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51. CMB 2008 (vol 51 pp. 67)

Kalton, Nigel; Sukochev, Fyodor
 Rearrangement-Invariant Functionals with Applications to Traces on Symmetrically Normed Ideals We present a construction of singular rearrangement invariant functionals on Marcinkiewicz function/operator spaces. The functionals constructed differ from all previous examples in the literature in that they fail to be symmetric. In other words, the functional $\phi$ fails the condition that if $x\pprec y$ (Hardy-Littlewood-Polya submajorization) and $0\leq x,y$, then $0\le \phi(x)\le \phi(y).$ We apply our results to singular traces on symmetric operator spaces (in particular on symmetrically-normed ideals of compact operators), answering questions raised by Guido and Isola. Categories:46L52, 47B10, 46E30

52. CMB 2007 (vol 50 pp. 172)

Aron, Richard; Gorkin, Pamela
 An Infinite Dimensional Vector Space of Universal Functions for $H^\infty$ of the Ball We show that there exists a closed infinite dimensional subspace of $H^\infty(B^n)$ such that every function of norm one is universal for some sequence of automorphisms of $B^n$. Categories:47B38, 47B33, 46J10

53. CMB 2007 (vol 50 pp. 85)

Han, Deguang
 Classification of Finite Group-Frames and Super-Frames Given a finite group $G$, we examine the classification of all frame representations of $G$ and the classification of all $G$-frames, \emph{i.e.,} frames induced by group representations of $G$. We show that the exact number of equivalence classes of $G$-frames and the exact number of frame representations can be explicitly calculated. We also discuss how to calculate the largest number $L$ such that there exists an $L$-tuple of strongly disjoint $G$-frames. Keywords:frames, group-frames, frame representations, disjoint framesCategories:42C15, 46C05, 47B10

54. CMB 2006 (vol 49 pp. 117)

Levene, R. H.
 A Double Triangle Operator Algebra From $SL_2(\R)$ We consider the w$^*$-closed operator algebra $\cA_+$ generated by the image of the semigroup $SL_2(\R_+)$ under a unitary representation $\rho$ of $SL_2(\R)$ on the Hilbert~space $L_2(\R)$. We show that $\cA_+$ is a reflexive operator algebra and $\cA_+=\Alg\cD$ where $\cD$ is a double triangle subspace lattice. Surprisingly, $\cA_+$ is also generated as a w$^*$-closed algebra by the image under $\rho$ of a strict subsemigroup of $SL_2(\R_+)$. Categories:46K50, 47L55

55. CMB 2005 (vol 48 pp. 607)

Park, Efton
 Toeplitz Algebras and Extensions of\\Irrational Rotation Algebras For a given irrational number $\theta$, we define Toeplitz operators with symbols in the irrational rotation algebra ${\mathcal A}_\theta$, and we show that the $C^*$-algebra $\mathcal T({\mathcal A}_\theta)$ generated by these Toeplitz operators is an extension of ${\mathcal A}_\theta$ by the algebra of compact operators. We then use these extensions to explicitly exhibit generators of the group $KK^1({\mathcal A}_\theta,\mathbb C)$. We also prove an index theorem for $\mathcal T({\mathcal A}_\theta)$ that generalizes the standard index theorem for Toeplitz operators on the circle. Keywords:Toeplitz operators, irrational rotation algebras, index theoryCategories:47B35, 46L80

56. CMB 2005 (vol 48 pp. 409)

Gauthier, P. M.; Xiao, J.
 The Existence of Universal Inner Functions on the Unit Ball of $\mathbb{C}^n$ It is shown that there exists an inner function $I$ defined on the unit ball ${\bf B}^n$ of ${\mathbb C}^n$ such that each function holomorphic on ${\bf B}^n$ and bounded by $1$ can be approximated by non-Euclidean translates" of $I$. Keywords:universal inner functionsCategories:32A35, 30D50, 47B38

57. CMB 2005 (vol 48 pp. 251)

Murphy, G. J.
 The Index Theory Associated to a Non-Finite Trace on a $C^\ast$-Algebra The index theory considered in this paper, a generalisation of the classical Fredholm index theory, is obtained in terms of a non-finite trace on a unital $C^\ast$-algebra. We relate it to the index theory of M.~Breuer, which is developed in a von~Neumann algebra setting, by means of a representation theorem. We show how our new index theory can be used to obtain an index theorem for Toeplitz operators on the compact group $\mathbf{U}(2)$, where the classical index theory does not give any interesting result. Categories:46L, 47B35, 47L80

58. CMB 2005 (vol 48 pp. 97)

Katavolos, Aristides; Paulsen, Vern I.
 On the Ranges of Bimodule Projections We develop a symbol calculus for normal bimodule maps over a masa that is the natural analogue of the Schur product theory. Using this calculus we are easily able to give a complete description of the ranges of contractive normal bimodule idempotents that avoids the theory of J*-algebras. We prove that if $P$ is a normal bimodule idempotent and $\|P\| < 2/\sqrt{3}$ then $P$ is a contraction. We finish with some attempts at extending the symbol calculus to non-normal maps. Categories:46L15, 47L25

59. CMB 2004 (vol 47 pp. 615)

Randrianantoanina, Narcisse
 $C^*$-Algebras and Factorization Through Diagonal Operators Let $\cal A$ be a $C^*$-algebra and $E$ be a Banach space with the Radon-Nikodym property. We prove that if $j$ is an embedding of $E$ into an injective Banach space then for every absolutely summing operator $T:\mathcal{A}\longrightarrow E$, the composition $j \circ T$ factors through a diagonal operator from $l^{2}$ into $l^{1}$. In particular, $T$ factors through a Banach space with the Schur property. Similarly, we prove that for $2 Keywords:$C^*$-algebras, summing operators, diagonal operators,, Radon-Nikodym propertyCategories:46L50, 47D15 60. CMB 2004 (vol 47 pp. 504) Cardoso, Fernando; Vodev, Georgi  High Frequency Resolvent Estimates and Energy Decay of Solutions to the Wave Equation We prove an uniform H\"older continuity of the resolvent of the Laplace-Beltrami operator on the real axis for a class of asymptotically Euclidean Riemannian manifolds. As an application we extend a result of Burq on the behaviour of the local energy of solutions to the wave equation. Categories:35B37, 35J15, 47F05 61. CMB 2004 (vol 47 pp. 456) Seto, Michio  On the Berger-Coburn-Lebow Problem for Hardy Submodules In this paper we shall give an affirmative solution to a problem, posed by Berger, Coburn and Lebow, for$C^{\ast}$-algebras on Hardy submodules. Keywords:Hardy submodulesCategory:47B38 62. CMB 2004 (vol 47 pp. 369) Fošner, Ajda; Šemrl, Peter  Spectrally Bounded Linear Maps on${\cal B}(X)$We characterize surjective linear maps on${\cal B}(X)$that are spectrally bounded and spectrally bounded below. Keywords:spectrally bounded linear map.Category:47B49 63. CMB 2004 (vol 47 pp. 257) Marwaha, Alka  A Geometric Characterization of Nonnegative Bands A band is a semigroup of idempotent operators. A nonnegative band$\cls$in$\clb(\cll^2 (\clx))$having at least one element of finite rank and with rank$(S) > 1 $for all$S$in$\cls$is known to have a special kind of common invariant subspace which is termed a standard subspace (defined below). Such bands are called decomposable. Decomposability has helped to understand the structure of nonnegative bands with constant finite rank. In this paper, a geometric characterization of maximal, rank-one, indecomposable nonnegative bands is obtained which facilitates the understanding of their geometric structure. Categories:47D03, 47A15 64. CMB 2004 (vol 47 pp. 298) Yahaghi, Bamdad R.  Near Triangularizability Implies Triangularizability In this paper we consider collections of compact operators on a real or complex Banach space including linear operators on finite-dimensional vector spaces. We show that such a collection is simultaneously triangularizable if and only if it is arbitrarily close to a simultaneously triangularizable collection of compact operators. As an application of these results we obtain an invariant subspace theorem for certain bounded operators. We further prove that in finite dimensions near reducibility implies reducibility whenever the ground field is$\BR$or$\BC$. Keywords:Linear transformation, Compact operator,, Triangularizability, Banach space, Hilbert, spaceCategories:47A15, 47D03, 20M20 65. CMB 2004 (vol 47 pp. 215) Jaworski, Wojciech  Countable Amenable Identity Excluding Groups A discrete group$G$is called \emph{identity excluding\/} if the only irreducible unitary representation of$G$which weakly contains the$1$-dimensional identity representation is the$1$-dimensional identity representation itself. Given a unitary representation$\pi$of$G$and a probability measure$\mu$on$G$, let$P_\mu$denote the$\mu$-average$\int\pi(g) \mu(dg)$. The goal of this article is twofold: (1)~to study the asymptotic behaviour of the powers$P_\mu^n$, and (2)~to provide a characterization of countable amenable identity excluding groups. We prove that for every adapted probability measure$\mu$on an identity excluding group and every unitary representation$\pi$there exists and orthogonal projection$E_\mu$onto a$\pi$-invariant subspace such that$s$-$\lim_{n\to\infty}\bigl(P_\mu^n- \pi(a)^nE_\mu\bigr)=0$for every$a\in\supp\mu$. This also remains true for suitably defined identity excluding locally compact groups. We show that the class of countable amenable identity excluding groups coincides with the class of$\FC$-hypercentral groups; in the finitely generated case this is precisely the class of groups of polynomial growth. We also establish that every adapted random walk on a countable amenable identity excluding group is ergodic. Categories:22D10, 22D40, 43A05, 47A35, 60B15, 60J50 66. CMB 2004 (vol 47 pp. 100) Seto, Michio  Invariant Subspaces on$\mathbb{T}^N$and$\mathbb{R}^N$Let$N$be an integer which is larger than one. In this paper we study invariant subspaces of$L^2 (\mathbb{T}^N)$under the double commuting condition. A main result is an$N$-dimensional version of the theorem proved by Mandrekar and Nakazi. As an application of this result, we have an$N$-dimensional version of Lax's theorem. Keywords:invariant subspacesCategories:47A15, 47B47 67. CMB 2004 (vol 47 pp. 144) Xia, Jingbo  On the Uniqueness of Wave Operators Associated With Non-Trace Class Perturbations Voiculescu has previously established the uniqueness of the wave operator for the problem of$\mathcal{C}^{(0)}$-perturbation of commuting tuples of self-adjoint operators in the case where the norm ideal$\mathcal{C}$has the property$\lim_{n\rightarrow\infty} n^{-1/2}\|P_n\|_{\mathcal{C}}=0$, where$\{P_n\}$is any sequence of orthogonal projections with$\rank(P_n)=n$. We prove that the same uniqueness result holds true so long as$\mathcal{C}$is not the trace class. (It is well known that there is no such uniqueness in the case of trace-class perturbation.) Categories:47A40, 47B10 68. CMB 2004 (vol 47 pp. 49) Lindström, Mikael; Makhmutov, Shamil; Taskinen, Jari  The Essential Norm of a Bloch-to-$Q_p$Composition Operator The$Q_p$spaces coincide with the Bloch space for$p>1$and are subspaces of$\BMOA$for$0 Keywords:Bloch space, little Bloch space, $\BMOA$, $\VMOA$, $Q_p$ spaces,, composition operator, compact operator, essential normCategories:47B38, 47B10, 46E40, 46E15

69. CMB 2003 (vol 46 pp. 538)

Borwein, Jonathan; Fitzpatrick, Simon; Girgensohn, Roland
 Subdifferentials Whose Graphs Are Not Norm$\times$Weak* Closed In this note we give examples of convex functions whose subdifferentials have unpleasant properties. Particularly, we exhibit a proper lower semicontinuous convex function on a separable Hilbert space such that the graph of its subdifferential is not closed in the product of the norm and bounded weak topologies. We also exhibit a set whose sequential normal cone is not norm closed. Categories:46N10, 47H05

70. CMB 2003 (vol 46 pp. 632)

Runde, Volker
 The Operator Amenability of Uniform Algebras We prove a quantized version of a theorem by M.~V.~She\u{\i}nberg: A uniform algebra equipped with its canonical, {\it i.e.}, minimal, operator space structure is operator amenable if and only if it is a commutative $C^\ast$-algebra. Keywords:uniform algebras, amenable Banach algebras, operator amenability, minimal, operator spaceCategories:46H20, 46H25, 46J10, 46J40, 47L25

71. CMB 2003 (vol 46 pp. 216)

Li, Chi-Kwong; Rodman, Leiba; Šemrl, Peter
 Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range Let $H$ be a complex Hilbert space, and $\HH$ be the real linear space of bounded selfadjoint operators on $H$. We study linear maps $\phi\colon \HH \to \HH$ leaving invariant various properties such as invertibility, positive definiteness, numerical range, {\it etc}. The maps $\phi$ are not assumed {\it a priori\/} continuous. It is shown that under an appropriate surjective or injective assumption $\phi$ has the form $X \mapsto \xi TXT^*$ or $X \mapsto \xi TX^tT^*$, for a suitable invertible or unitary $T$ and $\xi\in\{1, -1\}$, where $X^t$ stands for the transpose of $X$ relative to some orthonormal basis. Examples are given to show that the surjective or injective assumption cannot be relaxed. The results are extended to complex linear maps on the algebra of bounded linear operators on $H$. Similar results are proved for the (real) linear space of (selfadjoint) operators of the form $\alpha I+K$, where $\alpha$ is a scalar and $K$ is compact. Keywords:linear map, selfadjoint operator, invertible, positive definite, numerical rangeCategories:47B15, 47B49

72. CMB 2003 (vol 46 pp. 113)

Lee, Jaesung; Rim, Kyung Soo
 Properties of the $\mathcal{M}$-Harmonic Conjugate Operator We define the $\mathcal{M}$-harmonic conjugate operator $K$ and prove that it is bounded on the nonisotropic Lipschitz space and on $\BMO$. Then we show $K$ maps Dini functions into the space of continuous functions on the unit sphere. We also prove the boundedness and compactness properties of $\mathcal{M}$-harmonic conjugate operator with $L^p$ symbol. Keywords:$\mathcal{M}$-harmonic conjugate operatorCategories:32A70, 47G10

73. CMB 2003 (vol 46 pp. 59)

Constantinescu, T.; Johnson, J. L.
 A Note on Noncommutative Interpolation In this paper we formulate and solve Nevanlinna-Pick and Carath\'eodory type problems for tensor algebras with data given on the $N$-dimensional operator unit ball of a Hilbert space. We develop an approach based on the displacement structure theory. Categories:47A57, 47A20

74. CMB 2002 (vol 45 pp. 309)

Xia, Jingbo
 Joint Mean Oscillation and Local Ideals in the Toeplitz Algebra II: Local Commutivity and Essential Commutant A well-known theorem of Sarason [11] asserts that if $[T_f,T_h]$ is compact for every $h \in H^\infty$, then $f \in H^\infty + C(T)$. Using local analysis in the full Toeplitz algebra $\calT = \calT (L^\infty)$, we show that the membership $f \in H^\infty + C(T)$ can be inferred from the compactness of a much smaller collection of commutators $[T_f,T_h]$. Using this strengthened result and a theorem of Davidson [2], we construct a proper $C^\ast$-subalgebra $\calT (\calL)$ of $\calT$ which has the same essential commutant as that of $\calT$. Thus the image of $\calT (\calL)$ in the Calkin algebra does not satisfy the double commutant relation [12], [1]. We will also show that no {\it separable} subalgebra $\calS$ of $\calT$ is capable of conferring the membership $f \in H^\infty + C(T)$ through the compactness of the commutators $\{[T_f,S] : S \in \calS\}$. Categories:46H10, 47B35, 47C05

75. CMB 2001 (vol 44 pp. 469)

Marcoux, Laurent W.
 Sums and Products of Weighted Shifts In this article it is shown that every bounded linear operator on a complex, infinite dimensional, separable Hilbert space is a sum of at most eighteen unilateral (alternatively, bilateral) weighted shifts. As well, we classify products of weighted shifts, as well as sums and limits of the resulting operators. Categories:47B37, 47A99
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