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1. CMB Online first

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

2. 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

3. 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

4. CMB Online first

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

5. CMB Online first

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

6. CMB Online first

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

7. CMB Online first

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

8. CMB Online first

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

9. CMB Online first

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

10. CMB Online first

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

11. CMB Online first

Sickel, Winfried; Yang, Dachun; Yuan, Wen; Zhuo, Ciqiang
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

12. 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

13. 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

14. 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

15. 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

16. 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

17. CMB Online first

Liu, Feng; Wu, Huoxiong
Endpoint Regularity of Multisublinear Fractional Maximal Functions
In this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy-Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on vector-valued function $\vec{f}=(f_1,\dots,f_m)$ with all $f_j$ being $BV$-functions.

Keywords:multisublinear fractional maximal operators, Sobolev spaces, bounded variation
Categories:42B25, 46E35

18. CMB 2016 (vol 60 pp. 77)

Christ, Michael; Rieffel, Marc A.
Nilpotent Group C*-algebras as Compact Quantum Metric Spaces
Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$. Following Connes, $M_\mathbb{L}$ can be used as a ``Dirac'' operator for the reduced group C*-algebra $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a ``compact quantum metric space''). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

Keywords:group C*-algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growth
Categories:46L87, 20F65, 22D15, 53C23, 58B34

19. CMB 2016 (vol 59 pp. 769)

García-Pacheco, Francisco Javier; Hill, Justin R.
Geometric Characterizations of Hilbert Spaces
We study some geometric properties related to the set $\Pi_X:= \{ (x,x^* )\in\mathsf{S}_X\times \mathsf{S}_{X^*}:x^* (x )=1 \}$ obtaining two characterizations of Hilbert spaces in the category of Banach spaces. We also compute the distance of a generic element $ (h,k )\in H\oplus_2 H$ to $\Pi_H$ for $H$ a Hilbert space.

Keywords:Hilbert space, extreme point, smooth, $\mathsf{L}^2$-summands
Categories:46B20, 46C05

20. CMB 2016 (vol 59 pp. 606)

Mihăilescu, Mihai; Moroşanu, Gheorghe
Eigenvalues of $ -\Delta_p -\Delta_q $ Under Neumann Boundary Condition
The eigenvalue problem $-\Delta_p u-\Delta_q u=\lambda|u|^{q-2}u$ with $p\in(1,\infty)$, $q\in(2,\infty)$, $p\neq q$ subject to the corresponding homogeneous Neumann boundary condition is investigated on a bounded open set with smooth boundary from $\mathbb{R}^N$ with $N\geq 2$. A careful analysis of this problem leads us to a complete description of the set of eigenvalues as being a precise interval $(\lambda_1, +\infty )$ plus an isolated point $\lambda =0$. This comprehensive result is strongly related to our framework which is complementary to the well-known case $p=q\neq 2$ for which a full description of the set of eigenvalues is still unavailable.

Keywords:eigenvalue problem, Sobolev space, Nehari manifold, variational methods
Categories:35J60, 35J92, 46E30, 49R05

21. CMB 2016 (vol 59 pp. 878)

Wang, Jianfei
The Carleson Measure Problem Between Analytic Morrey Spaces
The purpose of this paper is to characterize positive measure $\mu$ on the unit disk such that the analytic Morrey space $\mathcal{AL}_{p,\eta}$ is boundedly and compactly embedded to the tent space $\mathcal{T}_{q,1-\frac{q}{p}(1-\eta)}^{\infty}(\mu)$ for the case $1\leq q\leq p\lt \infty$ respectively. As an application, these results are used to establish the boundedness and compactness of integral operators and multipliers between analytic Morrey spaces.

Keywords:Morrey space, Carleson measure problem, boundedness, compactness
Categories:30H35, 28A12, 47B38, 46E15

22. CMB 2016 (vol 59 pp. 320)

Ino, Shoji
Perturbations of Von Neumann Subalgebras with Finite Index
In this paper, we study uniform perturbations of von Neumann subalgebras of a von Neumann algebra. Let $M$ and $N$ be von Neumann subalgebras of a von Neumann algebra with finite probabilistic index in the sense of Pimsner-Popa. If $M$ and $N$ are sufficiently close, then $M$ and $N$ are unitarily equivalent. The implementing unitary can be chosen as being close to the identity.

Keywords:von Neumann algebras, perturbations
Categories:46L10, 46L37

23. CMB 2015 (vol 59 pp. 435)

Yao, Hongliang
On Extensions of Stably Finite C*-algebras (II)
For any $C^*$-algebra $A$ with an approximate unit of projections, there is a smallest ideal $I$ of $A$ such that the quotient $A/I$ is stably finite. In this paper, a sufficient and necessary condition is obtained for an ideal of a $C^*$-algebra with real rank zero is this smallest ideal by $K$-theory.

Keywords:extension, stably finite C*-algebra, index map
Categories:46L05, 46L80

24. CMB 2015 (vol 59 pp. 104)

He, Ziyi; Yang, Dachun; Yuan, Wen
Littlewood-Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls
In this paper, the authors characterize second-order Sobolev spaces $W^{2,p}({\mathbb R}^n)$, with $p\in [2,\infty)$ and $n\in\mathbb N$ or $p\in (1,2)$ and $n\in\{1,2,3\}$, via the Lusin area function and the Littlewood-Paley $g_\lambda^\ast$-function in terms of ball means.

Keywords:Sobolev space, ball means, Lusin-area function, $g_\lambda^*$-function
Categories:46E35, 42B25, 42B20, 42B35

25. CMB 2015 (vol 58 pp. 846)

Sundar, S.
A Computation with the Connes-Thom Isomorphism
Let $A \in M_{n}(\mathbb{R})$ be an invertible matrix. Consider the semi-direct product $\mathbb{R}^{n} \rtimes \mathbb{Z}$ where the action of $\mathbb{Z}$ on $\mathbb{R}^{n}$ is induced by the left multiplication by $A$. Let $(\alpha,\tau)$ be a strongly continuous action of $\mathbb{R}^{n} \rtimes \mathbb{Z}$ on a $C^{*}$-algebra $B$ where $\alpha$ is a strongly continuous action of $\mathbb{R}^{n}$ and $\tau$ is an automorphism. The map $\tau$ induces a map $\widetilde{\tau}$ on $B \rtimes_{\alpha} \mathbb{R}^{n}$. We show that, at the $K$-theory level, $\tau$ commutes with the Connes-Thom map if $\det(A)\gt 0$ and anticommutes if $\det(A)\lt 0$. As an application, we recompute the $K$-groups of the Cuntz-Li algebra associated to an integer dilation matrix.

Keywords:K-theory, Connes-Thom isomorphism, Cuntz-Li algebras
Categories:46L80, 58B34
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