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101. CMB 2006 (vol 49 pp. 389)

Hiai, Fumio; Petz, Dénes; Ueda, Yoshimichi
 A Free Logarithmic Sobolev Inequality on the Circle Free analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed. Categories:46L54, 60E15, 94A17

102. CMB 2006 (vol 49 pp. 414)

Jiang, Liya; Jia, Houyu; Xu, Han
 Commutators Estimates on Triebel--Lizorkin Spaces In this paper, we consider the behavior of the commutators of convolution operators on the Triebel--Lizorkin spaces $\dot{F}^{s, q} _p$. Keywords:commutators, Triebel--Lizorkin spaces, paraproductCategories:42B, 46F

103. CMB 2006 (vol 49 pp. 371)

Floricel, Remus
 Inner $E_0$-Semigroups on Infinite Factors This paper is concerned with the structure of inner $E_0$-semigroups. We show that any inner $E_0$-semigroup acting on an infinite factor $M$ is completely determined by a continuous tensor product system of Hilbert spaces in $M$ and that the product system associated with an inner $E_0$-semigroup is a complete cocycle conjugacy invariant. Keywords:von Neumann algebras, semigroups of endomorphisms, product systems, cocycle conjugacyCategories:46L40, 46L55

104. CMB 2006 (vol 49 pp. 213)

Dean, Andrew J.
 On Inductive Limit Type Actions of the Euclidean Motion Group on Stable UHF Algebras An invariant is presented which classifies, up to equivariant isomorphism, $C^*$-dynamical systems arising as limits from inductive systems of elementary $C^*$-algebras on which the Euclidean motion group acts by way of unitary representations that decompose into finite direct sums of irreducibles. Keywords:classification, $C^*$-dynamical systemCategories:46L57, 46L35

105. CMB 2006 (vol 49 pp. 313)

Wagner, Roy
 On the Relation Between the Gaussian Orthogonal Ensemble and Reflections, or a Self-Adjoint Version of the Marcus--Pisier Inequality We prove a self-adjoint analogue of the Marcus--Pisier inequality, comparing the expected value of convex functionals on randomreflection matrices and on elements of the Gaussian orthogonal (or unitary) ensemble. Categories:15A52, 46B09, 46L54

106. 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

107. 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

108. CMB 2006 (vol 49 pp. 82)

Gogatishvili, Amiran; Pick, Luboš
 Embeddings and Duality Theorem for Weak Classical Lorentz Spaces We characterize the weight functions $u,v,w$ on $(0,\infty)$ such that $$\left(\int_0^\infty f^{*}(t)^ qw(t)\,dt\right)^{1/q} \leq C \sup_{t\in(0,\infty)}f^{**}_u(t)v(t),$$ where $$f^{**}_u(t):=\left(\int_{0}^{t}u(s)\,ds\right)^{-1} \int_{0}^{t}f^*(s)u(s)\,ds.$$ As an application we present a~new simple characterization of the associate space to the space $\Gamma^ \infty(v)$, determined by the norm $$\|f\|_{\Gamma^ \infty(v)}=\sup_{t\in(0,\infty)}f^{**}(t)v(t),$$ where $$f^{**}(t):=\frac1t\int_{0}^{t}f^*(s)\,ds.$$ Keywords:Discretizing sequence, antidiscretization, classical Lorentz spaces, weak Lorentz spaces, embeddings, duality, Hardy's inequalityCategories:26D10, 46E20

109. 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 theoryCategories:47B35, 46L80

110. CMB 2005 (vol 48 pp. 481)

Azagra, D.; Fabian, M.; Jiménez-Sevilla, M.
 Exact Filling of Figures with the Derivatives of Smooth Mappings Between Banach Spaces We establish sufficient conditions on the shape of a set $A$ included in the space $\mathcal L _s^n(X,Y)$ of the $n$-linear symmetric mappings between Banach spaces $X$ and $Y$, to ensure the existence of a $C^n$\nobreakdash-smooth mapping $f\colon X \rightarrow Y$, with bounded support, and such that $f^{(n)}(X)=A$, provided that $X$ admits a $C^{n}$-smooth bump with bounded $n$-th derivative and $\dens X=\dens \mathcal L ^n(X,Y)$. For instance, when $X$ is infinite-dimensional, every bounded connected and open set $U$ containing the origin is the range of the $n$-th derivative of such a mapping. The same holds true for the closure of $U$, provided that every point in the boundary of $U$ is the end point of a path within $U$. In the finite-dimensional case, more restrictive conditions are required. We also study the Fr\'echet smooth case for mappings from $\mathbb R^n$ to a separable infinite-dimensional Banach space and the G\^ateaux smooth case for mappings defined on a separable infinite-dimensional Banach space and with values in a separable Banach space. Category:46B20

111. CMB 2005 (vol 48 pp. 455)

Rychtář, Jan
 On GÃ¢teaux Differentiability of Convex Functions in WCG Spaces It is shown, using the Borwein--Preiss variational principle that for every continuous convex function $f$ on a weakly compactly generated space $X$, every $x_0\in X$ and every weakly compact convex symmetric set $K$ such that $\cspan K=X$, there is a point of G\^ateaux differentiability of $f$ in $x_0+K$. This extends a Klee's result for separable spaces. Keywords:GÃ¢teaux smoothness, Borwein--Preiss variational principle,, weakly compactly generated spacesCategory:46B20

112. CMB 2005 (vol 48 pp. 340)

Andruchow, Esteban
 Short Geodesics of Unitaries in the $L^2$ Metric Let $\M$ be a type II$_1$ von Neumann algebra, $\tau$ a trace in $\M$, and $\l2$ the GNS Hilbert space of $\tau$. We regard the unitary group $U_\M$ as a subset of $\l2$ and characterize the shortest smooth curves joining two fixed unitaries in the $L^2$ metric. As a consequence of this we obtain that $U_\M$, though a complete (metric) topological group, is not an embedded riemannian submanifold of $\l2$ Keywords:unitary group, short geodesics, infinite dimensional riemannian manifolds.Categories:46L51, 58B10, 58B25

113. CMB 2005 (vol 48 pp. 251)

Murphy, G. J.
 The Index Theory Associated to a Non-Finite 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 non-finite 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

114. CMB 2005 (vol 48 pp. 283)

Thibault, Lionel; Zagrodny, Dariusz
 Enlarged Inclusion of Subdifferentials This paper studies the integration of inclusion of subdifferentials. Under various verifiable conditions, we obtain that if two proper lower semicontinuous functions $f$ and $g$ have the subdifferential of $f$ included in the $\gamma$-enlargement of the subdifferential of $g$, then the difference of those functions is $\gamma$-Lipschitz over their effective domain. Keywords:subdifferential,, directionally regular function,, approximate convex function,, subdifferentially and directionally stable functionCategories:49J52, 46N10, 58C20

115. CMB 2005 (vol 48 pp. 161)

Betancor, Jorge J.
 Hankel Convolution Operators on Spaces of Entire Functions of Finite Order In this paper we study Hankel transforms and Hankel convolution operators on spaces of entire functions of finite order and their duals. Keywords:Hankel transform, convolution, entire functions, finite orderCategory:46F12

116. CMB 2005 (vol 48 pp. 69)

Fabian, M.; Montesinos, V.; Zizler, V.
 Biorthogonal Systems in Weakly LindelÃ¶f Spaces We study countable splitting of Markushevich bases in weakly Lindel\"of Banach spaces in connection with the geometry of these spaces. Keywords:Weak compactness, projectional resolutions,, Markushevich bases, Eberlein compacts, Va\v sÃ¡k spacesCategories:46B03, 46B20., 46B26

117. 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 non-normal maps. Categories:46L15, 47L25

118. CMB 2005 (vol 48 pp. 50)

Elliott, George A.; Gong, Guihua; Li, Liangqing
 Injectivity of the Connecting Maps in AH Inductive Limit Systems Let $A$ be the inductive limit of a system $$A_{1}\xrightarrow{\phi_{1,2}}A_{2} \xrightarrow{\phi_{2,3}} A_{3}\longrightarrow \cd$$ with $A_n = \bigoplus_{i=1}^{t_n} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$, where $~X_{n,i}$ is a finite simplicial complex, and $P_{n,i}$ is a projection in $M_{[n,i]}(C(X_{n,i}))$. In this paper, we will prove that $A$ can be written as another inductive limit $$B_1\xrightarrow{\psi_{1,2}} B_2 \xrightarrow{\psi_{2,3}} B_3\longrightarrow \cd$$ with $B_n = \bigoplus_{i=1}^{s_n} Q_{n,i}M_{\{n,i\}}(C(Y_{n,i}))Q_{n,i}$, where $Y_{n,i}$ is a finite simplicial complex, and $Q_{n,i}$ is a projection in $M_{\{n,i\}}(C(Y_{n,i}))$, with the extra condition that all the maps $\psi_{n,n+1}$ are \emph{injective}. (The result is trivial if one allows the spaces $Y_{n,i}$ to be arbitrary compact metrizable spaces.) This result is important for the classification of simple AH algebras (see \cite{G5,G6,EGL}. The special case that the spaces $X_{n,i}$ are graphs is due to the third named author \cite{Li1}. Categories:46L05, 46L35, 19K14

119. CMB 2004 (vol 47 pp. 540)

Jain, Pankaj; Jain, Pawan K.; Gupta, Babita
 Compactness of Hardy-Type Operators over Star-Shaped Regions in $\mathbb{R}^N$ We study a compactness property of the operators between weighted Lebesgue spaces that average a function over certain domains involving a star-shaped region. The cases covered are (i) when the average is taken over a difference of two dilations of a star-shaped region in $\RR^N$, and (ii) when the average is taken over all dilations of star-shaped regions in $\RR^N$. These cases include, respectively, the average over annuli and the average over balls centered at origin. Keywords:Hardy operator, Hardy-Steklov operator, compactness, boundedness, star-shaped regionsCategories:46E35, 26D10

120. 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 Radon-Nikodym 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^*$-algebras, summing operators, diagonal operators,, Radon-Nikodym propertyCategories:46L50, 47D15 121. CMB 2004 (vol 47 pp. 553) Kerr, David  A Geometric Approach to Voiculescu-Brown Entropy A basic problem in dynamics is to identify systems with positive entropy, i.e., systems which are chaotic.'' While there is a vast collection of results addressing this issue in topological dynamics, the phenomenon of positive entropy remains by and large a mystery within the broader noncommutative domain of$C^*$-algebraic dynamics. To shed some light on the noncommutative situation we propose a geometric perspective inspired by work of Glasner and Weiss on topological entropy. This is a written version of the author's talk at the Winter 2002 Meeting of the Canadian Mathematical Society in Ottawa, Ontario. Categories:46L55, 37B40 122. CMB 2004 (vol 47 pp. 481) Bekjan, Turdebek N.  A New Characterization of Hardy Martingale Cotype Space We give a new characterization of Hardy martingale cotype property of complex quasi-Banach space by using the existence of a kind of plurisubharmonic functions. We also characterize the best constants of Hardy martingale inequalities with values in the complex quasi-Banach space. Keywords:Hardy martingale, Hardy martingale cotype,, plurisubharmonic functionCategories:46B20, 52A07, 60G44 123. CMB 2004 (vol 47 pp. 445) Pirkovskii, A. Yu.  Biprojectivity and Biflatness for Convolution Algebras of Nuclear Operators For a locally compact group$G$, the convolution product on the space$\nN(L^p(G))$of nuclear operators was defined by Neufang \cite{Neuf_PhD}. We study homological properties of the convolution algebra$\nN(L^p(G))$and relate them to some properties of the group$G$, such as compactness, finiteness, discreteness, and amenability. Categories:46M10, 46H25, 43A20, 16E65 124. CMB 2004 (vol 47 pp. 206) Hurri-Syrjänen, Ritva  The PoincarÃ© Inequality and Reverse Doubling Weights We show that Poincar\'e inequalities with reverse doubling weights hold in a large class of irregular domains whenever the weights satisfy certain conditions. Examples of these domains are John domains. Keywords:reverse doubling weights, PoincarÃ© inequality, John domainsCategory:46E35 125. CMB 2004 (vol 47 pp. 108) Śliwa, Wiesław  On Universal Schauder Bases in Non-Archimedean FrÃ©chet Spaces It is known that any non-archimedean Fr\'echet space of countable type is isomorphic to a subspace of$c_0^{\mathbb{N}}$. In this paper we prove that there exists a non-archimedean Fr\'echet space$U$with a basis$(u_n)$such that any basis$(x_n)$in a non-archimedean Fr\'echet space$X$is equivalent to a subbasis$(u_{k_n})$of$(u_n)$. Then any non-archimedean Fr\'echet space with a basis is isomorphic to a complemented subspace of$U$. In contrast to this, we show that a non-archimedean Fr\'echet space$X$with a basis$(x_n)$is isomorphic to a complemented subspace of$c_0^{\mathbb{N}}$if and only if$X$is isomorphic to one of the following spaces:$c_0$,$c_0 \times \mathbb{K}^{\mathbb{N}}$,$\mathbb{K}^{\mathbb{N}}$,$c_0^{\mathbb{N}}$. Finally, we prove that there is no nuclear non-archimedean Fr\'echet space$H$with a basis$(h_n)$such that any basis$(y_n)$in a nuclear non-archimedean Fr\'echet space$Y$is equivalent to a subbasis$(h_{k_n})$of$(h_n)\$. Keywords:universal bases, complemented subspaces with basesCategories:46S10, 46A35
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