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51. 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 Radó-Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet 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, Radó-Reichelderfer property, UDM mapping, differentiability
Categories:26B05, 46G05

52. 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, classification
Categories:46L35, 19K14, 46L05, 46L08

53. 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 theorem

54. 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

55. 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 completeness
Categories:43A15, 43A10, 46J10, 43A45

56. 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 arrays
Categories:46L53, 60F05

57. 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 group
Categories:46L54, 15A52, 16W30

58. 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-divisors
Categories:30H05, 46E15

59. 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 spaces
Categories:46E40, 42B25, 46B20

60. 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 algebra
Categories:43A20, 43A30, 46H25, 46H10, 46H20, 46L07

61. 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

62. 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

63. 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, Cowen—Douglas operators, commutant algebra, similarity classification
Categories:47A05, 47A55, 46H20

64. 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, reducibility
Categories:46J15, 19B10, 30H05, 93D15

65. 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 manifold
Categories:58E30, 49J52, 46T05, 47J30, 58B20

66. 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, expectation
Categories:47L55, 47L30, 47L75, 46L05

67. 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 reflexive

68. 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 Theorem
Categories:46E20, 30D15, 46E22

69. 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

70. 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

71. 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 spaces
Categories:28C10, 46B20, 54F65

72. 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

73. 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, quasimultipliers
Categories:47A60, 47D03, 46J15, 26A33, 47L60, 47B48, 43A22

74. 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

75. 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
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