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51. 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 minimal-volume sufficient enlargement that is a parallelepiped; some spaces have ``exotic'' minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having ``exotic'' minimal-volume sufficient enlargements in terms of Auerbach bases.

Keywords:Banach space, Auerbach basis, sufficient enlargement
Categories:46B07, 52A21, 46B15

52. 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

53. 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 Rørdam. 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 AF-algebras 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

54. CMB 2011 (vol 54 pp. 680)

Jiménez-Vargas, A.; Villegas-Vallecillos, 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 scalar-valued 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

55. CMB 2011 (vol 54 pp. 338)

Nakazi, Takahiko
Szegö's Theorem and Uniform Algebras
We study Szegö's theorem for a uniform algebra. In particular, we do it for the disc algebra or the bidisc algebra.

Keywords:Szegö's theorem, uniform algebras, disc algebra, weighted Bergman space
Categories:32A35, 46J15, 60G25

56. 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 non-commutative $L^p$-space).

Category:46L52

57. CMB 2011 (vol 54 pp. 302)

Kurka, Ondřej
Structure of the Set of Norm-attaining Functionals on Strictly Convex Spaces
Let $X$ be a separable non-reflexive Banach space. We show that there is no Borel class which contains the set of norm-attaining functionals for every strictly convex renorming of $X$.

Keywords:separable non-reflexive space, set of norm-attaining functionals, strictly convex norm, Borel class
Categories:46B20, 54H05, 46B10

58. 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 Cuntz--Krieger 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

59. 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

60. CMB 2010 (vol 54 pp. 68)

Eilers, Søren; Restorff, Gunnar; Ruiz, Efren
Non-splitting in Kirchberg's Ideal-related $KK$-Theory
A. Bonkat obtained a universal coefficient theorem in the setting of Kirchberg's ideal-related $KK$-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 non-trivial ideal.

Keywords:KK-theory, UCT
Category:46L35

61. 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

62. 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 quasi-Baer and the right FI-extending 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 quasi-Baer 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 quasi-Baer 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 torsion-free Abelian group $G$ over a commutative semiprime quasi-continuous 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

63. CMB 2010 (vol 53 pp. 550)

Shalit, Orr Moshe
Representing a Product System Representation as a Contractive Semigroup and Applications to Regular Isometric Dilations
In this paper we propose a new technical tool for analyzing representations of Hilbert $C^*$-product systems. Using this tool, we give a new proof that every doubly commuting representation over $\mathbb{N}^k$ has a regular isometric dilation, and we also prove sufficient conditions for the existence of a regular isometric dilation of representations over more general subsemigroups of $\mathbb R_{+}^k$.

Categories:47A20, 46L08

64. 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

65. CMB 2010 (vol 53 pp. 256)

Fang, Xiaochun; Wang, Lin
Equivalent Definitions of Infinite Positive Elements in Simple C*-algebras
We prove the equivalence of three definitions given by different comparison relations for infiniteness of positive elements in simple $C^*$-algebras.

Keywords:Infinite positive element, Comparison relation
Category:46L99

66. CMB 2010 (vol 53 pp. 466)

Dubarbie, Luis
Separating Maps between Spaces of Vector-Valued Absolutely Continuous Functions
In this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and biseparating in the finite-dimensional case. The infinite-dimensional case is also studied.

Keywords:separating maps, disjointness preserving, vector-valued absolutely continuous functions, automatic continuity
Categories:47B38, 46E15, 46E40, 46H40, 47B33

67. CMB 2009 (vol 53 pp. 37)

Choi, Man-Duen; Latrémolière, Frédéric
$C^*$-Crossed-Products by an Order-Two Automorphism
We describe the representation theory of $C^*$-crossed-products 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 crossed-product: 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 K-theory of several crossed-products of the free group on two generators.

Categories:46L55, 46L80

68. CMB 2009 (vol 53 pp. 133)

Moritoh, Shinya; Tomoeda, Kyoko
A Further Decay Estimate for the Dziubański-Hernández Wavelets
We give a further decay estimate for the Dziubański-Hernández wavelets that are band-limited and have subexponential decay. This is done by constructing an appropriate bell function and using the Paley-Wiener theorem for ultradifferentiable functions.

Keywords:wavelets, ultradifferentiable functions
Categories:42C40, 46E10

69. 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

70. CMB 2009 (vol 53 pp. 278)

Galego, Elói M.
Cantor-Bernstein 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 Schroeder--Bernstein (or Cantor--Bernstein) 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, $\Phi=(p-u)(s-v)-(q+u)(r+v)$ is different from zero and $\Phi$ divides $p+q$ and $r+s$. These sextuples are called Cantor--Bernstein sextuples for Banach spaces. The simplest case $(1, 0, 0, 1, 1, 1)$ indicates the well-known Pełczyński's decomposition method in Banach space. On the other hand, by interchanging some Banach spaces in the above decomposition scheme, refinements of the Schroeder--Bernstein problem become evident.

Keywords:Pel czyński's decomposition method, Schroeder-Bernstein problem
Categories:46B03, 46B20

71. CMB 2009 (vol 53 pp. 239)

Dong, Z.
A Note on the Exactness of Operator Spaces
In this paper, we give two characterizations of the exactness of operator spaces.

Keywords:operator space, exactness
Category:46L07

72. 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 weakly-null 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

73. CMB 2009 (vol 53 pp. 51)

Cobos, Fernando; Fernández-Cabrera, Luz M.
On the Relationship Between Interpolation of Banach Algebras and Interpolation of Bilinear Operators
We show that if the general real method $(\cdot ,\cdot )_\Gamma$ preserves the Banach-algebra structure, then a bilinear interpolation theorem holds for $(\cdot ,\cdot )_\Gamma$.

Keywords:real interpolation, bilinear operators, Banach algebras
Categories:46B70, 46M35, 46H05

74. 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 2-squashed 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

75. 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 side-lengths 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
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