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101. 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

102. 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

103. 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

104. 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

105. 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

106. 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

107. 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

108. 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

109. 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

110. 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

111. 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

112. 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

113. 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

114. 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

115. 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

116. 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

117. 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 $s Keywords:multipliers, convolution operators, Lorentz spaces, Lorentz-improving multipliersCategories:43A22, 42A45, 46E30 118. CJM 2001 (vol 53 pp. 631) Walters, Samuel G.  K-Theory of Non-Commutative Spheres Arising from the Fourier Automorphism For a dense$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 119. CJM 2001 (vol 53 pp. 355) Nica, Alexandru; Shlyakhtenko, Dimitri; Speicher, Roland $R$-Diagonal Elements and Freeness With Amalgamation The concept of$R$-diagonal element was introduced in \cite{NS2}, and was subsequently found to have applications to several problems in free probability. In this paper we describe a new approach to$R$-diagonality, which relies on freeness with amalgamation. The class of$R$-diagonal elements is enlarged to contain examples living in non-tracial$*$-probability spaces, such as the generalized circular elements of \cite{Sh1}. Category:46L54 120. 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 121. CJM 2001 (vol 53 pp. 51) Dean, Andrew  A Continuous Field of Projectionless$C^*$-Algebras We use some results about stable relations to show that some of the simple, stable, projectionless crossed products of$O_2$by$\bR$considered by Kishimoto and Kumjian are inductive limits of type I$C^*$-algebras. The type I$C^*$-algebras that arise are pullbacks of finite direct sums of matrix algebras over the continuous functions on the unit interval by finite dimensional$C^*$-algebras. Categories:46L35, 46L57 122. CJM 2001 (vol 53 pp. 161) Lin, Huaxin  Classification of Simple Tracially AF$C^*$-Algebras We prove that pre-classifiable (see 3.1) simple nuclear tracially AF \CA s (TAF) are classified by their$K$-theory. As a consequence all simple, locally AH and TAF \CA s are in fact AH algebras (it is known that there are locally AH algebras that are not AH). We also prove the following Rationalization Theorem. Let$A$and$B$be two unital separable nuclear simple TAF \CA s with unique normalized traces satisfying the Universal Coefficient Theorem. If$A$and$B$have the same (ordered and scaled)$K$-theory and$K_0 (A)_+$is locally finitely generated, then$A \otimes Q \cong B \otimes Q$, where$Q$is the UHF-algebra with the rational$K_0$. Classification results (with restriction on$K_0$-theory) for the above \CA s are also obtained. For example, we show that, if$A$and$B$are unital nuclear separable simple TAF \CA s with the unique normalized trace satisfying the UCT and with$K_1(A) = K_1(B)$, and$A$and$B$have the same rational (scaled ordered)$K_0$, then$A \cong B$. Similar results are also obtained for some cases in which$K_0$is non-divisible such as$K_0(A) = \mathbf{Z} [1/2]$. Categories:46L05, 46L35 123. CJM 2000 (vol 52 pp. 1164) Elliott, George A.; Villadsen, Jesper  Perforated Ordered$\K_0$-Groups A simple$\C^*$-algebra is constructed for which the Murray-von Neumann equivalence classes of projections, with the usual addition---induced by addition of orthogonal projections---form the additive semi-group $$\{0,2,3,\dots\}.$$ (This is a particularly simple instance of the phenomenon of perforation of the ordered$\K_0$-group, which has long been known in the commutative case---for instance, in the case of the four-sphere---and was recently observed by the second author in the case of a simple$\C^*$-algebra.) Categories:46L35, 46L80 124. CJM 2000 (vol 52 pp. 920) Evans, W. D.; Opic, B.  Real Interpolation with Logarithmic Functors and Reiteration We present reiteration theorems'' with limiting values$\theta=0$and$\theta = 1$for a real interpolation method involving broken-logarithmic functors. The resulting spaces lie outside of the original scale of spaces and to describe them new interpolation functors are introduced. For an ordered couple of (quasi-) Banach spaces similar results were presented without proofs by Doktorskii in [D]. Keywords:real interpolation, broken-logarithmic functors, reiteration, weighted inequalitiesCategories:46B70, 26D10, 46E30 125. CJM 2000 (vol 52 pp. 999) Mankiewicz, Piotr  Compact Groups of Operators on Subproportional Quotients of$l^m_1$It is proved that a typical''$n$-dimensional quotient$X_n$of$l^m_1$with$n = m^{\sigma}$,$0 < \sigma < 1$, has the property $$\Average \int_G \|Tx\|_{X_n} \,dh_G(T) \geq \frac{c}{\sqrt{n\log^3 n}} \biggl( n - \int_G |\tr T| \,dh_G (T) \biggr),$$ for every compact group$G$of operators acting on$X_n$, where$d_G(T)$stands for the normalized Haar measure on$G$and the average is taken over all extreme points of the unit ball of$X_n\$. Several consequences of this estimate are presented. Categories:46B20, 46B09
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