76. 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 

77. 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 

78. 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 

79. 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 

80. CMB 2011 (vol 54 pp. 302)
81. 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 

82. 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 

83. 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 

84. CMB 2010 (vol 53 pp. 690)
 Puerta, M. E.; Loaiza, G.

On the Maximal Operator Ideal Associated with a Tensor Norm Defined by Interpolation Spaces
The classical approach to studying operator ideals using tensor
norms mainly focuses on those tensor norms and operator ideals
defined by means of $\ell_p$ spaces. In a previous paper,
an interpolation space, defined via the real method
and using
$\ell_p$ spaces, was used to define a tensor
norm, and the associated minimal operator ideals were characterized.
In this paper, the next natural step is taken, that is, the
corresponding maximal operator
ideals are characterized. As an application, necessary and sufficient
conditions for the coincidence of
the maximal and minimal ideals are given.
Finally, the previous results are used in order to find some new
metric properties of the mentioned tensor norm.
Keywords:maximal operator ideals, ultraproducts of spaces, interpolation spaces Categories:46M05, 46M35, 46A32 

85. CMB 2010 (vol 53 pp. 587)
 Birkenmeier, Gary F.; Park, Jae Keol; Rizvi, S. Tariq

Hulls of Ring Extensions
We investigate the behavior of the quasiBaer and the
right FIextending right ring hulls under various ring extensions
including group ring extensions, full and triangular matrix ring
extensions, and infinite matrix ring extensions. As a consequence,
we show that for semiprime rings $R$ and $S$, if $R$ and $S$ are
Morita equivalent, then so are the quasiBaer right ring hulls
$\widehat{Q}_{\mathfrak{qB}}(R)$ and $\widehat{Q}_{\mathfrak{qB}}(S)$ of
$R$ and $S$, respectively. As an application, we prove that if
unital $C^*$algebras $A$ and $B$ are Morita equivalent as rings,
then the bounded central closure of $A$ and that of $B$ are
strongly Morita equivalent as $C^*$algebras. Our results show
that the quasiBaer property is always preserved by infinite
matrix rings, unlike the Baer property. Moreover, we give an
affirmative answer to an open question of Goel and Jain for the
commutative group ring $A[G]$ of a torsionfree Abelian group $G$
over a commutative semiprime quasicontinuous ring $A$. Examples
that illustrate and delimit the results of this paper are provided.
Keywords:(FI)extending, Morita equivalent, ring of quotients, essential overring, (quasi)Baer ring, ring hull, u.p.monoid, $C^*$algebra Categories:16N60, 16D90, 16S99, 16S50, 46L05 

86. CMB 2010 (vol 53 pp. 550)
87. CMB 2010 (vol 53 pp. 447)
 Choi, Yemon

Injective Convolution Operators on l^{∞}(Γ) are Surjective
Let $\Gamma$ be a discrete group and let $f \in \ell^{1}(\Gamma)$. We observe that if the natural convolution operator $\rho_f: \ell^{\infty}(\Gamma)\to \ell^{\infty}(\Gamma)$ is injective, then $f$ is invertible in $\ell^{1}(\Gamma)$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt by appealing to the direct finiteness of the algebra $\ell^{1}(\Gamma)$. We give simple examples to show that in general one cannot replace $\ell^{\infty}$ with $\ell^{p}$, $1\leq p< \infty$, nor with $L^{\infty}(G)$ for nondiscrete $G$. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on $\Gamma$, and give some partial results.
Categories:43A20, 46L05, 43A22 

88. CMB 2010 (vol 53 pp. 256)
89. CMB 2010 (vol 53 pp. 466)
 Dubarbie, Luis

Separating Maps between Spaces of VectorValued Absolutely Continuous Functions
In this paper we give a description of separating or disjointness preserving linear bijections on spaces of vectorvalued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and biseparating in the finitedimensional case. The infinitedimensional case is also studied.
Keywords:separating maps, disjointness preserving, vectorvalued absolutely continuous functions, automatic continuity Categories:47B38, 46E15, 46E40, 46H40, 47B33 

90. CMB 2009 (vol 53 pp. 37)
 Choi, ManDuen; Latrémolière, Frédéric

$C^*$CrossedProducts by an OrderTwo Automorphism
We describe the representation theory of $C^*$crossedproducts of a unital $C^*$algebra A by the cyclic group of order~2. We prove that there are two main types of irreducible representations for the crossedproduct: those whose restriction to A is irreducible and those who are the sum of two unitarily unequivalent representations of~A. We characterize each class in term of the restriction of the representations to the fixed point $C^*$subalgebra of~A. We apply our results to compute the Ktheory of several crossedproducts of the free group on two generators.
Categories:46L55, 46L80 

91. CMB 2009 (vol 53 pp. 133)
92. CMB 2009 (vol 53 pp. 118)
 Lewis, Paul

The Uncomplemented Spaces $W(X,Y)$ and $K(X,Y)$
Classical results of Kalton and techniques of Feder are used to study the complementation of the space $W(X, Y)$ of weakly compact operators and the space $K(X,Y)$ of compact operators in the space $L(X,Y)$ of all bounded linear maps from X to Y.
Keywords:spaces of operators, complemented subspace, weakly compact operator, basic sequence Categories:46B28, 46B15, 46B20 

93. CMB 2009 (vol 53 pp. 278)
 Galego, Elói M.

CantorBernstein Sextuples for Banach Spaces
Let $X$ and $Y$ be Banach spaces isomorphic
to complemented subspaces of each other with supplements $A$ and
$B$. In 1996, W. T. Gowers solved the SchroederBernstein (or
CantorBernstein) problem for Banach spaces by showing that $X$ is not
necessarily isomorphic to $Y$. In this paper, we obtain a necessary
and sufficient condition on the sextuples $(p, q, r, s, u, v)$ in
$\mathbb N$
with $p+q \geq 1$, $r+s \geq 1$ and $u, v \in \mathbb N^*$, to provide that
$X$ is isomorphic to $Y$, whenever these spaces satisfy the following
decomposition scheme
$$
A^u \sim X^p \oplus Y^q, \quad
B^v \sim X^r \oplus Y^s.
$$
Namely, $\Phi=(pu)(sv)(q+u)(r+v)$ is different from zero and $\Phi$
divides $p+q$ and $r+s$. These sextuples are called CantorBernstein
sextuples for Banach spaces. The simplest case $(1, 0, 0, 1, 1, 1)$
indicates the wellknown PeÅczyÅski's decomposition method in
Banach space. On the other hand, by interchanging some Banach spaces
in the above decomposition scheme, refinements of
the SchroederBernstein problem become evident.
Keywords:Pel czyÅski's decomposition method, SchroederBernstein problem Categories:46B03, 46B20 

94. CMB 2009 (vol 53 pp. 239)
95. CMB 2009 (vol 53 pp. 64)
 Dodos, Pandelis

On Antichains of Spreading Models of Banach Spaces
We show that for every separable Banach space $X$,
either $\mathrm{SP_w}(X)$ (the set of all spreading models
of $X$ generated by weaklynull sequences in $X$, modulo
equivalence) is countable, or $\mathrm{SP_w}(X)$ contains an
antichain of the size of the continuum. This answers
a question of S.~J. Dilworth, E. Odell, and B. Sari.
Categories:46B20, 03E15 

96. CMB 2009 (vol 53 pp. 51)
97. CMB 2009 (vol 52 pp. 598)
 Moreno, M. A.; Nicola, J.; Pardo, E.; Thomas, H.

Numerical Semigroups That Are Not Intersections of $d$Squashed Semigroups
We say that a numerical semigroup is \emph{$d$squashed} if it can
be written in the form
$$ S=\frac 1 N \langle a_1,\dots,a_d \rangle \cap \mathbb{Z}$$
for $N,a_1,\dots,a_d$ positive integers with
$\gcd(a_1,\dots, a_d)=1$.
Rosales and Urbano have shown that a numerical semigroup is
2squashed if and only if it is proportionally modular.
Recent works by Rosales \emph{et al.} give a concrete example of a
numerical semigroup that cannot be written as an intersection of
$2$squashed semigroups. We will show the existence of infinitely
many numerical semigroups that cannot be written as an
intersection of $2$squashed semigroups. We also will prove the
same result for $3$squashed semigroups. We conjecture that there
are numerical semigroups that cannot be written as the
intersection of $d$squashed semigroups for any fixed $d$, and we
prove some partial results towards this conjecture.
Keywords:numerical semigroup, squashed semigroup, proportionally modular semigroup Categories:20M14, 06F05, 46L80 

98. CMB 2009 (vol 52 pp. 424)
 Martini, Horst; Spirova, Margarita

Covering Discs in Minkowski Planes
We investigate the following version of the circle covering
problem in strictly convex (normed or) Minkowski planes: to cover
a circle of largest possible diameter by $k$ unit circles. In
particular, we study the cases $k=3$, $k=4$, and $k=7$. For $k=3$
and $k=4$, the diameters under consideration are described in
terms of sidelengths and circumradii of certain inscribed regular
triangles or quadrangles. This yields also simple explanations of
geometric meanings that the corresponding homothety ratios have.
It turns out that basic notions from Minkowski geometry play an
essential role in our proofs, namely Minkowskian bisectors,
$d$segments, and the monotonicity lemma.
Keywords:affine regular polygon, bisector, circle covering problem, circumradius, $d$segment, Minkowski plane, (strictly convex) normed plane Categories:46B20, 52A21, 52C15 

99. CMB 2009 (vol 52 pp. 213)
 Ghenciu, Ioana; Lewis, Paul

DunfordPettis Properties and Spaces of Operators
J. Elton used an application of Ramsey theory to show that
if $X$ is an infinite dimensional Banach space,
then $c_0$ embeds in $X$, $\ell_1$ embeds in $X$, or there
is a subspace of $X$ that fails to have the DunfordPettis property.
Bessaga and Pelczynski showed that if $c_0$ embeds in $X^*$,
then $\ell_\infty$ embeds in $X^*$. Emmanuele and John showed
that if $c_0$ embeds in $K(X,Y)$, then $K(X,Y)$ is not
complemented in $L(X,Y)$. Classical results from Schauder basis theory
are used in a study of DunfordPettis sets and strong
DunfordPettis sets to extend each of the preceding theorems. The space
$L_{w^*}(X^* , Y)$ of $w^*w$ continuous operators is also studied.
Keywords:DunfordPettis property, DunfordPettis set, basic sequence, complemented spaces of operators Categories:46B20, 46B28 

100. CMB 2009 (vol 52 pp. 28)