05

38. CJM 2010 (vol 63 pp. 3)

Banica, T.; Belinschi, S. T.; Capitaine, M.; Collins, B.
 Free Bessel Laws We introduce and study a remarkable family of real probability measures $\pi_{st}$ that we call free Bessel laws. These are related to the free Poisson law $\pi$ via the formulae $\pi_{s1}=\pi^{\boxtimes s}$ and ${\pi_{1t}=\pi^{\boxplus t}}$. Our study includes definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups. Keywords:Poisson law, Bessel function, Wishart matrix, quantum groupCategories:46L54, 15A52, 16W30

39. CJM 2010 (vol 62 pp. 961)

Aleman, Alexandru; Duren, Peter; Martín, María J.; Vukotić, Dragan
 Multiplicative Isometries and Isometric Zero-Divisors For some Banach spaces of analytic functions in the unit disk (weighted Bergman spaces, Bloch space, Dirichlet-type spaces), the isometric pointwise multipliers are found to be unimodular constants. As a consequence, it is shown that none of those spaces have isometric zero-divisors. Isometric coefficient multipliers are also investigated. Keywords:Banach spaces of analytic functions, Hardy spaces, Bergman spaces, Bloch space, Dirichlet space, Dirichlet-type spaces, pointwise multipliers, coefficient multipliers, isometries, isometric zero-divisorsCategories:30H05, 46E15

40. CJM 2010 (vol 62 pp. 827)

Ouyang, Caiheng; Xu, Quanhua
 BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm. Keywords:BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spacesCategories:46E40, 42B25, 46B20

41. CJM 2010 (vol 62 pp. 845)

Samei, Ebrahim; Spronk, Nico; Stokke, Ross
 Biflatness and Pseudo-Amenability of Segal Algebras We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, $L^1(G)$, and the Fourier algebra, $A(G)$, of a locally compact group~$G$. Keywords:Segal algebra, pseudo-amenable Banach algebra, biflat Banach algebraCategories:43A20, 43A30, 46H25, 46H10, 46H20, 46L07

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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

48. 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

49. 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

50. 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
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