Expand all Collapse all  Results 51  75 of 92 
51. CMB 2005 (vol 48 pp. 251)
The Index Theory Associated to a NonFinite 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 nonfinite 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 
52. CMB 2005 (vol 48 pp. 97)
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 nonnormal maps.
Categories:46L15, 47L25 
53. CMB 2004 (vol 47 pp. 615)
$C^*$Algebras and Factorization Through Diagonal Operators Let $\cal A$ be a $C^*$algebra and $E$ be a Banach space with
the RadonNikodym 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

54. CMB 2004 (vol 47 pp. 504)
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 LaplaceBeltrami 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 
55. CMB 2004 (vol 47 pp. 456)
On the BergerCoburnLebow 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 submodules Category:47B38 
56. CMB 2004 (vol 47 pp. 369)
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 
57. CMB 2004 (vol 47 pp. 257)
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,
rankone, indecomposable nonnegative bands is obtained which
facilitates the understanding of their geometric structure.
Categories:47D03, 47A15 
58. CMB 2004 (vol 47 pp. 298)
Near Triangularizability Implies Triangularizability In this paper we consider collections of
compact operators on a real or
complex Banach space including linear operators
on finitedimensional 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, space Categories:47A15, 47D03, 20M20 
59. CMB 2004 (vol 47 pp. 215)
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 
60. CMB 2004 (vol 47 pp. 100)
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 subspaces Categories:47A15, 47B47 
61. CMB 2004 (vol 47 pp. 144)
On the Uniqueness of Wave Operators Associated With NonTrace Class Perturbations Voiculescu has previously established the uniqueness of the wave operator
for the problem of $\mathcal{C}^{(0)}$perturbation of commuting tuples
of selfadjoint 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 traceclass perturbation.)
Categories:47A40, 47B10 
62. CMB 2004 (vol 47 pp. 49)
The Essential Norm of a Blochto$Q_p$ Composition Operator The $Q_p$ spaces coincide with the Bloch space for $p>1$ and are
subspaces of $\BMOA$ for $0

63. CMB 2003 (vol 46 pp. 538)
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 
64. CMB 2003 (vol 46 pp. 632)
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 space Categories:46H20, 46H25, 46J10, 46J40, 47L25 
65. CMB 2003 (vol 46 pp. 216)
Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range 
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 range Categories:47B15, 47B49 
66. CMB 2003 (vol 46 pp. 59)
A Note on Noncommutative Interpolation In this paper we formulate and solve NevanlinnaPick 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 
67. CMB 2003 (vol 46 pp. 113)
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 operator Categories:32A70, 47G10 
68. CMB 2002 (vol 45 pp. 309)
Joint Mean Oscillation and Local Ideals in the Toeplitz Algebra II: Local Commutivity and Essential Commutant 
Joint Mean Oscillation and Local Ideals in the Toeplitz Algebra II: Local Commutivity and Essential Commutant A wellknown 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 
69. CMB 2001 (vol 44 pp. 469)
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 
70. CMB 2001 (vol 44 pp. 270)
Linear Operators Preserving Generalized Numerical Ranges and Radii on Certain Triangular Algebras of Matrices 
Linear Operators Preserving Generalized Numerical Ranges and Radii on Certain Triangular Algebras of Matrices Let $c = (c_1, \dots, c_n)$ be such that $c_1 \ge \cdots \ge c_n$.
The $c$numerical range of an $n \times n$ matrix $A$ is defined by
$$
W_c(A) = \Bigl\{ \sum_{j=1}^n c_j (Ax_j,x_j) : \{x_1, \dots, x_n\}
\text{ an orthonormal basis for } \IC^n \Bigr\},
$$
and the $c$numerical radius of $A$ is defined by $r_c (A) = \max
\{z : z \in W_c (A)\}$. We determine the structure of those linear
operators $\phi$ on algebras of block triangular matrices, satisfying
$$
W_c \bigl( \phi(A) \bigr) = W_c (A) \text{ for all } A \quad \text{or}
\quad r_c \bigl( \phi(A) \bigr) = r_c (A) \text{ for all } A.
$$
Keywords:linear operator, numerical range (radius), block triangular matrices Categories:15A04, 15A60, 47B49 
71. CMB 2000 (vol 43 pp. 406)
Weighted Mean Operators on $l_p$ The weighted mean matrix $M_a$ is the triangular matrix $\{a_k/A_n\}$,
where $a_n > 0$ and $A_n := a_1 + a_2 + \cdots + a_n$. It is proved
that, subject to $n^c a_n$ being eventually monotonic for each
constant $c$ and to the existence of $\alpha := \lim
\frac{A_n}{na_n}$, $M_a \in B(l_p)$ for $1 < p < \infty$ if and only
if $\alpha < p$.
Keywords:weighted means, operators on $l_p$, norm estimates Categories:47B37, 47A30, 40G05 
72. CMB 2000 (vol 43 pp. 193)
C$^*$Convexity and the Numerical Range If $A$ is a prime C$^*$algebra, $a \in A$ and $\lambda$ is in the
numerical range $W(a)$ of $a$, then for each $\varepsilon > 0$ there
exists an element $h \in A$ such that $\norm{h} = 1$ and $\norm{h^*
(a\lambda)h} < \varepsilon$. If $\lambda$ is an extreme point of
$W(a)$, the same conclusion holds without the assumption that $A$ is
prime. Given any element $a$ in a von Neumann algebra (or in a
general C$^*$algebra) $A$, all normal elements in the weak* closure
(the norm closure, respectively) of the C$^*$convex hull of $a$ are
characterized.
Categories:47A12, 46L05, 46L10 
73. CMB 2000 (vol 43 pp. 157)
A Larger Class of Ornstein Transformations with Mixing Property We prove that Ornstein transformations are almost surely totally
ergodic provided only that the cutting parameter is not bounded.
We thus obtain a larger class of Ornstein transformations with the
mixing property.
Categories:28D05, 47A35 
74. CMB 2000 (vol 43 pp. 21)
The Commutant of an Abstract Backward Shift A bounded linear operator $T$ on a Banach space $X$ is an abstract
backward shift if the nullspace of $T$ is one dimensional, and the
union of the null spaces of $T^k$ for all $k \geq 1$ is dense in
$X$. In this paper it is shown that the commutant of an abstract
backward shift is an integral domain. This result is used to
derive properties of operators in the commutant.
Keywords:backward shift, commutant Category:47A99 
75. CMB 2000 (vol 43 pp. 87)
Lomonosov's Techniques and Burnside's Theorem In this note we give a proof of Lomonosov's extension
of Burnside's theorem to infinite dimensional Banach spaces.
Category:47A15 