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26. CJM 2012 (vol 65 pp. 481)

Ara, Pere; Dykema, Kenneth J.; Rørdam, Mikael
Correction of Proofs in "Purely Infinite Simple $C^*$-algebras Arising from Free Product Constructions'' and a Subsequent Paper
The proofs of Theorem 2.2 of K. J. Dykema and M. Rørdam, Purely infinite simple $C^*$-algebras arising from free product constructions}, Canad. J. Math. 50 (1998), 323--341 and of Theorem 3.1 of K. J. Dykema, Purely infinite simple $C^*$-algebras arising from free product constructions, II, Math. Scand. 90 (2002), 73--86 are corrected.

Keywords:C*-algebras, purely infinite
Category:46L05

27. CJM 2012 (vol 65 pp. 559)

Helemskii, A. Ya.
Extreme Version of Projectivity for Normed Modules Over Sequence Algebras
We define and study the so-called extreme version of the notion of a projective normed module. The relevant definition takes into account the exact value of the norm of the module in question, in contrast with the standard known definition that is formulated in terms of norm topology. After the discussion of the case where our normed algebra $A$ is just $\mathbb{C}$, we concentrate on the case of the next degree of complication, where $A$ is a sequence algebra, satisfying some natural conditions. The main results give a full characterization of extremely projective objects within the subcategory of the category of non-degenerate normed $A$--modules, consisting of the so-called homogeneous modules. We consider two cases, `non-complete' and `complete', and the respective answers turn out to be essentially different. In particular, all Banach non-degenerate homogeneous modules, consisting of sequences, are extremely projective within the category of Banach non-degenerate homogeneous modules. However, neither of them, provided it is infinite-dimensional, is extremely projective within the category of all normed non-degenerate homogeneous modules. On the other hand, submodules of these modules, consisting of finite sequences, are extremely projective within the latter category.

Keywords:extremely projective module, sequence algebra, homogeneous module
Category:46H25

28. CJM 2012 (vol 65 pp. 331)

Kadets, Vladimir; Martín, Miguel; Merí, Javier; Werner, Dirk
Lushness, Numerical Index 1 and the Daugavet Property in Rearrangement Invariant Spaces
We show that for spaces with 1-unconditional bases lushness, the alternative Daugavet property and numerical index 1 are equivalent. In the class of rearrangement invariant (r.i.) sequence spaces the only examples of spaces with these properties are $c_0$, $\ell_1$ and $\ell_\infty$. The only lush r.i. separable function space on $[0,1]$ is $L_1[0,1]$; the same space is the only r.i. separable function space on $[0,1]$ with the Daugavet property over the reals.

Keywords:lush space, numerical index, Daugavet property, Köthe space, rearrangement invariant space
Categories:46B04, 46E30

29. CJM 2012 (vol 65 pp. 52)

Christensen, Erik; Sinclair, Allan M.; Smith, Roger R.; White, Stuart
C$^*$-algebras Nearly Contained in Type $\mathrm{I}$ Algebras
In this paper we consider near inclusions $A\subseteq_\gamma B$ of C$^*$-algebras. We show that if $B$ is a separable type $\mathrm{I}$ C$^*$-algebra and $A$ satisfies Kadison's similarity problem, then $A$ is also type $\mathrm{I}$ and use this to obtain an embedding of $A$ into $B$.

Keywords:C$^*$-algebras, near inclusions, perturbations, type I C$^*$-algebras, similarity problem
Category:46L05

30. CJM 2012 (vol 65 pp. 485)

Bice, Tristan Matthew
Filters in C*-Algebras
In this paper we analyze states on C*-algebras and their relationship to filter-like structures of projections and positive elements in the unit ball. After developing the basic theory we use this to investigate the Kadison-Singer conjecture, proving its equivalence to an apparently quite weak paving conjecture and the existence of unique maximal centred extensions of projections coming from ultrafilters on the natural numbers. We then prove that Reid's positive answer to this for q-points in fact also holds for rapid p-points, and that maximal centred filters are obtained in this case. We then show that consistently such maximal centred filters do not exist at all meaning that, for every pure state on the Calkin algebra, there exists a pair of projections on which the state is 1, even though the state is bounded strictly below 1 for projections below this pair. Lastly we investigate towers, using cardinal invariant equalities to construct towers on the natural numbers that do and do not remain towers when canonically embedded into the Calkin algebra. Finally we show that consistently all towers on the natural numbers remain towers under this embedding.

Keywords:C*-algebras, states, Kadinson-Singer conjecture, ultrafilters, towers
Categories:46L03, 03E35

31. CJM 2011 (vol 64 pp. 755)

Brown, Lawrence G.; Lee, Hyun Ho
Homotopy Classification of Projections in the Corona Algebra of a Non-simple $C^*$-algebra
We study projections in the corona algebra of $C(X)\otimes K$, where K is the $C^*$-algebra of compact operators on a separable infinite dimensional Hilbert space and $X=[0,1],[0,\infty),(-\infty,\infty)$, or $[0,1]/\{ 0,1 \}$. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in $K_0$, Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct.

Keywords:essential codimension, continuous field of Hilbert spaces, Corona algebra
Categories:46L05, 46L80

32. CJM 2011 (vol 64 pp. 705)

Thomsen, Klaus
Pure Infiniteness of the Crossed Product of an AH-Algebra by an Endomorphism
It is shown that simplicity of the crossed product of a unital AH-algebra with slow dimension growth by an endomorphism implies that the algebra is also purely infinite, provided only that the endomorphism leaves no trace state invariant and takes the unit to a full projection.

Keywords:purely infinite $C^*$-algebras, crossed products
Category:46-xx

33. CJM 2011 (vol 64 pp. 544)

Li, Zhiqiang
On the Simple Inductive Limits of Splitting Interval Algebras with Dimension Drops
A K-theoretic classification is given of the simple inductive limits of finite direct sums of the type I $C^*$-algebras known as splitting interval algebras with dimension drops. (These are the subhomogeneous $C^*$-algebras, each having spectrum a finite union of points and an open interval, and torsion $\textrm{K}_1$-group.)

Categories:46L05, 46L35

34. CJM 2011 (vol 64 pp. 805)

Chapon, François; Defosseux, Manon
Quantum Random Walks and Minors of Hermitian Brownian Motion
Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam, and van Moerbeke that the process of eigenvalues of two consecutive minors of a Hermitian Brownian motion is a Markov process; whereas, if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion.

Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor process
Categories:46L53, 60B20, 14L24

35. CJM 2011 (vol 63 pp. 1161)

Neuwirth, Stefan; Ricard, Éric
Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group
We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group $\varGamma$ and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum $\varLambda\subseteq\varGamma$, the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than 1.

Keywords:Fourier multiplier, Toeplitz Schur multiplier, lacunary set, unconditional approximation property, Hilbert transform, Riesz projection
Categories:47B49, 43A22, 43A46, 46B28

36. CJM 2011 (vol 64 pp. 573)

Nawata, Norio
Fundamental Group of Simple $C^*$-algebras with Unique Trace III
We introduce the fundamental group ${\mathcal F}(A)$ of a simple $\sigma$-unital $C^*$-algebra $A$ with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of ``Fundamental Group of Simple $C^*$-algebras with Unique Trace I and II'' by Nawata and Watatani. Our definition in this paper makes sense for stably projectionless $C^*$-algebras. We show that there exist separable stably projectionless $C^*$-algebras such that their fundamental groups are equal to $\mathbb{R}_+^\times$ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian.

Keywords:fundamental group, Picard group, Hilbert module, countable basis, stably projectionless algebra, dimension function
Categories:46L05, 46L08, 46L35

37. CJM 2011 (vol 64 pp. 455)

Sherman, David
On Cardinal Invariants and Generators for von Neumann Algebras
We demonstrate how most common cardinal invariants associated with a von Neumann algebra $\mathcal M$ can be computed from the decomposability number, $\operatorname{dens}(\mathcal M)$, and the minimal cardinality of a generating set, $\operatorname{gen}(\mathcal M)$. Applications include the equivalence of the well-known generator problem, ``Is every separably-acting von Neumann algebra singly-generated?", with the formally stronger questions, ``Is every countably-generated von Neumann algebra singly-generated?" and ``Is the $\operatorname{gen}$ invariant monotone?" Modulo the generator problem, we determine the range of the invariant $\bigl( \operatorname{gen}(\mathcal M), \operatorname{dens}(\mathcal M) \bigr)$, which is mostly governed by the inequality $\operatorname{dens}(\mathcal M) \leq \mathfrak C^{\operatorname{gen}(\mathcal M)}$.

Keywords:von Neumann algebra, cardinal invariant, generator problem, decomposability number, representation density
Category:46L10

38. CJM 2011 (vol 63 pp. 798)

Daws, Matthew
Representing Multipliers of the Fourier Algebra on Non-Commutative $L^p$ Spaces
We show that the multiplier algebra of the Fourier algebra on a locally compact group $G$ can be isometrically represented on a direct sum on non-commutative $L^p$ spaces associated with the right von Neumann algebra of $G$. The resulting image is the idealiser of the image of the Fourier algebra. If these spaces are given their canonical operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the non-commutative $L^p$ spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca-Herz algebra built out of these non-commutative $L^p$ spaces, say $A_p(\widehat G)$. It is shown that $A_2(\widehat G)$ is isometric to $L^1(G)$, generalising the abelian situation.

Keywords:multiplier, Fourier algebra, non-commutative $L^p$ space, complex interpolation
Categories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52

39. CJM 2011 (vol 63 pp. 1188)

Śliwa, Wiesław; Ziemkowska, Agnieszka
On Complemented Subspaces of Non-Archimedean Power Series Spaces
The non-archimedean power series spaces, $A_1(a)$ and $A_\infty(b)$, are the best known and most important examples of non-archimedean nuclear Fréchet spaces. We prove that the range of every continuous linear map from $A_p(a)$ to $A_q(b)$ has a Schauder basis if either $p=1$ or $p=\infty$ and the set $M_{b,a}$ of all bounded limit points of the double sequence $(b_i/a_j)_{i,j\in\mathbb{N}}$ is bounded. It follows that every complemented subspace of a power series space $A_p(a)$ has a Schauder basis if either $p=1$ or $p=\infty$ and the set $M_{a,a}$ is bounded.

Keywords:non-archimedean Köthe space, range of a continuous linear map, Schauder basis
Categories:46S10, 47S10, 46A35

40. CJM 2011 (vol 63 pp. 551)

Hadwin, Don; Li, Qihui; Shen, Junhao
Topological Free Entropy Dimensions in Nuclear C$^*$-algebras and in Full Free Products of Unital C$^*$-algebras
In the paper, we introduce a new concept, topological orbit dimension of an $n$-tuple of elements in a unital C$^{\ast}$-algebra. Using this concept, we conclude that Voiculescu's topological free entropy dimension of every finite family of self-adjoint generators of a nuclear C$^{\ast}$-algebra is less than or equal to $1$. We also show that the Voiculescu's topological free entropy dimension is additive in the full free product of some unital C$^{\ast}$-algebras. We show that the unital full free product of Blackadar and Kirchberg's unital MF algebras is also an MF algebra. As an application, we obtain that $\mathop{\textrm{Ext}}(C_{r}^{\ast}(F_{2})\ast_{\mathbb{C}}C_{r}^{\ast}(F_{2}))$ is not a group.

Keywords: topological free entropy dimension, unital C$^{*}$-algebra
Categories:46L10, 46L54

41. CJM 2011 (vol 63 pp. 460)

Pavlíček, Libor
Monotonically Controlled Mappings
We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó-Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

Keywords: monotone mapping, DM mapping, Radó-Reichelderfer property, UDM mapping, differentiability
Categories:26B05, 46G05

42. CJM 2011 (vol 63 pp. 381)

Ji, Kui ; Jiang, Chunlan
A Complete Classification of AI Algebras with the Ideal Property
Let $A$ be an AI algebra; that is, $A$ is the $\mbox{C}^{*}$-algebra inductive limit of a sequence $$ A_{1}\stackrel{\phi_{1,2}}{\longrightarrow}A_{2}\stackrel{\phi_{2,3}}{\longrightarrow}A_{3} \longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots, $$ where $A_{n}=\bigoplus_{i=1}^{k_n}M_{[n,i]}(C(X^{i}_n))$, $X^{i}_n$ are $[0,1]$, $k_n$, and $[n,i]$ are positive integers. Suppose that $A$ has the ideal property: each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two-sided ideal. In this article, we give a complete classification of AI algebras with the ideal property.

Keywords:AI algebras, K-group, tracial state, ideal property, classification
Categories:46L35, 19K14, 46L05, 46L08

43. CJM 2011 (vol 63 pp. 500)

Dadarlat, Marius; Elliott, George A.; Niu, Zhuang
One-Parameter Continuous Fields of Kirchberg Algebras. II
Parallel to the first two authors' earlier classification of separable, unita one-parameter, continuous fields of Kirchberg algebras with torsion free $\mathrm{K}$-groups supported in one dimension, one-parameter, separable, uni continuous fields of AF-algebras are classified by their ordered $\mathrm{K}_0$-sheaves. Effros-Handelman-Shen type theorems are pr separable unital one-parameter continuous fields of AF-algebras and Kirchberg algebras.

Keywords:continuous fields of C$^*$-algebras, $\mathrm{K}_0$-presheaves, Effros--Handeen type theorem
Category:46L35

44. CJM 2010 (vol 63 pp. 436)

Mine, Kotaro; Sakai, Katsuro
Simplicial Complexes and Open Subsets of Non-Separable LF-Spaces
Let $F$ be a non-separable LF-space homeomorphic to the direct sum $\sum_{n\in\mathbb{N}} \ell_2(\tau_n)$, where $\aleph_0 < \tau_1 < \tau_2 < \cdots$. It is proved that every open subset $U$ of $F$ is homeomorphic to the product $|K| \times F$ for some locally finite-dimensional simplicial complex $K$ such that every vertex $v \in K^{(0)}$ has the star $\operatorname{St}(v,K)$ with $\operatorname{card} \operatorname{St}(v,K)^{(0)} < \tau = \sup\tau_n$ (and $\operatorname{card} K^{(0)} \le \tau$), and, conversely, if $K$ is such a simplicial complex, then the product $|K| \times F$ can be embedded in $F$ as an open set, where $|K|$ is the polyhedron of $K$ with the metric topology.

Keywords:LF-space, open set, simplicial complex, metric topology, locally finite-dimensional, star, small box product, ANR, $\ell_2(\tau)$, $\ell_2(\tau)$-manifold, open embedding, $\sum_{i\in\mathbb{N}}\ell_2(\tau_i)$
Categories:57N20, 46A13, 46T05, 57N17, 57Q05, 57Q40

45. CJM 2010 (vol 63 pp. 123)

Granirer, Edmond E.
Strong and Extremely Strong Ditkin sets for the Banach Algebras $A_p^r(G)=A_p\cap L^r(G)$
Let $A_p(G)$ be the Figa-Talamanca, Herz Banach Algebra on $G$; thus $A_2(G)$ is the Fourier algebra. Strong Ditkin (SD) and Extremely Strong Ditkin (ESD) sets for the Banach algebras $A_p^r(G)$ are investigated for abelian and nonabelian locally compact groups $G$. It is shown that SD and ESD sets for $A_p(G)$ remain SD and ESD sets for $A_p^r(G)$, with strict inclusion for ESD sets. The case for the strict inclusion of SD sets is left open. A result on the weak sequential completeness of $A_2(F)$ for ESD sets $F$ is proved and used to show that Varopoulos, Helson, and Sidon sets are not ESD sets for $A_2(G)$, yet they are such for $A_2^r(G)$ for discrete groups $G$, for any $1\le r\le 2$. A result is given on the equivalence of the sequential and the net definitions of SD or ESD sets for $\sigma$-compact groups. The above results are new even if $G$ is abelian.

Keywords:Fourier algebra, Figa-Talamanca-Herz algebra, locally compact group, Ditkin sets, Helson sets, Sidon sets, weak sequential completeness
Categories:43A15, 43A10, 46J10, 43A45

46. CJM 2010 (vol 63 pp. 222)

Wang, Jiun-Chau
Limit Theorems for Additive Conditionally Free Convolution
In this paper we determine the limiting distributional behavior for sums of infinitesimal conditionally free random variables. We show that the weak convergence of classical convolution and that of conditionally free convolution are equivalent for measures in an infinitesimal triangular array, where the measures may have unbounded support. Moreover, we use these limit theorems to study the conditionally free infinite divisibility. These results are obtained by complex analytic methods without reference to the combinatorics of c-free convolution.

Keywords:additive c-free convolution, limit theorems, infinitesimal arrays
Categories:46L53, 60F05

47. CJM 2010 (vol 63 pp. 3)

Banica, T.; Belinschi, S. T.; Capitaine, M.; Collins, B.
Free Bessel Laws
We introduce and study a remarkable family of real probability measures $\pi_{st}$ that we call free Bessel laws. These are related to the free Poisson law $\pi$ via the formulae $\pi_{s1}=\pi^{\boxtimes s}$ and ${\pi_{1t}=\pi^{\boxplus t}}$. Our study includes definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.

Keywords:Poisson law, Bessel function, Wishart matrix, quantum group
Categories:46L54, 15A52, 16W30

48. CJM 2010 (vol 62 pp. 961)

Aleman, Alexandru; Duren, Peter; Martín, María J.; Vukotić, Dragan
Multiplicative Isometries and Isometric Zero-Divisors
For some Banach spaces of analytic functions in the unit disk (weighted Bergman spaces, Bloch space, Dirichlet-type spaces), the isometric pointwise multipliers are found to be unimodular constants. As a consequence, it is shown that none of those spaces have isometric zero-divisors. Isometric coefficient multipliers are also investigated.

Keywords:Banach spaces of analytic functions, Hardy spaces, Bergman spaces, Bloch space, Dirichlet space, Dirichlet-type spaces, pointwise multipliers, coefficient multipliers, isometries, isometric zero-divisors
Categories:30H05, 46E15

49. CJM 2010 (vol 62 pp. 827)

Ouyang, Caiheng; Xu, Quanhua
BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces
This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.

Keywords:BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spaces
Categories:46E40, 42B25, 46B20

50. CJM 2010 (vol 62 pp. 845)

Samei, Ebrahim; Spronk, Nico; Stokke, Ross
Biflatness and Pseudo-Amenability of Segal Algebras
We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, $L^1(G)$, and the Fourier algebra, $A(G)$, of a locally compact group~$G$.

Keywords:Segal algebra, pseudo-amenable Banach algebra, biflat Banach algebra
Categories:43A20, 43A30, 46H25, 46H10, 46H20, 46L07
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