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26. CMB 2007 (vol 50 pp. 268)

Manuilov, V.; Thomsen, K.
 On the Lack of Inverses to $C^*$-Extensions Related to Property T Groups Using ideas of S. Wassermann on non-exact $C^*$-algebras and property T groups, we show that one of his examples of non-invertible $C^*$-extensions is not semi-invertible. To prove this, we show that a certain element vanishes in the asymptotic tensor product. We also show that a modification of the example gives a $C^*$-extension which is not even invertible up to homotopy. Keywords:$C^*$-algebra extension, property T group, asymptotic tensor $C^*$-norm, homotopyCategories:19K33, 46L06, 46L80, 20F99

27. CMB 2006 (vol 49 pp. 185)

 On the Inequality for Volume and Minkowskian Thickness Given a centrally symmetric convex body $B$ in $\E^d,$ we denote by $\M^d(B)$ the Minkowski space ({\em i.e.,} finite dimensional Banach space) with unit ball $B.$ Let $K$ be an arbitrary convex body in $\M^d(B).$ The relationship between volume $V(K)$ and the Minkowskian thickness ($=$ minimal width) $\thns_B(K)$ of $K$ can naturally be given by the sharp geometric inequality $V(K) \ge \alpha(B) \cdot \thns_B(K)^d,$ where $\alpha(B)>0.$ As a simple corollary of the Rogers--Shephard inequality we obtain that $\binom{2d}{d}{}^{-1} \le \alpha(B)/V(B) \le 2^{-d}$ with equality on the left attained if and only if $B$ is the difference body of a simplex and on the right if $B$ is a cross-polytope. The main result of this paper is that for $d=2$ the equality on the right implies that $B$ is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach--Mazur distance to the regular hexagon. Keywords:convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, Banach-Mazur compactum, (modified) Banach-Mazur distance, volume ratioCategories:52A40, 46B20

28. CMB 2005 (vol 48 pp. 195)

Daniel, D.; Nikiel, J.; Treybig, L. B.; Tuncali, H. M.; Tymchatyn, E. D.
 On Suslinian Continua A continuum is said to be Suslinian if it does not contain uncountably many mutually exclusive nondegenerate subcontinua. We prove that Suslinian continua are perfectly normal and rim-metrizable. Locally connected Suslinian continua have weight at most $\omega_1$ and under appropriate set-theoretic conditions are metrizable. Non-separable locally connected Suslinian continua are rim-finite on some open set. Keywords:Suslinian continuum, Souslin line, locally connected, rim-metrizable,, perfectly normal, rim-finiteCategories:54F15, 54D15, 54F50

29. CMB 2005 (vol 48 pp. 121)

Mollin, R. A.
 Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$ We look at the simple continued fraction expansion of $\sqrt{D}$ for any $D=2^hc$ where $c>1$ is odd with a goal of determining necessary and sufficient conditions for the central norm (as determined by the infrastructure of the underlying real quadratic order therein) to be $2^h$. At the end of the paper, we also address the case where $D=c$ is odd and the central norm of $\sqrt{D}$ is equal to $2$. Keywords:quadratic Diophantine equations, simple continued fractions,, norms of ideals, infrastructure of real quadratic fieldsCategories:11A55, 11D09, 11R11

30. CMB 2004 (vol 47 pp. 49)

Lindström, Mikael; Makhmutov, Shamil; Taskinen, Jari
 The Essential Norm of a Bloch-to-$Q_p$ 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$spaces,, composition operator, compact operator, essential normCategories:47B38, 47B10, 46E40, 46E15 31. CMB 2001 (vol 44 pp. 323) Schuman, Bertrand  Une classe d'hamiltoniens polynomiaux isochrones Soit$H_0 = \frac{x^2+y^2}{2}$un hamiltonien isochrone du plan$\Rset^2$. On met en \'evidence une classe d'hamiltoniens isochrones qui sont des perturbations polynomiales de$H_0$. On obtient alors une condition n\'ecessaire d'isochronisme, et un crit\ere de choix pour les hamiltoniens isochrones. On voit ce r\'esultat comme \'etant une g\'en\'eralisation du caract\ere isochrone des perturbations hamiltoniennes homog\`enes consid\'er\'ees dans [L], [P], [S]. Let$H_0 = \frac{x^2+y^2}{2}$be an isochronous Hamiltonian of the plane$\Rset^2$. We obtain a necessary condition for a system to be isochronous. We can think of this result as a generalization of the isochronous behaviour of the homogeneous polynomial perturbation of the Hamiltonian$H_0$considered in [L], [P], [S]. Keywords:Hamiltonian system, normal forms, resonance, linearizationCategories:34C20, 58F05, 58F22, 58F30 32. CMB 2001 (vol 44 pp. 370) Weston, Anthony  On Locating Isometric$\ell_{1}^{(n)}$Motivated by a question of Per Enflo, we develop a hypercube criterion for locating linear isometric copies of$\lone$in an arbitrary real normed space$X$. The said criterion involves finding$2^{n}$points in$X$that satisfy one metric equality. This contrasts nicely to the standard classical criterion wherein one seeks$n$points that satisfy$2^{n-1}$metric equalities. Keywords:normed spaces, hypercubesCategories:46B04, 05C10, 05B99 33. 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 estimatesCategories:47B37, 47A30, 40G05 34. CMB 1999 (vol 42 pp. 118) Rao, T. S. S. R. K.  Points of Weak$^\ast$-Norm Continuity in the Unit Ball of the Space$\WC(K,X)^\ast$For a compact Hausdorff space with a dense set of isolated points, we give a complete description of points of weak$^\ast$-norm continuity in the dual unit ball of the space of Banach space valued functions that are continuous when the range has the weak topology. As an application we give a complete description of points of weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property. Keywords:Points of weak$^\ast$-norm continuity, space of vector valued weakly continuous functions,$M$-idealsCategories:46B20, 46E40 35. CMB 1999 (vol 42 pp. 87) Kittaneh, Fuad  Some norm inequalities for operators Let$A_i$,$B_i$and$X_i(i=1, 2, \dots, n)$be operators on a separable Hilbert space. It is shown that if$f$and$g$are nonnegative continuous functions on$[0,\infty)$which satisfy the relation$f(t)g(t) =t$for all$t$in$[0,\infty)$, then $$\Biglvert \,\Bigl|\sum^n_{i=1} A^*_i X_i B_i \Bigr|^r \,\Bigrvert^2 \leq \Biglvert \Bigl( \sum^n_{i=1} A^*_i f (|X^*_i|)^2 A_i \Bigr)^r \Bigrvert \, \Biglvert \Bigl( \sum^n_{i=1} B^*_i g (|X_i|)^2 B_i \Bigr)^r \Bigrvert$$ for every$r>0\$ and for every unitarily invariant norm. This result improves some known Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained. Keywords:Unitarily invariant norm, positive operator, arithmetic-geometric mean inequality, Cauchy-Schwarz inequality, Heinz inequalityCategories:47A30, 47B10, 47B15, 47B20
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