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1. CMB Online first

Figiel, Tadeusz; Johnson, William
Quotients of Essentially Euclidean Spaces
A precise quantitative version of the following qualitative statement is proved: If a finite dimensional normed space contains approximately Euclidean subspaces of all proportional dimensions, then every proportional dimensional quotient space has the same property.

Keywords:essentially euclidean space
Categories:46B20, 46B07, 46B99

2. CMB Online first

Liu, Zhichao
Injectivity of the connecting homomorphisms in inductive limits of Elliott-Thomsen algebras
Let $A$ be the inductive limit of a sequence $$ A_1\,\xrightarrow{\phi_{1,2}}\,A_2\,\xrightarrow{\phi_{2,3}}\,A_3\rightarrow\cdots $$ with $A_n=\bigoplus_{i=1}^{n_i}A_{[n,i]}$, where all the $A_{[n,i]}$ are Elliott-Thomsen algebras and $\phi_{n,n+1}$ are homomorphisms. In this paper, we will prove that $A$ can be written as another inductive limit $$ B_1\,\xrightarrow{\psi_{1,2}}\,B_2\,\xrightarrow{\psi_{2,3}}\,B_3\rightarrow\cdots $$ with $B_n=\bigoplus_{i=1}^{n_i'}B_{[n,i]'}$, where all the $B_{[n,i]'}$ are Elliott-Thomsen algebras and with the extra condition that all the $\psi_{n,n+1}$ are injective.

Keywords:injective, inductive limit, Elliott-Thomsen algebra
Categories:46L05, 46L35

3. CMB Online first

Loring, Terry A.; Schulz-Baldes, Hermann
Spectral flow argument localizing an odd index pairing
An odd Fredholm module for a given invertible operator on a Hilbert space is specified by an unbounded so-called Dirac operator with compact resolvent and bounded commutator with the given invertible. Associated to this is an index pairing in terms of a Fredholm operator with Noether index. Here it is shown by a spectral flow argument how this index can be calculated as the signature of a finite dimensional matrix called the spectral localizer.

Keywords:index pairing, spectral flow, topological materials
Categories:19K56, 46L80

4. CMB 2018 (vol 61 pp. 738)

Cruz-Uribe, David; Rodney, Scott; Rosta, Emily
Poincaré Inequalities and Neumann Problems for the $p$-Laplacian
We prove an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a Neumann problem related to a degenerate $p$-Laplacian. The Poincaré inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the $p$-Laplacian.

Keywords:degenerate Sobolev space, $p$-Laplacian, Poincaré inequalities
Categories:30C65, 35B65, 35J70, 42B35, 42B37, 46E35

5. CMB 2018 (vol 61 pp. 483)

Banica, Teodor
Tannakian duality for affine homogeneous spaces
Associated to any closed quantum subgroup $G\subset U_N^+$ and any index set $I\subset\{1,\dots,N\}$ is a certain homogeneous space $X_{G,I}\subset S^{N-1}_{\mathbb C,+}$, called affine homogeneous space. We discuss here the abstract axiomatization of the algebraic manifolds $X\subset S^{N-1}_{\mathbb C,+}$ which can appear in this way, by using Tannakian duality methods.

Keywords:quantum isometry, noncommutative manifold
Categories:46L65, 46L89

6. CMB 2018 (vol 61 pp. 848)

Schmidt, Simon; Weber, Moritz
Quantum Symmetries of Graph $C^*$-algebras
The study of graph $C^*$-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have never been computed so far. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph $C^*$-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph $C^*$-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

Keywords:finite graph, graph automorphism, automorphism group, quantum automorphism, graph C*-algebra, quantum group, quantum symmetry
Categories:46LXX, 05CXX, 20B25

7. CMB 2018 (vol 61 pp. 236)

Boutonnet, Remi; Roydor, Jean
A Note on Uniformly Bounded Cocycles into Finite von Neumann Algebras
We give a short proof of a result of T. Bates and T. Giordano stating that any uniformly bounded Borel cocycle into a finite von Neumann algebra is cohomologous to a unitary cocycle. We also point out a separability issue in their proof. Our approach is based on the existence of a non-positive curvature metric on the positive cone of a finite von Neumann algebra.

Keywords:Borel cocycle, von Neumann algebra
Categories:46L55, 46L40, 22D40

8. CMB 2017 (vol 61 pp. 704)

Bénéteau, Catherine Anne; Fleeman, Matthew C.; Khavinson, Dmitry S.; Seco, Daniel; Sola, Alan
Remarks on Inner Functions and Optimal Approximants
We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modified to produce inner functions.

Keywords:inner function, reproducing Kernel Hilbert Space, operator-theoretic function theory
Categories:46E22, 30J05

9. CMB 2017 (vol 61 pp. 449)

Abrahamsen, Trond A.; Hájek, Petr; Nygaard, Olav; Troyanski, Stanimir L.
Strongly extreme points and approximation properties
We show that if $x$ is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at $x$, then $x$ is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the sufficient conditions mentioned.

Keywords:denting point, strongly extreme point, unconditional compact approximation property
Categories:46B20, 46B04

10. CMB 2017 (vol 60 pp. 855)

Suárez de la Fuente, Jesús
The Kottman Constant for $\alpha$-Hölder Maps
We investigate the role of the Kottman constant of a Banach space $X$ in the extension of $\alpha$-Hölder continuous maps for every $\alpha\in (0,1]$.

Keywords:Kottman constant, $\alpha$-Hölder map
Categories:46B60, 46B80

11. CMB 2017 (vol 60 pp. 673)

Abtahi, Fatemeh; Azizi, Mohsen; Rejali, Ali
Character Amenability of the Intersection of Lipschitz Algebras
Let $(X,d)$ be a metric space and $J\subseteq [0,\infty)$ be nonempty. We study the structure of the arbitrary intersections of Lipschitz algebras, and define a special Banach subalgebra of $\bigcap_{\gamma\in J}\operatorname{Lip}_\gamma X$, denoted by $\operatorname{ILip}_J X$. Mainly, we investigate $C$-character amenability of $\operatorname{ILip}_J X$, in particular Lipschitz algebras. We address a gap in the proof of a recent result in this field. Then we remove this gap, and obtain a necessary and sufficient condition for $C$-character amenability of $\operatorname{ILip}_J X$, specially Lipschitz algebras, under an additional assumption.

Keywords:amenability, character amenability, Lipschitz algebra, metric space
Categories:46H05, 46J10, 11J83

12. CMB 2017 (vol 61 pp. 225)

Bichon, Julien; Kyed, David; Raum, Sven
Higher $\ell^2$-Betti Numbers of Universal Quantum Groups
We calculate all $\ell^2$-Betti numbers of the universal discrete Kac quantum groups $\hat{\mathrm U}^+_n$ as well as their half-liberated counterparts $\hat{\mathrm U}^*_n$.

Keywords:$\ell^2$-Betti number, free unitary quantum group, half-liberated unitary quantum group, free product formula, extension
Categories:16T05, 46L65, 20G42

13. CMB 2017 (vol 60 pp. 402)

Shravan Kumar, N.
Invariant Means on a Class of von Neumann Algebras Related to Ultraspherical Hypergroups II
Let $K$ be an ultraspherical hypergroup associated to a locally compact group $G$ and a spherical projector $\pi$ and let $VN(K)$ denote the dual of the Fourier algebra $A(K)$ corresponding to $K.$ In this note, we show that the set of invariant means on $VN(K)$ is singleton if and only if $K$ is discrete. Here $K$ need not be second countable. We also study invariant means on the dual of the Fourier algebra $A_0(K),$ the closure of $A(K)$ in the $cb$-multiplier norm. Finally, we consider generalized translations and generalized invariant means.

Keywords:ultraspherical hypergroup, Fourier algebra, Fourier-Stieltjes algebra, invariant mean, generalized translation, generalized invariant mean
Categories:43A62, 46J10, 43A30, 20N20

14. CMB 2017 (vol 61 pp. 301)

Józiak, Paweł
Remarks on Hopf Images and Quantum Permutation Groups $S_n^+$
Motivated by a question of A. Skalski and P.M. Sołtan (2016) about inner faithfulness of the S. Curran's map of extending a quantum increasing sequence to a quantum permutation, we revisit the results and techniques of T. Banica and J. Bichon (2009) and study some group-theoretic properties of the quantum permutation group on $4$ points. This enables us not only to answer the aforementioned question in positive in case $n=4, k=2$, but also to classify the automorphisms of $S_4^+$, describe all the embeddings $O_{-1}(2)\subset S_4^+$ and show that all the copies of $O_{-1}(2)$ inside $S_4^+$ are conjugate. We then use these results to show that the converse to the criterion we applied to answer the aforementioned question is not valid.

Keywords:Hopf image, quantum permutation group, compact quantum group
Categories:20G42, 81R50, 46L89, 16W35

15. CMB 2017 (vol 61 pp. 114)

Haralampidou, Marina; Oudadess, Mohamed; Palacios, Lourdes; Signoret, Carlos
A characterization of $C^{\ast}$-normed algebras via positive functionals
We give a characterization of $C^{\ast}$-normed algebras, among certain involutive normed ones. This is done through the existence of enough specific positive functionals. The same question is also examined in some non normed (topological) algebras.

Keywords:$C^{\ast}$-normed algebra, $C^*$-algebra, (pre-)locally $C^*$-algebra, pre-$C^*$-bornological algebra, positive functional, locally uniformly $A$-convex algebra, perfect locally $m$-convex algebra, $C^*$-(resp. $^*$-) subnormable algebra
Categories:46H05, 46K05

16. CMB 2017 (vol 60 pp. 449)

Alaghmandan, Mahmood; Crann, Jason
Character Density in Central Subalgebras of Compact Quantum Groups
We investigate quantum group generalizations of various density results from Fourier analysis on compact groups. In particular, we establish the density of characters in the space of fixed points of the conjugation action on $L^2(\mathbb{G})$, and use this result to show the weak* density and norm density of characters in $ZL^\infty(\mathbb{G})$ and $ZC(\mathbb{G})$, respectively. As a corollary, we partially answer an open question of Woronowicz. At the level of $L^1(\mathbb{G})$, we show that the center $\mathcal{Z}(L^1(\mathbb{G}))$ is precisely the closed linear span of the quantum characters for a large class of compact quantum groups, including arbitrary compact Kac algebras. In the latter setting, we show, in addition, that $\mathcal{Z}(L^1(\mathbb{G}))$ is a completely complemented $\mathcal{Z}(L^1(\mathbb{G}))$-submodule of $L^1(\mathbb{G})$.

Keywords:compact quantum group, irreducible character
Categories:43A20, 43A40, 46J40

17. CMB 2017 (vol 60 pp. 791)

Jiang, Chunlan
Reduction to Dimension Two of Local Spectrum for $AH$ Algebra with Ideal Property
A $C^{*}$-algebra $A$ has the ideal property if any ideal $I$ of $A$ is generated as a closed two sided ideal by the projections inside the ideal. Suppose that the limit $C^{*}$-algebra $A$ of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has ideal property. In this paper we will prove that $A$ can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension drop interval algebras and matrix algebras over 2-dimensional spaces with torsion $H^{2}$ groups.

Keywords:AH algebra, reduction, local spectrum, ideal property
Category:46L35

18. CMB 2017 (vol 60 pp. 690)

Bao, Guanlong; Göğüş, Nıhat Gökhan; Pouliasis, Stamatis
$\mathcal{Q}_p$ Spaces and Dirichlet Type Spaces
In this paper, we show that the Möbius invariant function space $\mathcal {Q}_p$ can be generated by variant Dirichlet type spaces $\mathcal{D}_{\mu, p}$ induced by finite positive Borel measures $\mu$ on the open unit disk. A criterion for the equality between the space $\mathcal{D}_{\mu, p}$ and the usual Dirichlet type space $\mathcal {D}_p$ is given. We obtain a sufficient condition to construct different $\mathcal{D}_{\mu, p}$ spaces and we provide examples. We establish decomposition theorems for $\mathcal{D}_{\mu, p}$ spaces, and prove that the non-Hilbert space $\mathcal {Q}_p$ is equal to the intersection of Hilbert spaces $\mathcal{D}_{\mu, p}$. As an application of the relation between $\mathcal {Q}_p$ and $\mathcal{D}_{\mu, p}$ spaces, we also obtain that there exist different $\mathcal{D}_{\mu, p}$ spaces; this is a trick to prove the existence without constructing examples.

Keywords:$\mathcal {Q}_p$ space, Dirichlet type space, Möbius invariant function space
Categories:30H25, 31C25, 46E15

19. CMB 2017 (vol 60 pp. 816)

Moslehian, Mohammad Sal; Zamani, Ali
Characterizations of Operator Birkhoff--James Orthogonality
In this paper, we obtain some characterizations of the (strong) Birkhoff--James orthogonality for elements of Hilbert $C^*$-modules and certain elements of $\mathbb{B}(\mathscr{H})$. Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for $T\in \mathbb{B}(\mathscr{H})$ we prove that if the norm attaining set $\mathbb{M}_T$ is a unit sphere of some finite dimensional subspace $\mathscr{H}_0$ of $\mathscr{H}$ and $\|T\|_{{{\mathscr{H}}_0}^\perp} \lt \|T\|$, then for every $S\in\mathbb{B}(\mathscr{H})$, $T$ is the strong Birkhoff--James orthogonal to $S$ if and only if there exists a unit vector $\xi\in {\mathscr{H}}_0$ such that $\|T\|\xi = |T|\xi$ and $S^*T\xi = 0$. Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product $C^*$-modules.

Keywords:Hilbert $C^*$-module, Birkhoff--James orthogonality, strong Birkhoff--James orthogonality, approximate orthogonality
Categories:46L05, 46L08, 46B20

20. CMB 2016 (vol 60 pp. 655)

Zhuo, Ciqiang; Sickel, Winfried; Yang, Dachun; Yuan, Wen
Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces via Averages on Balls
Let $\ell\in\mathbb N$ and $\alpha\in (0,2\ell)$. In this article, the authors establish equivalent characterizations of Besov-type spaces, Triebel-Lizorkin-type spaces and Besov-Morrey spaces via the sequence $\{f-B_{\ell,2^{-k}}f\}_{k}$ consisting of the difference between $f$ and the ball average $B_{\ell,2^{-k}}f$. These results give a way to introduce Besov-type spaces, Triebel-Lizorkin-type spaces and Besov-Morrey spaces with any smoothness order on metric measure spaces. As special cases, the authors obtain a new characterization of Morrey-Sobolev spaces and $Q_\alpha$ spaces with $\alpha\in(0,1)$, which are of independent interest.

Keywords:Besov space, Triebel-Lizorkin space, ball average, Calderón reproducing formula
Categories:42B25, 46E35, 42B35

21. CMB 2016 (vol 60 pp. 350)

Ma, Yumei
Isometry on Linear $n$-G-quasi Normed Spaces
This paper generalizes the Aleksandrov problem: the Mazur-Ulam theorem on $n$-G-quasi normed spaces. It proves that a one-$n$-distance preserving mapping is an $n$-isometry if and only if it has the zero-$n$-G-quasi preserving property, and two kinds of $n$-isometries on $n$-G-quasi normed space are equivalent; we generalize the Benz theorem to n-normed spaces with no restrictions on the dimension of spaces.

Keywords:$n$-G-quasi norm, Mazur-Ulam theorem, Aleksandrov problem, $n$-isometry, $n$-0-distance
Categories:46B20, 46B04, 51K05

22. CMB 2016 (vol 60 pp. 217)

Wang, Yuanyi
Condition $C'_{\wedge}$ of Operator Spaces
In this paper, we study condition $C'_{\wedge}$ which is a projective tensor product analogue of condition $C'$. We show that the finite-dimensional OLLP operator spaces have condition $C'_{\wedge}$ and $M_{n}$ $(n\gt 2)$ does not have that property.

Keywords:operator space, local theory, tensor product
Category:46L07

23. CMB 2016 (vol 60 pp. 104)

Diestel, Geoff
An Extension of Nikishin's Factorization Theorem
A Nikishin-Maurey characterization is given for bounded subsets of weak-type Lebesgue spaces. New factorizations for linear and multilinear operators are shown to follow.

Keywords:factorization, type, cotype, Banach spaces
Categories:46E30, 28A25

24. CMB 2016 (vol 60 pp. 122)

Ghanei, Mohammad Reza; Nasr-Isfahani, Rasoul; Nemati, Mehdi
A Homological Property and Arens Regularity of Locally Compact Quantum Groups
We characterize two important notions of amenability and compactness of a locally compact quantum group ${\mathbb G}$ in terms of certain homological properties. For this, we show that ${\mathbb G}$ is character amenable if and only if it is both amenable and co-amenable. We finally apply our results to Arens regularity problems of the quantum group algebra $L^1({\mathbb G})$; in particular, we improve an interesting result by Hu, Neufang and Ruan.

Keywords:amenability, Arens regularity, co-amenability, locally compact quantum group, homological property
Categories:46L89, 43A07, 46H20, 46M10, 58B32

25. CMB 2016 (vol 60 pp. 173)

Oubbi, Lahbib
On Ulam Stability of a Functional Equation in Banach Modules
Let $X$ and $Y$ be Banach spaces and $f : X \to Y$ an odd mapping. For any rational number $r \ne 2$, C. Baak, D. H. Boo, and Th. M. Rassias have proved the Hyers-Ulam stability of the following functional equation: \begin{align*} r f \left(\frac{\sum_{j=1}^d x_j}{r} \right) & + \sum_{\substack{i(j) \in \{0,1\} \\ \sum_{j=1}^d i(j)=\ell}} r f \left( \frac{\sum_{j=1}^d (-1)^{i(j)}x_j}{r} \right) = (C^\ell_{d-1} - C^{\ell -1}_{d-1} + 1) \sum_{j=1}^d f(x_j) \end{align*} where $d$ and $\ell$ are positive integers so that $1 \lt \ell \lt \frac{d}{2}$, and $C^p_q := \frac{q!}{(q-p)!p!}$, $p, q \in \mathbb{N}$ with $p \le q$. In this note we solve this equation for arbitrary nonzero scalar $r$ and show that it is actually Hyers-Ulam stable. We thus extend and generalize Baak et al.'s result. Different questions concerning the *-homomorphisms and the multipliers between C*-algebras are also considered.

Keywords:linear functional equation, Hyers-Ulam stability, Banach modules, C*-algebra homomorphisms.
Categories:39A30, 39B10, 39A06, 46Hxx
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