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101. CJM 2004 (vol 56 pp. 699)

Gaspari, Thierry
 Bump Functions with HÃ¶lder Derivatives We study the range of the gradients of a $C^{1,\al}$-smooth bump function defined on a Banach space. We find that this set must satisfy two geometrical conditions: It can not be too flat and it satisfies a strong compactness condition with respect to an appropriate distance. These notions are defined precisely below. With these results we illustrate the differences with the case of $C^1$-smooth bump functions. Finally, we give a sufficient condition on a subset of $X^{\ast}$ so that it is the set of the gradients of a $C^{1,1}$-smooth bump function. In particular, if $X$ is an infinite dimensional Banach space with a $C^{1,1}$-smooth bump function, then any convex open bounded subset of $X^{\ast}$ containing $0$ is the set of the gradients of a $C^{1,1}$-smooth bump function. Keywords:Banach space, bump function, range of the derivativeCategories:46T20, 26E15, 26B05

102. CJM 2004 (vol 56 pp. 843)

Ruan, Zhong-Jin
 Type Decomposition and the Rectangular AFD Property for $W^*$-TRO's We study the type decomposition and the rectangular AFD property for $W^*$-TRO's. Like von Neumann algebras, every $W^*$-TRO can be uniquely decomposed into the direct sum of $W^*$-TRO's of type $I$, type $II$, and type $III$. We may further consider $W^*$-TRO's of type $I_{m, n}$ with cardinal numbers $m$ and $n$, and consider $W^*$-TRO's of type $II_{\lambda, \mu}$ with $\lambda, \mu = 1$ or $\infty$. It is shown that every separable stable $W^*$-TRO (which includes type $I_{\infty,\infty}$, type $II_{\infty, \infty}$ and type $III$) is TRO-isomorphic to a von Neumann algebra. We also introduce the rectangular version of the approximately finite dimensional property for $W^*$-TRO's. One of our major results is to show that a separable $W^*$-TRO is injective if and only if it is rectangularly approximately finite dimensional. As a consequence of this result, we show that a dual operator space is injective if and only if its operator predual is a rigid rectangular ${\OL}_{1, 1^+}$ space (equivalently, a rectangular Categories:46L07, 46L08, 46L89

103. CJM 2004 (vol 56 pp. 472)

 Infinite-Dimensional Polyhedrality This paper deals with generalizations of the notion of a polytope to infinite dimensions. The most general definition is the following: a bounded closed convex subset of a Banach space is called a \emph{polytope} if each of its finite-dimensional affine sections is a (standard) polytope. We study the relationships between eight known definitions of infinite-dimensional polyhedrality. We provide a complete isometric classification of them, which gives solutions to several open problems. An almost complete isomorphic classification is given as well (only one implication remains open). Categories:46B20, 46B03, 46B04, 52B99

104. CJM 2004 (vol 56 pp. 225)

Blower, Gordon; Ransford, Thomas
 Complex Uniform Convexity and Riesz Measure The norm on a Banach space gives rise to a subharmonic function on the complex plane for which the distributional Laplacian gives a Riesz measure. This measure is calculated explicitly here for Lebesgue $L^p$ spaces and the von~Neumann-Schatten trace ideals. Banach spaces that are $q$-uniformly $\PL$-convex in the sense of Davis, Garling and Tomczak-Jaegermann are characterized in terms of the mass distribution of this measure. This gives a new proof that the trace ideals $c^p$ are $2$-uniformly $\PL$-convex for $1\leq p\leq 2$. Keywords:subharmonic functions, Banach spaces, Schatten trace idealsCategories:46B20, 46L52

105. CJM 2004 (vol 56 pp. 3)

Amini, Massoud
 Locally Compact Pro-$C^*$-Algebras Let $X$ be a locally compact non-compact Hausdorff topological space. Consider the algebras $C(X)$, $C_b(X)$, $C_0(X)$, and $C_{00}(X)$ of respectively arbitrary, bounded, vanishing at infinity, and compactly supported continuous functions on $X$. Of these, the second and third are $C^*$-algebras, the fourth is a normed algebra, whereas the first is only a topological algebra (it is indeed a pro-$C^\ast$-algebra). The interesting fact about these algebras is that if one of them is given, the others can be obtained using functional analysis tools. For instance, given the $C^\ast$-algebra $C_0(X)$, one can get the other three algebras by $C_{00}(X)=K\bigl(C_0(X)\bigr)$, $C_b(X)=M\bigl(C_0(X)\bigr)$, $C(X)=\Gamma\bigl( K(C_0(X))\bigr)$, where the right hand sides are the Pedersen ideal, the multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of $C_0(X)$, respectively. In this article we consider the possibility of these transitions for general $C^\ast$-algebras. The difficult part is to start with a pro-$C^\ast$-algebra $A$ and to construct a $C^\ast$-algebra $A_0$ such that $A=\Gamma\bigl(K(A_0)\bigr)$. The pro-$C^\ast$-algebras for which this is possible are called {\it locally compact\/} and we have characterized them using a concept similar to that of an approximate identity. Keywords:pro-$C^\ast$-algebras, projective limit, multipliers of Pedersen's idealCategories:46L05, 46M40

106. CJM 2003 (vol 55 pp. 1302)

Katsura, Takeshi
 The Ideal Structures of Crossed Products of Cuntz Algebras by Quasi-Free Actions of Abelian Groups We completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A.~Kishimoto's result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra and $K$-groups of our algebras. Categories:46L05, 46L55, 46L45

107. CJM 2003 (vol 55 pp. 969)

Glöckner, Helge
 Lie Groups of Measurable Mappings We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space $(X,\Sigma,\mu)$ and (possibly infinite-dimensional) Lie group $G$, we construct a Lie group $L^\infty (X,G)$, which is a Fr\'echet-Lie group if $G$ is so. We also show that the weak direct product $\prod^*_{i\in I} G_i$ of an arbitrary family $(G_i)_{i\in I}$ of Lie groups can be made a Lie group, modelled on the locally convex direct sum $\bigoplus_{i\in I} L(G_i)$. Categories:22E65, 46E40, 46E30, 22E67, 46T20, 46T25

108. CJM 2003 (vol 55 pp. 204)

Yan, Yaqiang
 On the Nonsquare Constants of Orlicz Spaces with Orlicz Norm Let $l^{\Phi}$ and $L^\Phi (\Omega)$ be the Orlicz sequence space and function space generated by $N$-function $\Phi(u)$ with Orlicz norm. We give equivalent expressions for the nonsquare constants $C_J (l^\Phi)$, $C_J \bigl( L^\Phi (\Omega) \bigr)$ in sense of James and $C_S (l^\Phi)$, $C_S \bigl( L^\Phi(\Omega) \bigr)$ in sense of Sch\"affer. We are devoted to get practical computational formulas giving estimates of these constants and to obtain their exact value in a class of spaces $l^{\Phi}$ and $L^\Phi (\Omega)$. Keywords:James nonsquare constant, SchÃ¤ffer nonsquare constant, Orlicz sequence space, Orlicz function spaceCategory:46E30

109. CJM 2002 (vol 54 pp. 1280)

Skrzypczak, Leszek
 Besov Spaces and Hausdorff Dimension For Some Carnot-CarathÃ©odory Metric Spaces We regard a system of left invariant vector fields $\mathcal{X}=\{X_1,\dots,X_k\}$ satisfying the H\"ormander condition and the related Carnot-Carath\'eodory metric on a unimodular Lie group $G$. We define Besov spaces corresponding to the sub-Laplacian $\Delta=\sum X_i^2$ both with positive and negative smoothness. The atomic decomposition of the spaces is given. In consequence we get the distributional characterization of the Hausdorff dimension of Borel subsets with the Haar measure zero. Keywords:Besov spaces, sub-elliptic operators, Carnot-CarathÃ©odory metric, Hausdorff dimensionCategories:46E35, 43A15, 28A78

110. CJM 2002 (vol 54 pp. 1165)

Blasco, Oscar; Arregui, José Luis
 Multipliers on Vector Valued Bergman Spaces Let $X$ be a complex Banach space and let $B_p(X)$ denote the vector-valued Bergman space on the unit disc for $1\le p<\infty$. A sequence $(T_n)_n$ of bounded operators between two Banach spaces $X$ and $Y$ defines a multiplier between $B_p(X)$ and $B_q(Y)$ (resp.\ $B_p(X)$ and $\ell_q(Y)$) if for any function $f(z) = \sum_{n=0}^\infty x_n z^n$ in $B_p(X)$ we have that $g(z) = \sum_{n=0}^\infty T_n (x_n) z^n$ belongs to $B_q(Y)$ (resp.\ $\bigl( T_n (x_n) \bigr)_n \in \ell_q(Y)$). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces $X$ and $Y$. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in $B_p(X)$ are introduced. Categories:42A45, 46E40

111. CJM 2002 (vol 54 pp. 1100)

Wood, Peter J.
 The Operator Biprojectivity of the Fourier Algebra In this paper, we investigate projectivity in the category of operator spaces. In particular, we show that the Fourier algebra of a locally compact group $G$ is operator biprojective if and only if $G$ is discrete. Keywords:locally compact group, Fourier algebra, operator space, projectiveCategories:13D03, 18G25, 43A95, 46L07, 22D99

112. CJM 2002 (vol 54 pp. 694)

Gabriel, Michael J.
 Cuntz Algebra States Defined by Implementers of Endomorphisms of the $\CAR$ Algebra We investigate representations of the Cuntz algebra $\mathcal{O}_2$ on antisymmetric Fock space $F_a (\mathcal{K}_1)$ defined by isometric implementers of certain quasi-free endomorphisms of the CAR algebra in pure quasi-free states $\varphi_{P_1}$. We pay corresponding to these representations and the Fock special attention to the vector states on $\mathcal{O}_2$ vacuum, for which we obtain explicit formulae. Restricting these states to the gauge-invariant subalgebra $\mathcal{F}_2$, we find that for natural choices of implementers, they are again pure quasi-free and are, in fact, essentially the states $\varphi_{P_1}$. We proceed to consider the case for an arbitrary pair of implementers, and deduce that these Cuntz algebra representations are irreducible, as are their restrictions to $\mathcal{F}_2$. The endomorphisms of $B \bigl( F_a (\mathcal{K}_1) \bigr)$ associated with these representations of $\mathcal{O}_2$ are also considered. Categories:46L05, 46L30

113. CJM 2002 (vol 54 pp. 634)

Weber, Eric
 Frames and Single Wavelets for Unitary Groups We consider a unitary representation of a discrete countable abelian group on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We extend Robertson's theorem to apply to frames generated by the action of the group. Within this setup we use Stone's theorem and the theory of projection valued measures to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multi\-resolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on $L^2({\mathbb R})$. Keywords:wavelet, multiresolution analysis, unitary group representation, frameCategories:42C40, 43A25, 42C15, 46N99

114. CJM 2002 (vol 54 pp. 225)

Arslan, Bora; Goncharov, Alexander P.; Kocatepe, Mefharet
 Spaces of Whitney Functions on Cantor-Type Sets We introduce the concept of logarithmic dimension of a compact set. In terms of this magnitude, the extension property and the diametral dimension of spaces $\calE(K)$ can be described for Cantor-type compact sets. Categories:46E10, 31A15, 46A04

115. CJM 2002 (vol 54 pp. 303)

Ghahramani, Fereidoun; Grabiner, Sandy
 Convergence Factors and Compactness in Weighted Convolution Algebras We study convergence in weighted convolution algebras $L^1(\omega)$ on $R^+$, with the weights chosen such that the corresponding weighted space $M(\omega)$ of measures is also a Banach algebra and is the dual space of a natural related space of continuous functions. We determine convergence factor $\eta$ for which weak$^\ast$-convergence of $\{\lambda_n\}$ to $\lambda$ in $M(\omega)$ implies norm convergence of $\lambda_n \ast f$ to $\lambda \ast f$ in $L^1 (\omega\eta)$. We find necessary and sufficent conditions which depend on $\omega$ and $f$ and also find necessary and sufficent conditions for $\eta$ to be a convergence factor for all $L^1(\omega)$ and all $f$ in $L^1(\omega)$. We also give some applications to the structure of weighted convolution algebras. As a preliminary result we observe that $\eta$ is a convergence factor for $\omega$ and $f$ if and only if convolution by $f$ is a compact operator from $M(\omega)$ (or $L^1(\omega)$) to $L^1 (\omega\eta)$. Categories:43A10, 43A15, 46J45, 46J99

116. CJM 2002 (vol 54 pp. 138)

Razak, Shaloub
 On the Classification of Simple Stably Projectionless $\C^*$-Algebras It is shown that simple stably projectionless $\C^S*$-algebras which are inductive limits of certain specified building blocks with trivial $\K$-theory are classified by their cone of positive traces with distinguished subset. This is the first example of an isomorphism theorem verifying the conjecture of Elliott for a subclass of the stably projectionless algebras. Categories:46L35, 46L05

117. CJM 2001 (vol 53 pp. 1223)

Mygind, Jesper
 Classification of Certain Simple $C^*$-Algebras with Torsion in $K_1$ We show that the Elliott invariant is a classifying invariant for the class of $C^*$-algebras that are simple unital infinite dimensional inductive limits of finite direct sums of building blocks of the form $$\{f \in C(\T) \otimes M_n : f(x_i) \in M_{d_i}, i = 1,2,\dots,N\},$$ where $x_1,x_2,\dots,x_N \in \T$, $d_1,d_2,\dots,d_N$ are integers dividing $n$, and $M_{d_i}$ is embedded unitally into $M_n$. Furthermore we prove existence and uniqueness theorems for $*$-homomorphisms between such algebras and we identify the range of the invariant. Categories:46L80, 19K14, 46L05

118. CJM 2001 (vol 53 pp. 979)

Nagisa, Masaru; Osaka, Hiroyuki; Phillips, N. Christopher
 Ranks of Algebras of Continuous $C^*$-Algebra Valued Functions We prove a number of results about the stable and particularly the real ranks of tensor products of \ca s under the assumption that one of the factors is commutative. In particular, we prove the following: {\raggedright \begin{enumerate}[(5)] \item[(1)] If $X$ is any locally compact $\sm$-compact Hausdorff space and $A$ is any \ca, then\break $\RR \bigl( C_0 (X) \otimes A \bigr) \leq \dim (X) + \RR(A)$. \item[(2)] If $X$ is any locally compact Hausdorff space and $A$ is any \pisca, then $\RR \bigl( C_0 (X) \otimes A \bigr) \leq 1$. \item[(3)] $\RR \bigl( C ([0,1]) \otimes A \bigr) \geq 1$ for any nonzero \ca\ $A$, and $\sr \bigl( C ([0,1]^2) \otimes A \bigr) \geq 2$ for any unital \ca\ $A$. \item[(4)] If $A$ is a unital \ca\ such that $\RR(A) = 0$, $\sr (A) = 1$, and $K_1 (A) = 0$, then\break $\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$. \item[(5)] There is a simple separable unital nuclear \ca\ $A$ such that $\RR(A) = 1$ and\break $\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$. \end{enumerate}} Categories:46L05, 46L52, 46L80, 19A13, 19B10

119. CJM 2001 (vol 53 pp. 1031)

Sampson, G.; Szeptycki, P.
 The Complete $(L^p,L^p)$ Mapping Properties of Some Oscillatory Integrals in Several Dimensions We prove that the operators $\int_{\mathbb{R}_+^2} e^{ix^a \cdot y^b} \varphi (x,y) f(y)\, dy$ map $L^p(\mathbb{R}^2)$ into itself for $p \in J =\bigl[\frac{a_l+b_l}{a_l+(\frac{b_l r}{2})},\frac{a_l+b_l} {a_l(1-\frac{r}{2})}\bigr]$ if $a_l,b_l\ge 1$ and $\varphi(x,y)=|x-y|^{-r}$, $0\le r <2$, the result is sharp. Generalizations to dimensions $d>2$ are indicated. Categories:42B20, 46B70, 47G10

120. CJM 2001 (vol 53 pp. 809)

Robertson, Guyan; Steger, Tim
 Asymptotic $K$-Theory for Groups Acting on $\tA_2$ Buildings Let $\Gamma$ be a torsion free lattice in $G=\PGL(3, \mathbb{F})$ where $\mathbb{F}$ is a nonarchimedean local field. Then $\Gamma$ acts freely on the affine Bruhat-Tits building $\mathcal{B}$ of $G$ and there is an induced action on the boundary $\Omega$ of $\mathcal{B}$. The crossed product $C^*$-algebra $\mathcal{A}(\Gamma)=C(\Omega) \rtimes \Gamma$ depends only on $\Gamma$ and is classified by its $K$-theory. This article shows how to compute the $K$-theory of $\mathcal{A}(\Gamma)$ and of the larger class of rank two Cuntz-Krieger algebras. Keywords:$K$-theory, $C^*$-algebra, affine buildingCategories:46L80, 51E24

121. CJM 2001 (vol 53 pp. 565)

Hare, Kathryn E.; Sato, Enji
 Spaces of Lorentz Multipliers We study when the spaces of Lorentz multipliers from $L^{p,t} \rightarrow L^{p,s}$ are distinct. Our main interest is the case when $s Keywords:multipliers, convolution operators, Lorentz spaces, Lorentz-improving multipliersCategories:43A22, 42A45, 46E30 122. CJM 2001 (vol 53 pp. 592) Perera, Francesc  Ideal Structure of Multiplier Algebras of Simple$C^*$-algebras With Real Rank Zero We give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of$\sigma$-unital simple$C^\ast$-algebras$A$with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of the multiplier algebra$\mul$, is therefore analyzed. In important cases it is shown that, if$A$has finite scale then the quotient of$\mul$modulo any closed ideal$I$that properly contains$A$has stable rank one. The intricacy of the ideal structure of$\mul$is reflected in the fact that$\mul$can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion. Keywords:$C^\ast$-algebra, multiplier algebra, real rank zero, stable rank, refinement monoidCategories:46L05, 46L80, 06F05 123. CJM 2001 (vol 53 pp. 631) Walters, Samuel G.  K-Theory of Non-Commutative Spheres Arising from the Fourier Automorphism For a dense$G_\delta$set of real parameters$\theta$in$[0,1]$(containing the rationals) it is shown that the group$K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4)$is isomorphic to$\mathbb{Z}^9$, where$A_\theta$is the rotation C*-algebra generated by unitaries$U$,$V$satisfying$VU = e^{2\pi i\theta} UV$and$\sigma$is the Fourier automorphism of$A_\theta$defined by$\sigma(U) = V$,$\sigma(V) = U^{-1}$. More precisely, an explicit basis for$K_0$consisting of nine canonical modules is given. (A slight generalization of this result is also obtained for certain separable continuous fields of unital C*-algebras over$[0,1]$.) The Connes Chern character$\ch \colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)^*$is shown to be injective for a dense$G_\delta$set of parameters$\theta$. The main computational tool in this paper is a group homomorphism$\vtr \colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to \mathbb{R}^8 \times \mathbb{Z}$obtained from the Connes Chern character by restricting the functionals in its codomain to a certain nine-dimensional subspace of$H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)$. The range of$\vtr$is fully determined for each$\theta$. (We conjecture that this subspace is all of$H^{\ev}$.) Keywords:C*-algebras, K-theory, automorphisms, rotation algebras, unbounded traces, Chern charactersCategories:46L80, 46L40, 19K14 124. CJM 2001 (vol 53 pp. 546) Erlijman, Juliana  Multi-Sided Braid Type Subfactors We generalise the two-sided construction of examples of pairs of subfactors of the hyperfinite II$_1$factor$R$in [E1]---which arise by considering unitary braid representations with certain properties---to multi-sided pairs. We show that the index for the multi-sided pair can be expressed as a power of that for the two-sided pair. This construction can be applied to the natural examples---where the braid representations are obtained in connection with the representation theory of Lie algebras of types$A$,$B$,$C$,$D$. We also compute the (first) relative commutants. Category:46L37 125. CJM 2001 (vol 53 pp. 325) Matui, Hiroki  Ext and OrderExt Classes of Certain Automorphisms of$C^*$-Algebras Arising from Cantor Minimal Systems Giordano, Putnam and Skau showed that the transformation group$C^*$-algebra arising from a Cantor minimal system is an$AT$-algebra, and classified it by its$K$-theory. For approximately inner automorphisms that preserve$C(X)\$, we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto's result. Categories:46L40, 46L80, 54H20
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