76. 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 

77. 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 theory Categories:47B35, 46L80 

78. CMB 2005 (vol 48 pp. 409)
79. CMB 2005 (vol 48 pp. 251)
 Murphy, G. J.

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 

80. 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 nonnormal maps.
Categories:46L15, 47L25 

81. 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 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 $2
Keywords:$C^*$algebras, summing operators, diagonal operators,, RadonNikodym property Categories:46L50, 47D15 

82. CMB 2004 (vol 47 pp. 504)
83. CMB 2004 (vol 47 pp. 456)
84. CMB 2004 (vol 47 pp. 369)
85. 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 

86. 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 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 

87. 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,
rankone, indecomposable nonnegative bands is obtained which
facilitates the understanding of their geometric structure.
Categories:47D03, 47A15 

88. CMB 2004 (vol 47 pp. 49)
 Lindström, Mikael; Makhmutov, Shamil; Taskinen, Jari

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
Keywords:Bloch space, little Bloch space, $\BMOA$, $\VMOA$, $Q_p$ spaces,, composition operator, compact operator, essential norm Categories:47B38, 47B10, 46E40, 46E15 

89. 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 subspaces Categories:47A15, 47B47 

90. CMB 2004 (vol 47 pp. 144)
 Xia, Jingbo

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 

91. 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 space Categories:46H20, 46H25, 46J10, 46J40, 47L25 

92. CMB 2003 (vol 46 pp. 538)
93. CMB 2003 (vol 46 pp. 216)
 Li, ChiKwong; 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 range Categories:47B15, 47B49 

94. CMB 2003 (vol 46 pp. 59)
95. 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 operator Categories:32A70, 47G10 

96. 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 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 

97. 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 

98. CMB 2001 (vol 44 pp. 270)
 Cheung, WaiShun; Li, ChiKwong

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 

99. CMB 2000 (vol 43 pp. 406)
 Borwein, David

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 

100. CMB 2000 (vol 43 pp. 157)