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126. 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

127. 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

128. 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

129. 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 $s Keywords:multipliers, convolution operators, Lorentz spaces, Lorentz-improving multipliersCategories:43A22, 42A45, 46E30 130. CJM 2001 (vol 53 pp. 631) Walters, Samuel G.  K-Theory of Non-Commutative Spheres Arising from the Fourier Automorphism For a dense$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}$.) Keywords:C*-algebras, K-theory, automorphisms, rotation algebras, unbounded traces, Chern charactersCategories:46L80, 46L40, 19K14 131. CJM 2001 (vol 53 pp. 592) Perera, Francesc  Ideal Structure of Multiplier Algebras of Simple$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$-algebra, multiplier algebra, real rank zero, stable rank, refinement monoidCategories:46L05, 46L80, 06F05 132. CJM 2001 (vol 53 pp. 546) Erlijman, Juliana  Multi-Sided Braid Type Subfactors We generalise the two-sided construction of examples of pairs of subfactors of the hyperfinite II$_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 133. 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 134. 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 135. 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 136. 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 137. 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 138. 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 139. 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 140. CJM 2000 (vol 52 pp. 789) Kamińska, Anna; Mastyło, Mieczysław  The Dunford-Pettis Property for Symmetric Spaces A complete description of symmetric spaces on a separable measure space with the Dunford-Pettis property is given. It is shown that$\ell^1$,$c_0$and$\ell^{\infty}$are the only symmetric sequence spaces with the Dunford-Pettis property, and that in the class of symmetric spaces on$(0, \alpha)$,$0 < \alpha \leq \infty$, the only spaces with the Dunford-Pettis property are$L^1$,$L^{\infty}$,$L^1 \cap L^{\infty}$,$L^1 + L^{\infty}$,$(L^{\infty})^\circ$and$(L^1 + L^{\infty})^\circ$, where$X^\circ$denotes the norm closure of$L^1 \cap L^{\infty}$in$X$. It is also proved that all Banach dual spaces of$L^1 \cap L^{\infty}$and$L^1 + L^{\infty}$have the Dunford-Pettis property. New examples of Banach spaces showing that the Dunford-Pettis property is not a three-space property are also presented. As applications we obtain that the spaces$(L^1 + L^{\infty})^\circ$and$(L^{\infty})^\circ$have a unique symmetric structure, and we get a characterization of the Dunford-Pettis property of some K\"othe-Bochner spaces. Categories:46E30, 46B42 141. CJM 2000 (vol 52 pp. 849) Sukochev, F. A.  Operator Estimates for Fredholm Modules We study estimates of the type $$\Vert \phi(D) - \phi(D_0) \Vert_{\emt} \leq C \cdot \Vert D - D_0 \Vert^{\alpha}, \quad \alpha = \frac12, 1$$ where$\phi(t) = t(1 + t^2)^{-1/2}$,$D_0 = D_0^*$is an unbounded linear operator affiliated with a semifinite von Neumann algebra$\calM$,$D - D_0$is a bounded self-adjoint linear operator from$\calM$and$(1 + D_0^2)^{-1/2} \in \emt$, where$\emt$is a symmetric operator space associated with$\calM$. In particular, we prove that$\phi(D) - \phi(D_0)$belongs to the non-commutative$L_p$-space for some$p \in (1,\infty)$, provided$(1 + D_0^2)^{-1/2}$belongs to the non-commutative weak$L_r$-space for some$r \in [1,p)$. In the case$\calM = \calB (\calH)$and$1 \leq p \leq 2$, we show that this result continues to hold under the weaker assumption$(1 + D_0^2)^{-1/2} \in \calC_p$. This may be regarded as an odd counterpart of A.~Connes' result for the case of even Fredholm modules. Categories:46L50, 46E30, 46L87, 47A55, 58B15 142. CJM 2000 (vol 52 pp. 633) Walters, Samuel G.  Chern Characters of Fourier Modules Let$A_\theta$denote the rotation algebra---the universal$C^\ast$-algebra generated by unitaries$U,V$satisfying$VU=e^{2\pi i\theta}UV$, where$\theta$is a fixed real number. Let$\sigma$denote the Fourier automorphism of$A_\theta$defined by$U\mapsto V$,$V\mapsto U^{-1}$, and let$B_\theta = A_\theta \rtimes_\sigma \mathbb{Z}/4\mathbb{Z}$denote the associated$C^\ast$-crossed product. It is shown that there is a canonical inclusion$\mathbb{Z}^9 \hookrightarrow K_0(B_\theta)$for each$\theta$given by nine canonical modules. The unbounded trace functionals of$B_\theta$(yielding the Chern characters here) are calculated to obtain the cyclic cohomology group of order zero$\HC^0(B_\theta)$when$\theta$is irrational. The Chern characters of the nine modules---and more importantly, the Fourier module---are computed and shown to involve techniques from the theory of Jacobi's theta functions. Also derived are explicit equations connecting unbounded traces across strong Morita equivalence, which turn out to be non-commutative extensions of certain theta function equations. These results provide the basis for showing that for a dense$G_\delta$set of values of$\theta$one has$K_0(B_\theta)\cong\mathbb{Z}^9$and is generated by the nine classes constructed here. Keywords:$C^\ast$-algebras, unbounded traces, Chern characters, irrational rotation algebras,$K$-groupsCategories:46L80, 46L40 143. CJM 1999 (vol 51 pp. 850) Muhly, Paul S.; Solel, Baruch  Tensor Algebras, Induced Representations, and the Wold Decomposition Our objective in this sequel to \cite{MSp96a} is to develop extensions, to representations of tensor algebras over$C^{*}$-correspondences, of two fundamental facts about isometries on Hilbert space: The Wold decomposition theorem and Beurling's theorem, and to apply these to the analysis of the invariant subspace structure of certain subalgebras of Cuntz-Krieger algebras. Keywords:tensor algebras, correspondence, induced representation, Wold decomposition, Beurling's theoremCategories:46L05, 46L40, 46L89, 47D15, 47D25, 46M10, 46M99, 47A20, 47A45, 47B35 144. CJM 1999 (vol 51 pp. 745) Echterhoff, Siegfried; Quigg, John  Induced Coactions of Discrete Groups on$C^*$-Algebras Using the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a$C^*$-coaction$\delta\colon D\to D\otimes C^*(G/N)$of a quotient group$G/N$of a discrete group$G$to a$C^*$-coaction$\Ind\delta\colon\Ind D\to \Ind D\otimes C^*(G)$of$G$. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products$\Ind D\times_{\Ind\delta}G$and$D\times_{\delta}G/N$are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions. Category:46L55 145. CJM 1999 (vol 51 pp. 566) Ferenczi, V.  Quotient Hereditarily Indecomposable Banach Spaces A Banach space$X$is said to be {\it quotient hereditarily indecomposable\/} if no infinite dimensional quotient of a subspace of$X$is decomposable. We provide an example of a quotient hereditarily indecomposable space, namely the space$X_{\GM}$constructed by W.~T.~Gowers and B.~Maurey in \cite{GM}. Then we provide an example of a reflexive hereditarily indecomposable space$\hat{X}$whose dual is not hereditarily indecomposable; so$\hat{X}$is not quotient hereditarily indecomposable. We also show that every operator on$\hat{X}^*$is a strictly singular perturbation of an homothetic map. Categories:46B20, 47B99 146. CJM 1999 (vol 51 pp. 309) Leung, Denny H.; Tang, Wee-Kee  Symmetric sequence subspaces of$C(\alpha)$, II If$\alpha$is an ordinal, then the space of all ordinals less than or equal to$\alpha$is a compact Hausdorff space when endowed with the order topology. Let$C(\alpha)$be the space of all continuous real-valued functions defined on the ordinal interval$[0, \alpha]$. We characterize the symmetric sequence spaces which embed into$C(\alpha)$for some countable ordinal$\alpha$. A hierarchy$(E_\alpha)$of symmetric sequence spaces is constructed so that, for each countable ordinal$\alpha$,$E_\alpha$embeds into$C(\omega^{\omega^\alpha})$, but does not embed into$C(\omega^{\omega^\beta})$for any$\beta < \alpha$. Categories:03E13, 03E15, 46B03, 46B45, 46E15, 54G12 147. CJM 1999 (vol 51 pp. 147) Suárez, Daniel  Homeomorphic Analytic Maps into the Maximal Ideal Space of$H^\infty$Let$m$be a point of the maximal ideal space of$\papa$with nontrivial Gleason part$P(m)$. If$L_m \colon \disc \rr P(m)$is the Hoffman map, we show that$\papa \circ L_m$is a closed subalgebra of$\papa$. We characterize the points$m$for which$L_m$is a homeomorphism in terms of interpolating sequences, and we show that in this case$\papa \circ L_m$coincides with$\papa$. Also, if$I_m$is the ideal of functions in$\papa$that identically vanish on$P(m)$, we estimate the distance of any$f\in \papa$to$I_m$. Categories:30H05, 46J20 148. CJM 1999 (vol 51 pp. 26) Fabian, Marián; Mordukhovich, Boris S.  Separable Reduction and Supporting Properties of FrÃ©chet-Like Normals in Banach Spaces We develop a method of separable reduction for Fr\'{e}chet-like normals and$\epsilon$-normals to arbitrary sets in general Banach spaces. This method allows us to reduce certain problems involving such normals in nonseparable spaces to the separable case. It is particularly helpful in Asplund spaces where every separable subspace admits a Fr\'{e}chet smooth renorm. As an applicaton of the separable reduction method in Asplund spaces, we provide a new direct proof of a nonconvex extension of the celebrated Bishop-Phelps density theorem. Moreover, in this way we establish new characterizations of Asplund spaces in terms of$\epsilon$-normals. Keywords:nonsmooth analysis, Banach spaces, separable reduction, FrÃ©chet-like normals and subdifferentials, supporting properties, Asplund spacesCategories:49J52, 58C20, 46B20 149. CJM 1998 (vol 50 pp. 1138) Chalov, P. A.; Terzioğlu, T.; Zahariuta, V. P.  Compound invariants and mixed$F$-,$\DF$-power spaces The problems on isomorphic classification and quasiequivalence of bases are studied for the class of mixed$F$-,$\DF$-power series spaces, {\it i.e.} the spaces of the following kind $$G(\la,a)=\lim_{p \to \infty} \proj \biggl(\lim_{q \to \infty}\ind \Bigl(\ell_1\bigl(a_i (p,q)\bigr)\Bigr)\biggr),$$ where$a_i (p,q)=\exp\bigl((p-\la_i q)a_i\bigr)$,$p,q \in \N$, and$\la =( \la_i)_{i \in \N}$,$a=(a_i)_{i \in \N}$are some sequences of positive numbers. These spaces, up to isomorphisms, are basis subspaces of tensor products of power series spaces of$F$- and$\DF$-types, respectively. The$m$-rectangle characteristic$\mu_m^{\lambda,a}(\delta,\varepsilon; \tau,t)$,$m \in \N$of the space$G(\la,a)$is defined as the number of members of the sequence$(\la_i, a_i)_{i \in \N}$which are contained in the union of$m$rectangles$P_k = (\delta_k, \varepsilon_k] \times (\tau_k, t_k]$,$k = 1,2, \ldots, m$. It is shown that each$m$-rectangle characteristic is an invariant on the considered class under some proper definition of an equivalency relation. The main tool are new compound invariants, which combine some version of the classical approximative dimensions (Kolmogorov, Pe{\l}czynski) with appropriate geometrical and interpolational operations under neighborhoods of the origin (taken from a given basis). Categories:46A04, 46A45, 46M05 150. CJM 1998 (vol 50 pp. 1236) Kalton, N. J.; Tzafriri, L.  The behaviour of Legendre and ultraspherical polynomials in$L_p$-spaces We consider the analogue of the$\Lambda(p)-$problem for subsets of the Legendre polynomials or more general ultraspherical polynomials. We obtain the best possible'' result that if$2 Categories:42C10, 33C45, 46B07
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