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76. CJM 2009 (vol 61 pp. 124)

Dijkstra, Jan J.; Mill, Jan van
 Characterizing Complete Erd\H os Space The space now known as {\em complete Erd\H os space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the closed subspace of the Hilbert space $\ell^2$ consisting of all vectors such that every coordinate is in the convergent sequence $\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G. Oversteegen we present simple and useful topological characterizations of $\cerdos$. As an application we determine the class of factors of $\cerdos$. In another application we determine precisely which of the spaces that can be constructed in the Banach spaces $\ell^p$ according to the Erd\H os method' are homeomorphic to $\cerdos$. A novel application states that if $I$ is a Polishable $F_\sigma$-ideal on $\omega$, then $I$ with the Polish topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$, $\Z\times2^\omega$, or $\cerdos$. This last result answers a question that was asked by Stevo Todor{\v{c}}evi{\'c}. Keywords:Complete Erd\H os space, Lelek fan, almost zero-dimensional, nowhere zero-dimensional, Polishable ideals, submeasures on $\omega$, $\R$-trees, line-free groups in Banach spacesCategories:28C10, 46B20, 54F65

77. CJM 2008 (vol 60 pp. 1108)

 A Classification of Tsirelson Type Spaces We give a complete classification of mixed Tsirelson spaces $T[(\mathcal F_i,\theta_i)_{i=1}^{r}]$ for finitely many pairs of given compact and hereditary families $\mathcal F_i$ of finite sets of integers and $0<\theta_i<1$ in terms of the Cantor--Bendixson indices of the families $\mathcal F_i$, and $\theta_i$ ($1\le i\le r$). We prove that there are unique countable ordinal $\alpha$ and $0<\theta<1$ such that every block sequence of $T[(\mathcal F_i,\theta_i)_{i=1}^{r}]$ has a subsequence equivalent to a subsequence of the natural basis of the $T(\mathcal S_{\omega^\alpha},\theta)$. Finally, we give a complete criterion of comparison in between two of these mixed Tsirelson spaces. Categories:46B20, 05D10

78. CJM 2008 (vol 60 pp. 1010)

Galé, José E.; Miana, Pedro J.
 $H^\infty$ Functional Calculus and Mikhlin-Type Multiplier Conditions Let $T$ be a sectorial operator. It is known that the existence of a bounded (suitably scaled) $H^\infty$ calculus for $T$, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra $\Lambda_{\infty,1}^\alpha(\R^+)$. Such an algebra includes functions defined by Mikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for $T$. In this paper, we use fractional derivation to analyse in detail the relationship between $\Lambda_{\infty,1}^\alpha$ and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence. Keywords:functional calculus, fractional calculus, Mikhlin multipliers, analytic semigroups, unbounded operators, quasimultipliersCategories:47A60, 47D03, 46J15, 26A33, 47L60, 47B48, 43A22

79. CJM 2008 (vol 60 pp. 975)

Boca, Florin P.
 An AF Algebra Associated with the Farey Tessellation We associate with the Farey tessellation of the upper half-plane an AF algebra $\AA$ encoding the cutting sequences'' that define vertical geodesics. The Effros--Shen AF algebras arise as quotients of $\AA$. Using the path algebra model for AF algebras we construct, for each $\tau \in \big(0,\frac{1}{4}\big]$, projections $(E_n)$ in $\AA$ such that $E_n E_{n\pm 1}E_n \leq \tau E_n$. Categories:46L05, 11A55, 11B57, 46L55, 37E05, 82B20

80. CJM 2008 (vol 60 pp. 703)

Toms, Andrew S.; Winter, Wilhelm
 $\mathcal{Z}$-Stable ASH Algebras The Jiang--Su algebra $\mathcal{Z}$ has come to prominence in the classification program for nuclear $C^*$-algebras of late, due primarily to the fact that Elliott's classification conjecture in its strongest form predicts that all simple, separable, and nuclear $C^*$-algebras with unperforated $\mathrm{K}$-theory will absorb $\mathcal{Z}$ tensorially, i.e., will be $\mathcal{Z}$-stable. There exist counterexamples which suggest that the conjecture will only hold for simple, nuclear, separable and $\mathcal{Z}$-stable $C^*$-algebras. We prove that virtually all classes of nuclear $C^*$-algebras for which the Elliott conjecture has been confirmed so far consist of $\mathcal{Z}$-stable $C^*$-algebras. This follows in large part from the following result, also proved herein: separable and approximately divisible $C^*$-algebras are $\mathcal{Z}$-stable. Keywords:nuclear $C^*$-algebras, K-theory, classificationCategories:46L85, 46L35

81. CJM 2008 (vol 60 pp. 520)

Chen, Chang-Pao; Huang, Hao-Wei; Shen, Chun-Yen
 Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences Let $A=(a_{j,k})_{j,k \ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $\|A\|_{E, F}$ are determined by their actions on decreasing sequences, where $E$ and $F$ are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces: $\ell_p$, $d(w,p)$, and $\ell_p(w)$. The results established here generalize ones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour. Keywords:norms of matrices, normed Riesz spaces, weighted mean matrices, NÃ¶rlund mean matrices, summability matrices, matrices with row decreasingCategories:15A60, 40G05, 47A30, 47B37, 46B42

82. CJM 2008 (vol 60 pp. 189)

Lin, Huaxin
 Furstenberg Transformations and Approximate Conjugacy Let $\alpha$ and $\beta$ be two Furstenberg transformations on $2$-torus associated with irrational numbers $\theta_1,$ $\theta_2,$ integers $d_1, d_2$ and Lipschitz functions $f_1$ and $f_2$. It is shown that $\alpha$ and $\beta$ are approximately conjugate in a measure theoretical sense if (and only if) $\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z.$ Closely related to the classification of simple amenable \CAs, it is shown that $\af$ and $\bt$ are approximately $K$-conjugate if (and only if) $\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z$ and $|d_1|=|d_2|.$ This is also shown to be equivalent to the condition that the associated crossed product \CAs are isomorphic. Keywords:Furstenberg transformations, approximate conjugacyCategories:37A55, 46L35

83. CJM 2007 (vol 59 pp. 1135)

Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari
 Sobolev Extensions of HÃ¶lder Continuous and Characteristic Functions on Metric Spaces We study when characteristic and H\"older continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and H\"older continuous functions into globally defined Sobolev functions. Keywords:characteristic function, Newtonian function, metric space, resolutivity, HÃ¶lder continuous, Perron solution, $p$-harmonic, Sobolev extension, Whitney coveringCategories:46E35, 31C45

84. CJM 2007 (vol 59 pp. 897)

Bruneau, Laurent
 The Ground State Problem for a Quantum Hamiltonian Model Describing Friction In this paper, we consider the quantum version of a Hamiltonian model describing friction. This model consists of a particle which interacts with a bosonic reservoir representing a homogeneous medium through which the particle moves. We show that if the particle is confined, then the Hamiltonian admits a ground state if and only if a suitable infrared condition is satisfied. The latter is violated in the case of linear friction, but satisfied when the friction force is proportional to a higher power of the particle speed. Categories:81Q10, 46N50

85. CJM 2007 (vol 59 pp. 966)

Forrest, Brian E.; Runde, Volker; Spronk, Nico
 Operator Amenability of the Fourier Algebra in the $\cb$-Multiplier Norm Let $G$ be a locally compact group, and let $A_{\cb}(G)$ denote the closure of $A(G)$, the Fourier algebra of $G$, in the space of completely bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group such that $\cstar(G)$ is residually finite-dimensional, we show that $A_{\cb}(G)$ is operator amenable. In particular, $A_{\cb}(\free_2)$ is operator amenable even though $\free_2$, the free group in two generators, is not an amenable group. Moreover, we show that if $G$ is a discrete group such that $A_{\cb}(G)$ is operator amenable, a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$ if and only if it has an approximate identity bounded in the $\cb$-multiplier norm. Keywords:$\cb$-multiplier norm, Fourier algebra, operator amenability, weak amenabilityCategories:43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25

86. CJM 2007 (vol 59 pp. 1029)

 The Geometry of $L_0$ Suppose that we have the unit Euclidean ball in $\R^n$ and construct new bodies using three operations --- linear transformations, closure in the radial metric, and multiplicative summation defined by $\|x\|_{K+_0L} = \sqrt{\|x\|_K\|x\|_L}.$ We prove that in dimension $3$ this procedure gives all origin-symmetric convex bodies, while this is no longer true in dimensions $4$ and higher. We introduce the concept of embedding of a normed space in $L_0$ that naturally extends the corresponding properties of $L_p$-spaces with $p\ne0$, and show that the procedure described above gives exactly the unit balls of subspaces of $L_0$ in every dimension. We provide Fourier analytic and geometric characterizations of spaces embedding in $L_0$, and prove several facts confirming the place of $L_0$ in the scale of $L_p$-spaces. Categories:52A20, 52A21, 46B20

87. CJM 2007 (vol 59 pp. 614)

Labuschagne, C. C. A.
 Preduals and Nuclear Operators Associated with Bounded, $p$-Convex, $p$-Concave and Positive $p$-Summing Operators We use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of $p$-convex, $p$-concave and positive $p$-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators. Keywords:$p$-convex operator, $p$-concave operator, $p$-summing operator, Banach space, Banach lattice, nuclear operator, sequence spaceCategories:46B28, 47B10, 46B42, 46B45

88. CJM 2007 (vol 59 pp. 276)

Bernardis, A. L.; Martín-Reyes, F. J.; Salvador, P. Ortega
 Weighted Inequalities for Hardy--Steklov Operators We characterize the pairs of weights $(v,w)$ for which the operator $Tf(x)=g(x)\int_{s(x)}^{h(x)}f$ with $s$ and $h$ increasing and continuous functions is of strong type $(p,q)$ or weak type $(p,q)$ with respect to the pair $(v,w)$ in the case $0 Keywords:Hardy--Steklov operator, weights, inequalitiesCategories:26D15, 46E30, 42B25 89. CJM 2007 (vol 59 pp. 343) Lin, Huaxin  Weak Semiprojectivity in Purely Infinite Simple$C^*$-Algebras Let$A$be a separable amenable purely infinite simple \CA which satisfies the Universal Coefficient Theorem. We prove that$A$is weakly semiprojective if and only if$K_i(A)$is a countable direct sum of finitely generated groups ($i=0,1$). Therefore, if$A$is such a \CA, for any$\ep>0$and any finite subset${\mathcal F}\subset A$there exist$\dt>0$and a finite subset${\mathcal G}\subset A$satisfying the following: for any contractive positive linear map$L: A\to B$(for any \CA$B$) with$ \|L(ab)-L(a)L(b)\|<\dt$for$a, b\in {\mathcal G}$there exists a homomorphism$h\from A\to B$such that$ \|h(a)-L(a)\|<\ep$for$a\in {\mathcal F}$. Keywords:weakly semiprojective, purely infinite simple$C^*$-algebrasCategories:46L05, 46L80 90. CJM 2007 (vol 59 pp. 3) Biller, Harald  Holomorphic Generation of Continuous Inverse Algebras We study complex commutative Banach algebras (and, more generally, continuous inverse algebras) in which the holomorphic functions of a fixed$n$-tuple of elements are dense. In particular, we characterize the compact subsets of~$\C^n$which appear as joint spectra of such$n$-tuples. The characterization is compared with several established notions of holomorphic convexity by means of approximation conditions. Keywords:holomorphic functional calculus, commutative continuous inverse algebra, holomorphic convexity, Stein compacta, meromorphic convexity, holomorphic approximationCategories:46H30, 32A38, 32E30, 41A20, 46J15 91. CJM 2007 (vol 59 pp. 63) Ferenczi, Valentin; Galego, Elói Medina  Some Results on the Schroeder--Bernstein Property for Separable Banach Spaces We construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder--Bernstein Index of any of these spaces is$2^{\aleph_0}$. Our construction is based on a Banach space introduced by W. T. Gowers and B. Maurey in 1997. We also use classical descriptive set theory methods, as in some work of the first author and C. Rosendal, to improve some results of P. G. Casazza and of N. J. Kalton on the Schroeder--Bernstein Property for spaces with an unconditional finite-dimensional Schauder decomposition. Keywords:complemented subspaces,, Schroeder-Bernstein propertyCategories:46B03, 46B20 92. CJM 2006 (vol 58 pp. 1144) Hamana, Masamichi  Partial$*$-Automorphisms, Normalizers, and Submodules in Monotone Complete$C^*$-Algebras For monotone complete$C^*$-algebras$A\subset B$with$A$contained in$B$as a monotone closed$C^*$-subalgebra, the relation$X = AsA$gives a bijection between the set of all monotone closed linear subspaces$X$of$B$such that$AX + XA \subset X$and$XX^* + X^*X \subset A$and a set of certain partial isometries$s$in the normalizer" of$A$in$B$, and similarly for the map$s \mapsto \Ad s$between the latter set and a set of certain partial$*$-automorphisms" of$A$. We introduce natural inverse semigroup structures in the set of such$X$'s and the set of partial$*$-automorphisms of$A$, modulo a certain relation, so that the composition of these maps induces an inverse semigroup homomorphism between them. For a large enough$B$the homomorphism becomes surjective and all the partial$*$-automorphisms of$A$are realized via partial isometries in$B$. In particular, the inverse semigroup associated with a type${\rm II}_1$von Neumann factor, modulo the outer automorphism group, can be viewed as the fundamental group of the factor. We also consider the$C^*$-algebra version of these results. Categories:46L05, 46L08, 46L40, 20M18 93. CJM 2006 (vol 58 pp. 1268) Sims, Aidan  Gauge-Invariant Ideals in the$C^*$-Algebras of Finitely Aligned Higher-Rank Graphs We produce a complete description of the lattice of gauge-invariant ideals in$C^*(\Lambda)for a finitely alignedk$-graph$\Lambda$. We provide a condition on$\Lambda$under which every ideal is gauge-invariant. We give conditions on$\Lambda$under which$C^*(\Lambda)$satisfies the hypotheses of the Kirchberg--Phillips classification theorem. Keywords:Graphs as categories, graph algebra,$C^*$-algebraCategory:46L05 94. CJM 2006 (vol 58 pp. 691) Bendikov, A.; Saloff-Coste, L.  Hypoelliptic Bi-Invariant Laplacians on Infinite Dimensional Compact Groups On a compact connected group$G$, consider the infinitesimal generator$-L$of a central symmetric Gaussian convolution semigroup$(\mu_t)_{t>0}$. Using appropriate notions of distribution and smooth function spaces, we prove that$L$is hypoelliptic if and only if$(\mu_t)_{t>0} $is absolutely continuous with respect to Haar measure and admits a continuous density$x\mapsto \mu_t(x)$,$t>0$, such that$\lim_{t\ra 0} t\log \mu_t(e)=0$. In particular, this condition holds if and only if any Borel measure$u$which is solution of$Lu=0$in an open set$\Omega$can be represented by a continuous function in$\Omega$. Examples are discussed. Categories:60B15, 43A77, 35H10, 46F25, 60J45, 60J60 95. CJM 2006 (vol 58 pp. 859) Read, C. J.  Nonstandard Ideals from Nonstandard Dual Pairs for$L^1(\omega)$and$l^1(\omega)$The Banach convolution algebras$l^1(\omega)$and their continuous counterparts$L^1(\bR^+,\omega)$are much studied, because (when the submultiplicative weight function$\omega$is radical) they are pretty much the prototypic examples of commutative radical Banach algebras. In cases of nice'' weights$\omega$, the only closed ideals they have are the obvious, or standard'', ideals. But in the general case, a brilliant but very difficult paper of Marc Thomas shows that nonstandard ideals exist in$l^1(\omega)$. His proof was successfully exported to the continuous case$L^1(\bR^+,\omega)$by Dales and McClure, but remained difficult. In this paper we first present a small improvement: a new and easier proof of the existence of nonstandard ideals in$l^1(\omega)$and$L^1(\bR^+,\omega)$. The new proof is based on the idea of a `nonstandard dual pair'' which we introduce. We are then able to make a much larger improvement: we find nonstandard ideals in$L^1(\bR^+,\omega)$containing functions whose supports extend all the way down to zero in$\bR^+$, thereby solving what has become a notorious problem in the area. Keywords:Banach algebra, radical, ideal, standard ideal, semigroupCategories:46J45, 46J20, 47A15 96. CJM 2006 (vol 58 pp. 768) Hu, Zhiguo; Neufang, Matthias  Decomposability of von Neumann Algebras and the Mazur Property of Higher Level The decomposability number of a von Neumann algebra$\m$(denoted by$\dec(\m)$) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in$\m$. In this paper, we explore the close connection between$\dec(\m)$and the cardinal level of the Mazur property for the predual$\m_*$of$\m$, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group$G$as the group algebra$\lone$, the Fourier algebra$A(G)$, the measure algebra$M(G)$, the algebra$\luc^*$, etc. We show that for any of these von Neumann algebras, say$\m$, the cardinal number$\dec(\m)$and a certain cardinal level of the Mazur property of$\m_*$are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of$G$: the compact covering number$\kg$of$G$and the least cardinality$\bg$of an open basis at the identity of$G$. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra$\ag^{**}$. Keywords:Mazur property, predual of a von Neumann algebra, locally compact group and its cardinal invariants, group algebra, Fourier algebra, topological centreCategories:22D05, 43A20, 43A30, 03E55, 46L10 97. CJM 2006 (vol 58 pp. 820) Moreno, J. P.; Papini, P. L.; Phelps, R. R.  Diametrically Maximal and Constant Width Sets in Banach Spaces We characterize diametrically maximal and constant width sets in$C(K)$, where$K$is any compact Hausdorff space. These results are applied to prove that the sum of two diametrically maximal sets needs not be diametrically maximal, thus solving a question raised in a paper by Groemer. A~characterization of diametrically maximal sets in$\ell_1^3$is also given, providing a negative answer to Groemer's problem in finite dimensional spaces. We characterize constant width sets in$c_0(I)$, for every$I$, and then we establish the connections between the Jung constant of a Banach space and the existence of constant width sets with empty interior. Porosity properties of families of sets of constant width and rotundity properties of diametrically maximal sets are also investigated. Finally, we present some results concerning non-reflexive and Hilbert spaces. Categories:52A05, 46B20 98. CJM 2006 (vol 58 pp. 548) Galanopoulos, P.; Papadimitrakis, M.  Hausdorff and Quasi-Hausdorff Matrices on Spaces of Analytic Functions We consider Hausdorff and quasi-Hausdorff matrices as operators on classical spaces of analytic functions such as the Hardy and the Bergman spaces, the Dirichlet space, the Bloch spaces and$\BMOA$. When the generating sequence of the matrix is the moment sequence of a measure$\mu$, we find the conditions on$\mu$which are equivalent to the boundedness of the matrix on the various spaces. Categories:47B38, 46E15, 40G05, 42A20 99. CJM 2006 (vol 58 pp. 492) Chua, Seng-Kee  Extension Theorems on Weighted Sobolev Spaces and Some Applications We extend the extension theorems to weighted Sobolev spaces$L^p_{w,k}(\mathcal D)$on$(\varepsilon,\delta)$domains with doubling weight$w$that satisfies a Poincar\'e inequality and such that$w^{-1/p}$is locally$L^{p'}$. We also make use of the main theorem to improve weighted Sobolev interpolation inequalities. Keywords:PoincarÃ© inequalities,$A_p$weights, doubling weights,$(\ep,\delta)$domain,$(\ep,\infty)$domainCategory:46E35 100. CJM 2006 (vol 58 pp. 39) Exel, R.; Vershik, A. $C^*$-Algebras of Irreversible Dynamical Systems We show that certain$C^*\$-algebras which have been studied by, among others, Arzumanian, Vershik, Deaconu, and Renault, in connection with a measure-preserving transformation of a measure space or a covering map of a compact space, are special cases of the endomorphism crossed-product construction recently introduced by the first named author. As a consequence these algebras are given presentations in terms of generators and relations. These results come as a consequence of a general theorem on faithfulness of representations which are covariant with respect to certain circle actions. For the case of topologically free covering maps we prove a stronger result on faithfulness of representations which needs no covariance. We also give a necessary and sufficient condition for simplicity. Categories:46L55, 37A55
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