76. CMB 2011 (vol 56 pp. 65)
 Ghenciu, Ioana

The Uncomplemented Subspace $\mathbf K(X,Y) $
A vector measure result is used to study the complementation of the
space $K(X,Y)$ of compact operators in the spaces $W(X,Y)$ of weakly
compact operators, $CC(X,Y)$ of completely continuous operators, and
$U(X,Y)$ of unconditionally converging operators.
Results of Kalton and Emmanuele concerning the complementation of
$K(X,Y)$ in $L(X,Y)$ and in $W(X,Y)$ are generalized. The containment
of $c_0$ and $\ell_\infty$ in spaces of operators is also studied.
Keywords:compact operators, weakly compact operators, uncomplemented subspaces of operators Categories:46B20, 46B28 

77. CMB 2011 (vol 55 pp. 673)
 Aizenbud, Avraham; Gourevitch, Dmitry

Multiplicity Free Jacquet Modules
Let $F$ be a nonArchimedean local field or a finite field.
Let $n$ be a natural number and $k$ be $1$ or $2$.
Consider $G:=\operatorname{GL}_{n+k}(F)$ and let
$M:=\operatorname{GL}_n(F) \times \operatorname{GL}_k(F)\lt G$ be a maximal Levi subgroup.
Let $U\lt G$ be the corresponding unipotent subgroup and let $P=MU$ be the corresponding parabolic subgroup.
Let $J:=J_M^G: \mathcal{M}(G) \to \mathcal{M}(M)$ be the Jacquet functor, i.e., the functor of coinvariants with respect to $U$.
In this paper we prove that $J$ is a multiplicity free functor, i.e.,
$\dim \operatorname{Hom}_M(J(\pi),\rho)\leq 1$,
for any irreducible representations $\pi$ of $G$ and $\rho$ of $M$.
We adapt the classical method of Gelfand and Kazhdan, which proves the ``multiplicity free" property of certain representations to prove the ``multiplicity free" property of certain functors.
At the end we discuss whether other Jacquet functors are multiplicity free.
Keywords:multiplicity one, Gelfand pair, invariant distribution, finite group Categories:20G05, 20C30, 20C33, 46F10, 47A67 

78. CMB 2011 (vol 55 pp. 821)
 PerezGarcia, C.; Schikhof, W. H.

New Examples of NonArchimedean Banach Spaces and Applications
The study carried out in this paper about some new examples of
Banach spaces, consisting of certain valued fields extensions, is
a typical nonarchimedean feature. We determine whether these
extensions are of countable type, have $t$orthogonal bases, or are
reflexive.
As an application we construct, for a class of base fields, a norm
$\\cdot\$ on $c_0$, equivalent to the canonical supremum norm,
without nonzero vectors that are $\\cdot\$orthogonal and such
that there is a multiplication on $c_0$ making $(c_0,\\cdot\)$
into a valued field.
Keywords:nonarchimedean Banach spaces, valued field extensions, spaces of countable type, orthogonal bases Categories:46S10, 12J25 

79. CMB 2011 (vol 56 pp. 136)
 Munteanu, RaduBogdan

On Constructing Ergodic Hyperfinite Equivalence Relations of NonProduct Type
Product type equivalence relations are hyperfinite measured
equivalence relations, which, up to orbit equivalence, are generated
by product type odometer actions. We give a concrete example of a
hyperfinite equivalence relation of nonproduct type, which is the
tail equivalence on a Bratteli diagram.
In order to show that the equivalence relation constructed is not of
product type we will use a criterion called property A. This
property, introduced by Krieger for nonsingular transformations, is
defined directly for hyperfinite equivalence relations in this paper.
Keywords:property A, hyperfinite equivalence relation, nonproduct type Categories:37A20, 37A35, 46L10 

80. CMB 2011 (vol 55 pp. 697)
 Borwein, Jonathan M.; Vanderwerff, Jon

Constructions of Uniformly Convex Functions
We give precise conditions under which the composition
of a norm with a convex function yields a
uniformly convex function on a Banach space.
Various applications are given to functions of power type.
The results are dualized to study uniform smoothness
and several examples are provided.
Keywords:convex function, uniformly convex function, uniformly smooth function, power type, Fenchel conjugate, composition, norm Categories:52A41, 46G05, 46N10, 49J50, 90C25 

81. CMB 2011 (vol 55 pp. 767)
 Martini, Horst; Wu, Senlin

On Zindler Curves in Normed Planes
We extend the notion of Zindler curve from the Euclidean plane to
normed planes. A characterization of Zindler curves for general
normed planes is given, and the relation between Zindler curves and
curves of constant areahalving distances in such planes is
discussed.
Keywords:rc length, areahalving distance, Birkhoff orthogonality, convex curve, halving pair, halving distance, isosceles orthogonality, midpoint curve, Minkowski plane, normed plane, Zindler curve Categories:52A21, 52A10, 46C15 

82. CMB 2011 (vol 54 pp. 654)
 Forrest, Brian E.; Runde, Volker

Norm One Idempotent $cb$Multipliers with Applications to the Fourier Algebra in the $cb$Multiplier Norm
For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely
bounded multipliers of $A(G)$, and let $A_{\mathit{Mcb}}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We
characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm
one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we
describe the closed ideals of $A_{\mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize
those $G$ for which $A_{\mathit{Mcb}}(G)$ is $1$amenable in the sense of B. E. Johnson. (We can even slightly
relax the norm bounds.)
Keywords:amenability, bounded approximate identity, $cb$multiplier norm, Fourier algebra, norm one idempotent Categories:43A22, 20E05, 43A30, 46J10, 46J40, 46L07, 47L25 

83. CMB 2011 (vol 55 pp. 449)
 Bahreini, Manijeh; Bator, Elizabeth; Ghenciu, Ioana

Complemented Subspaces of Linear Bounded Operators
We study the complementation of the space $W(X,Y)$ of weakly compact operators, the space $K(X,Y)$ of compact operators, the space $U(X,Y)$ of unconditionally converging operators, and the space $CC(X,Y)$ of completely continuous operators in the space $L(X,Y)$ of bounded linear operators from $X$ to $Y$.
Feder proved that if $X$ is infinitedimensional and $c_0
\hookrightarrow Y$, then $K(X,Y)$ is uncomplemented in
$L(X,Y)$. Emmanuele and John showed that if $c_0 \hookrightarrow
K(X,Y)$, then $K(X,Y)$ is uncomplemented in $L(X,Y)$.
Bator and Lewis showed that if $X$ is not a Grothendieck space and
$c_0 \hookrightarrow Y$, then $W(X,Y)$ is uncomplemented in
$L(X,Y)$. In this paper, classical results of Kalton and separably
determined operator ideals with property $(*)$ are used to obtain
complementation results that yield these theorems as corollaries.
Keywords:spaces of operators, complemented subspaces, compact operators, weakly compact operators, completely continuous operators Categories:46B20, 46B28 

84. CMB 2011 (vol 55 pp. 548)
 Lewis, Paul; Schulle, Polly

Noncomplemented Spaces of Operators, Vector Measures, and $c_o$
The Banach spaces $L(X, Y)$, $K(X, Y)$, $L_{w^*}(X^*, Y)$, and
$K_{w^*}(X^*, Y)$ are studied to determine when they contain the
classical Banach spaces $c_o$ or $\ell_\infty$. The complementation of
the Banach space $K(X, Y)$ in $L(X, Y)$ is discussed as well as what
impact this complementation has on the embedding of $c_o$ or
$\ell_\infty$ in $K(X, Y)$ or $L(X, Y)$. Results of Kalton, Feder, and
Emmanuele concerning the complementation of $K(X, Y)$ in $L(X, Y)$ are
generalized. Results concerning the complementation of the Banach
space $K_{w^*}(X^*, Y)$ in $L_{w^*}(X^*, Y)$ are also explored as well
as how that complementation affects the embedding of $c_o$ or
$\ell_\infty$ in $K_{w^*}(X^*, Y)$ or $L_{w^*}(X^*, Y)$. The $\ell_p$
spaces for $1 = p < \infty$ are studied to determine when the space of
compact operators from one $\ell_p$ space to another contains
$c_o$. The paper contains a new result which classifies these spaces
of operators. A new result using vector measures is given to
provide more efficient proofs of theorems by Kalton, Feder, Emmanuele,
Emmanuele and John, and Bator and Lewis.
Keywords:spaces of operators, compact operators, complemented subspaces, $w^*w$compact operators Category:46B20 

85. CMB 2011 (vol 54 pp. 385)
 Blackadar, Bruce; Kirchberg, Eberhard

Irreducible Representations of Inner Quasidiagonal $C^*$Algebras
It is shown that a separable $C^*$algebra is inner quasidiagonal if and
only if it has a separating family of quasidiagonal irreducible
representations. As a consequence, a separable $C^*$algebra is a strong
NF algebra if and only if it is nuclear and has a separating family of
quasidiagonal irreducible representations.
We also obtain some permanence properties of the class of inner
quasidiagonal $C^*$algebras.
Category:46L05 

86. CMB 2011 (vol 55 pp. 260)
 Delvaux, L.; Van Daele, A.; Wang, Shuanhong

A Note on the Antipode for Algebraic Quantum Groups
Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a coFrobenius Hopf algebra.
In this note, we show that this formula can be proved for any regular multiplier Hopf
algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a
finitedimensional Hopf algebra, but also that of any
Hopf algebra with integrals (coFrobenius Hopf algebras). Moreover, it turns out that
the proof in this more general situation, in fact, follows in a few lines from wellknown formulas obtained earlier in the
theory of regular multiplier Hopf algebras with integrals.
We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.
Keywords:multiplier Hopf algebras, algebraic quantum groups, the antipode Categories:16W30, 46L65 

87. CMB 2011 (vol 55 pp. 410)
 Service, Robert

A Ramsey Theorem with an Application to Sequences in Banach Spaces
The notion of a maximally conditional sequence is introduced for sequences in a Banach space. It is then proved using
Ramsey theory that every basic sequence in a Banach space has a subsequence which is either an unconditional
basic sequence or a maximally conditional sequence. An apparently novel, purely combinatorial lemma in the spirit of
Galvin's theorem is used in the proof. An alternative proof
of the dichotomy result for sequences in Banach spaces is
also sketched,
using the GalvinPrikry theorem.
Keywords:Banach spaces, Ramsey theory Categories:46B15, 05D10 

88. CMB 2011 (vol 54 pp. 577)
 Aqzzouz, Belmesnaoui

Erratum: The Duality Problem For The Class of AMCompact Operators On Banach Lattices
It is proved that if a positive operator
$S: E \rightarrow F$ is AMcompact whenever its adjoint
$S': F' \rightarrow E'$ is AMcompact, then either the
norm of F is order continuous or $E'$ is discrete.
This note corrects an error in the proof of Theorem 2.3 of
B. Aqzzouz, R. Nouira, and L. Zraoula, The duality problem for
the class of AMcompact operators on Banach lattices. Canad. Math. Bull.
51(2008).
Categories:46A40, 46B40, 46B42 

89. CMB 2011 (vol 55 pp. 339)
 Loring, Terry A.

From Matrix to Operator Inequalities
We generalize LÃ¶wner's method for proving that matrix monotone
functions are operator monotone. The relation $x\leq y$ on bounded
operators is our model for a definition of $C^{*}$relations
being residually finite dimensional.
Our main result is a metatheorem about theorems involving relations
on bounded operators. If we can show there are residually finite dimensional
relations involved and verify a technical condition, then such a
theorem will follow from its restriction to matrices.
Applications are shown regarding norms of exponentials, the norms
of commutators, and "positive" noncommutative $*$polynomials.
Keywords:$C*$algebras, matrices, bounded operators, relations, operator norm, order, commutator, exponential, residually finite dimensional Categories:46L05, 47B99 

90. CMB 2011 (vol 55 pp. 73)
 Dean, Andrew J.

Classification of Inductive Limits of Outer Actions of ${\mathbb R}$ on Approximate Circle Algebras
In this paper we present a classification,
up to equivariant isomorphism, of $C^*$dynamical systems $(A,{\mathbb R},\alpha )$
arising as inductive limits of directed systems
$\{ (A_n,{\mathbb R},\alpha_n),\varphi_{nm}\}$, where each $A_n$
is a finite direct sum of matrix algebras over the continuous
functions on the unit circle, and the $\alpha_n$s are outer actions
generated by rotation of the spectrum.
Keywords:classification, $C^*$dynamical system Categories:46L57, 46L35 

91. CMB 2011 (vol 54 pp. 726)
 Ostrovskii, M. I.

Auerbach Bases and Minimal Volume Sufficient Enlargements
Let $B_Y$ denote the unit ball of a
normed linear space $Y$. A symmetric, bounded, closed, convex set
$A$ in a finite dimensional normed linear space $X$ is called a
sufficient enlargement for $X$ if, for an arbitrary
isometric embedding of $X$ into a Banach space $Y$, there exists a
linear projection $P\colon Y\to X$ such that $P(B_Y)\subset A$. Each
finite dimensional normed space has a minimalvolume sufficient
enlargement that is a parallelepiped; some spaces have ``exotic''
minimalvolume sufficient enlargements. The main result of the
paper is a characterization of spaces having ``exotic''
minimalvolume sufficient enlargements in terms of Auerbach
bases.
Keywords:Banach space, Auerbach basis, sufficient enlargement Categories:46B07, 52A21, 46B15 

92. CMB 2011 (vol 54 pp. 411)
 Davidson, Kenneth R.; Wright, Alex

Operator Algebras with Unique Preduals
We show that every free semigroup algebra has a (strongly) unique
Banach space predual. We also provide a new simpler proof that a
weak$*$ closed unital operator algebra containing a weak$*$
dense subalgebra of compact operators has a unique Banach space
predual.
Keywords:unique predual, free semigroup algebra, CSL algebra Categories:47L50, 46B04, 47L35 

93. CMB 2011 (vol 54 pp. 593)
 Boersema, Jeffrey L.; Ruiz, Efren

Stability of Real $C^*$Algebras
We will give a characterization of stable real $C^*$algebras
analogous to the one given for complex $C^*$algebras by Hjelmborg
and RÃ¸rdam. Using this result, we will prove
that any real $C^*$algebra satisfying the corona factorization
property is stable if and only if its complexification is stable.
Real $C^*$algebras satisfying the corona factorization property
include AFalgebras and purely infinite $C^*$algebras. We will also
provide an example of a simple unstable $C^*$algebra, the
complexification of which is stable.
Keywords:stability, real C*algebras Category:46L05 

94. CMB 2011 (vol 54 pp. 680)
 JiménezVargas, A.; VillegasVallecillos, Moisés

$2$Local Isometries on Spaces of Lipschitz Functions
Let $(X,d)$ be a metric space, and let $\mathop{\textrm{Lip}}(X)$ denote the Banach
space of all scalarvalued bounded Lipschitz functions $f$ on $X$
endowed with one of the natural norms
$
\ f\ =\max \{\ f\ _\infty ,L(f)\}$ or $\f\ =\
f\ _\infty +L(f),
$
where $L(f)$ is the
Lipschitz constant of $f.$ It is said that the isometry
group of $\mathop{\textrm{Lip}}(X)$ is canonical if every
surjective linear isometry of
$\mathop{\textrm{Lip}}(X) $ is induced by a surjective isometry of $X$.
In this paper
we prove that if $X$ is bounded separable and the isometry group of
$\mathop{\textrm{Lip}}(X)$ is canonical, then every $2$local isometry
of $\mathop{\textrm{Lip}}(X)$ is
a surjective linear isometry. Furthermore, we give a complete
description of all $2$local isometries of $\mathop{\textrm{Lip}}(X)$ when $X$ is
bounded.
Keywords:isometry, local isometry, Lipschitz function Categories:46B04, 46J10, 46E15 

95. CMB 2011 (vol 54 pp. 338)
 Nakazi, Takahiko

SzegÃ¶'s Theorem and Uniform Algebras
We study SzegÃ¶'s theorem for a uniform algebra.
In particular, we do it for the disc algebra or the bidisc algebra.
Keywords:SzegÃ¶'s theorem, uniform algebras, disc algebra, weighted Bergman space Categories:32A35, 46J15, 60G25 

96. CMB 2011 (vol 54 pp. 347)
 Potapov, D.; Sukochev, F.

The Haar System in the Preduals of Hyperfinite Factors
We shall present examples of Schauder bases in the preduals to the
hyperfinite factors of types~$\hbox{II}_1$, $\hbox{II}_\infty$,
$\hbox{III}_\lambda$, $0 < \lambda \leq 1$. In the semifinite
(respectively, purely infinite) setting, these systems form Schauder bases
in any associated separable symmetric space of measurable operators
(respectively, in any noncommutative $L^p$space).
Category:46L52 

97. CMB 2011 (vol 54 pp. 302)
98. CMB 2010 (vol 54 pp. 82)
 Emerson, Heath

Lefschetz Numbers for $C^*$Algebras
Using Poincar\'e duality, we obtain a formula of Lefschetz type
that computes the Lefschetz number of an endomorphism of a separable
nuclear $C^*$algebra satisfying Poincar\'e duality and the Kunneth
theorem. (The Lefschetz number of an endomorphism is the graded trace
of the induced map on $\textrm{K}$theory tensored with $\mathbb{C}$, as in the
classical case.) We then examine endomorphisms of CuntzKrieger
algebras $O_A$. An endomorphism has an invariant, which is a
permutation of an infinite set, and the contracting and expanding
behavior of this permutation describes the Lefschetz number of the
endomorphism. Using this description, we derive a closed polynomial
formula for the Lefschetz number depending on the matrix $A$ and the
presentation of the endomorphism.
Categories:19K35, 46L80 

99. CMB 2010 (vol 54 pp. 141)
 Kim, Sang Og; Park, Choonkil

Linear Maps on $C^*$Algebras Preserving the Set of Operators that are Invertible in $\mathcal{A}/\mathcal{I}$
For $C^*$algebras $\mathcal{A}$ of real rank zero, we describe
linear maps $\phi$ on $\mathcal{A}$ that are surjective up to ideals
$\mathcal{I}$, and $\pi(A)$ is invertible in $\mathcal{A}/\mathcal{I}$ if and only if
$\pi(\phi(A))$ is invertible in $\mathcal{A}/\mathcal{I}$, where $A\in\mathcal{A}$ and
$\pi:\mathcal{A}\to\mathcal{A}/\mathcal{I}$ is the quotient map. We also consider similar
linear maps preserving zero products on the Calkin algebra.
Keywords:preservers, Jordan automorphisms, invertible operators, zero products Categories:47B48, 47A10, 46H10 

100. CMB 2010 (vol 54 pp. 68)
 Eilers, Søren; Restorff, Gunnar; Ruiz, Efren

Nonsplitting in Kirchberg's Idealrelated $KK$Theory
A. Bonkat obtained a universal coefficient theorem in the setting of Kirchberg's
idealrelated $KK$theory in the fundamental case of a
$C^*$algebra with one
specified ideal. The universal coefficient sequence was shown to split, unnaturally, under certain
conditions. Employing certain $K$theoretical information derivable
from the given operator algebras using a method introduced here, we shall
demonstrate that Bonkat's UCT does not split in general. Related
methods lead to information on the complexity of the $K$theory which
must be used to
classify $*$isomorphisms for purely infinite $C^*$algebras with
one nontrivial ideal.
Keywords:KKtheory, UCT Category:46L35 
