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1. CJM Online first

Stokke, Ross Thomas
Fourier spaces and completely isometric representations of Arens product algebras
Motivated by the definition of a semigroup compactification of a locally compact group and a large collection of examples, we introduce the notion of an (operator) ``homogeneous left dual Banach algebra" (HLDBA) over a (completely contractive) Banach algebra $A$. We prove a Gelfand-type representation theorem showing that every HLDBA over $A$ has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^*$ with a compatible (matrix) norm and a type of left Arens product ${\scriptstyle\square}$. Examples include all left Arens product algebras over $A$, but also -- when $A$ is the group algebra of a locally compact group -- the dual of its Fourier algebra. Beginning with any (completely) contractive (operator) $A$-module action $Q$ on a space $X$, we introduce the (operator) Fourier space $(\mathcal F_Q(A^*), \| \cdot \|_Q)$ and prove that $(\mathcal F_Q(A^*)^*, {\scriptstyle\square})$ is the unique (operator) HLDBA over $A$ for which there is a weak$^*$-continuous completely isometric representation as completely bounded operators on $X^*$ extending the dual module representation. Applying our theory to several examples of (completely contractive) Banach algebras $A$ and module operations, we provide new characterizations of familiar HLDBAs over $A$ and we recover -- and often extend -- some (completely) isometric representation theorems concerning these HLDBAs.

Keywords:Banach algebra, operator space, Arens product, group algebra, Fourier algebra
Categories:47L10, 43A20, 43A30, 46H15, 46H25, 47L25

2. CJM Online first

Cameron, Jan; Smith, Roger R.
A Galois correspondence for reduced crossed products of unital simple C$^*$-algebras by discrete groups
Let a discrete group $G$ act on a unital simple C$^*$-algebra $A$ by outer automorphisms. We establish a Galois correspondence $H\mapsto A\rtimes_{\alpha,r}H$ between subgroups of $G$ and C$^*$-algebras $B$ satisfying $A\subseteq B \subseteq A\rtimes_{\alpha,r}G$, where $A\rtimes_{\alpha,r}G$ denotes the reduced crossed product. For a twisted dynamical system $(A,G,\alpha,\sigma)$, we also prove the corresponding result for the reduced twisted crossed product $A\rtimes^\sigma_{\alpha,r}G$.

Keywords:C$^*$-algebra, group, crossed product, bimodule, reduced, twisted
Categories:46L55, 46L40

3. CJM Online first

Georgescu, Magdalena Cecilia
Integral Formula for Spectral Flow for $p$-Summable Operators
Fix a von Neumann algebra $\mathcal{N}$ equipped with a suitable trace $\tau$. For a path of self-adjoint Breuer-Fredholm operators, the spectral flow measures the net amount of spectrum which moves from negative to non-negative. We consider specifically the case of paths of bounded perturbations of a fixed unbounded self-adjoint Breuer-Fredholm operator affiliated with $\mathcal{N}$. If the unbounded operator is p-summable (that is, its resolvents are contained in the ideal $L^p$), then it is possible to obtain an integral formula which calculates spectral flow. This integral formula was first proven by Carey and Phillips, building on earlier approaches of Phillips. Their proof was based on first obtaining a formula for the larger class of $\theta$-summable operators, and then using Laplace transforms to obtain a p-summable formula. In this paper, we present a direct proof of the p-summable formula, which is both shorter and simpler than theirs.

Keywords:spectral flow, $p$-summable Fredholm module
Categories:19k56, 46L87, , 58B34

4. CJM Online first

Glöckner, Helge
Completeness of infinite-dimensional Lie groups in their left uniformity
We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence $G_1\subseteq G_2\subseteq\cdots$ of topological groups~$G_n$ such that~$G_n$ is a subgroup of $G_{n+1}$ and the latter induces the given topology on~$G_n$, for each $n\in\mathbb{N}$. Let $G$ be the direct limit of the sequence in the category of topological groups. We show that $G$ induces the given topology on each~$G_n$ whenever $\bigcup_{n\in \mathbb{N}}V_1V_2\cdots V_n$ is an identity neighbourhood in~$G$ for all identity neighbourhoods $V_n\subseteq G_n$. If, moreover, each $G_n$ is complete, then~$G$ is complete. We also show that the weak direct product $\bigoplus_{j\in J}G_j$ is complete for each family $(G_j)_{j\in J}$ of complete Lie groups~$G_j$. As a consequence, every strict direct limit $G=\bigcup_{n\in \mathbb{N}}G_n$ of finite-dimensional Lie groups is complete, as well as the diffeomorphism group $\operatorname{Diff}_c(M)$ of a paracompact finite-dimensional smooth manifold~$M$ and the test function group $C^k_c(M,H)$, for each $k\in\mathbb{N}_0\cup\{\infty\}$ and complete Lie group~$H$ modelled on a complete locally convex space.

Keywords:infinite-dimensional Lie group, left uniform structure, completeness
Categories:22E65, 22A05, 22E67, 46A13, 46M40, 58D05

5. CJM Online first

Wang, Xing; Zhang, Chunjie
Pointwise convergence of solutions to the Schrödinger equation on manifolds
Let $(M^n,g)$ be a Riemannian manifold without boundary. We study the amount of initial regularity is required so that the solution to free Schrödinger equation converges pointwise to its initial data. Assume the initial data is in $H^\alpha(M)$. For Hyperbolic Space, standard Sphere and the 2 dimensional Torus, we prove that $\alpha\gt \frac{1}{2}$ is enough. For general compact manifolds, due to lacking of local smoothing effect, it is hard to beat the bound $\alpha\gt 1$ from interpolation. We managed to go below 1 for dimension $\leq 3$. The more interesting thing is that, for 1 dimensional compact manifold, $\alpha\gt \frac{1}{3}$ is sufficient.

Keywords:pointwise convergence, Schrödinger operator, manifold, Strichartz estimate
Categories:35L05, 46E35, 42B37

6. CJM Online first

Mingo, James A.; Popa, Mihai
Freeness and The Partial Transposes of Wishart Random Matrices
We show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives a example where the partial transpose produces freeness at the operator level. Finally we investigate the case of real Wishart matrices.

Keywords:free probability, random matrix, partial transpose, quantum information theory
Categories:15B52, 46L54, 60B20

7. CJM Online first

Astashkin, Sergey V.; Lesnik, Karol; Maligranda, Lech
Isomorphic structure of Cesàro and Tandori spaces
We investigate the isomorphic structure of the Cesàro spaces and their duals, the Tandori spaces. The main result states that the Cesàro function space $Ces_{\infty}$ and its sequence counterpart $ces_{\infty}$ are isomorphic, which answers the question posted previously. This is rather surprising since $Ces_{\infty}$ (like the known Talagrand's example) has no natural lattice predual. We prove that $ces_{\infty}$ is not isomorphic to ${\ell}_{\infty}$ nor is $Ces_{\infty}$ isomorphic to the Tandori space $\widetilde{L_1}$ with the norm $\|f\|_{\widetilde{L_1}}= \|\widetilde{f}\|_{L_1},$ where $\widetilde{f}(t):= \operatorname{esssup}_{s \geq t} |f(s)|.$ Our investigation involves also an examination of the Schur and Dunford-Pettis properties of Cesàro and Tandori spaces. In particular, using results of Bourgain we show that a wide class of Cesàro-Marcinkiewicz and Cesàro-Lorentz spaces have the latter property.

Keywords:Cesàro and Tandori sequence spaces, Cesàro and Tandori function spaces, Cesàro operator, Banach ideal space, symmetric space, Schur property, Dunford-Pettis property, isomorphism
Categories:46E30, 46B20, 46B42, 46B45

8. CJM Online first

Matsumoto, Kengo
Asymptotic continuous orbit equivalence of Smale spaces and Ruelle algebras
In the first part of the paper, we introduce notions of asymptotic continuous orbit equivalence and asymptotic conjugacy in Smale spaces and characterize them in terms of their asymptotic Ruelle algebras with their dual actions. In the second part, we introduce a groupoid $C^*$-algebra which is an extended version of the asymptotic Ruelle algebra from a Smale space and study the extended Ruelle algebras from the view points of Cuntz-Krieger algebras. As a result, the asymptotic Ruelle algebra is realized as a fixed point algebra of the extended Ruelle algebra under certain circle action.

Keywords:hyperbolic dynamics, Smale space, Ruelle algebra, groupoid, zeta function, continuous orbit equivalence, shifts of finite type, Cuntz-Krieger algebra
Categories:37D20, 46L35

9. CJM 2018 (vol 70 pp. 1236)

Clouâtre, Raphaël
Unperforated Pairs of Operator Spaces and Hyperrigidity of Operator Systems
We study restriction and extension properties for states on C$^*$-algebras with an eye towards hyperrigidity of operator systems. We use these ideas to provide supporting evidence for Arveson's hyperrigidity conjecture. Prompted by various characterizations of hyperrigidity in terms of states, we examine unperforated pairs of self-adjoint subspaces in a C$^*$-algebra. The configuration of the subspaces forming an unperforated pair is in some sense compatible with the order structure of the ambient C$^*$-algebra. We prove that commuting pairs are unperforated, and obtain consequences for hyperrigidity. Finally, by exploiting recent advances in the tensor theory of operator systems, we show how the weak expectation property can serve as a flexible relaxation of the notion of unperforated pairs.

Keywords:operator system, state, peak point, hyperrigidity conjecture
Categories:46L07, 46L30, 46L52

10. CJM 2018 (vol 70 pp. 1261)

Fricain, Emmanuel; Hartmann, Andreas; Ross, William T.
Range Spaces of Co-analytic Toeplitz Operators
In this paper we discuss the range of a co-analytic Toeplitz operator. These range spaces are closely related to de Branges-Rovnyak spaces (in some cases they are equal as sets). In order to understand its structure, we explore when the range space decomposes into the range of an associated analytic Toeplitz operator and an identifiable orthogonal complement. For certain cases, we compute this orthogonal complement in terms of the kernel of a certain Toeplitz operator on the Hardy space where we focus on when this kernel is a model space (backward shift invariant subspace). In the spirit of Ahern-Clark, we also discuss the non-tangential boundary behavior in these range spaces. These results give us further insight into the description of the range of a co-analytic Toeplitz operator as well as its orthogonal decomposition. Our Ahern-Clark type results, which are stated in a general abstract setting, will also have applications to related sub-Hardy Hilbert spaces of analytic functions such as the de Branges-Rovnyak spaces and the harmonically weighted Dirichlet spaces.

Keywords:Toeplitz operator, Hardy space, range space, de Branges-Rovnyak space, boundary behavior, kernel function, non-extreme point, corona pair
Categories:30J05, 30H10, 46E22

11. CJM 2018 (vol 70 pp. 1008)

Elazar, Boaz; Shaviv, Ary
Schwartz Functions on Real Algebraic Varieties
We define Schwartz functions, tempered functions and tempered distributions on (possibly singular) real algebraic varieties. We prove that all classical properties of these spaces, defined previously on affine spaces and on Nash manifolds, also hold in the case of affine real algebraic varieties, and give partial results for the non-affine case.

Keywords:real algebraic geometry, Schwartz function, tempered distribution
Categories:14P99, 14P05, 22E45, 46A11, 46F05

12. CJM 2017 (vol 70 pp. 400)

Osaka, Hiroyuki; Teruya, Tamotsu
The Jiang-Su absorption for inclusions of unital C*-algebras
We introduce the tracial Rokhlin property for a conditional expectation for an inclusion of unital C*-algebras $P \subset A$ with index finite, and show that an action $\alpha$ from a finite group $G$ on a simple unital C*-algebra $A$ has the tracial Rokhlin property in the sense of N. C. Phillips if and only if the canonical conditional expectation $E\colon A \rightarrow A^G$ has the tracial Rokhlin property. Let $\mathcal{C}$ be a class of infinite dimensional stably finite separable unital C*-algebras which is closed under the following conditions: (1) If $A \in {\mathcal C}$ and $B \cong A$, then $B \in \mathcal{C}$. (2) If $A \in \mathcal{C}$ and $n \in \mathbb{N}$, then $M_n(A) \in \mathcal{C}$. (3) If $A \in \mathcal{C}$ and $p \in A$ is a nonzero projection, then $pAp \in \mathcal{C}$. Suppose that any C*-algebra in $\mathcal{C}$ is weakly semiprojective. We prove that if $A$ is a local tracial $\mathcal{C}$-algebra in the sense of Fan and Fang and a conditional expectation $E\colon A \rightarrow P$ is of index-finite type with the tracial Rokhlin property, then $P$ is a unital local tracial $\mathcal{C}$-algebra. The main result is that if $A$ is simple, separable, unital nuclear, Jiang-Su absorbing and $E\colon A \rightarrow P$ has the tracial Rokhlin property, then $P$ is Jiang-Su absorbing. As an application, when an action $\alpha$ from a finite group $G$ on a simple unital C*-algebra $A$ has the tracial Rokhlin property, then for any subgroup $H$ of $G$ the fixed point algebra $A^H$ and the crossed product algebra $A \rtimes_{\alpha_{|H}} H$ is Jiang-Su absorbing. We also show that the strict comparison property for a Cuntz semigroup $W(A)$ is hereditary to $W(P)$ if $A$ is simple, separable, exact, unital, and $E\colon A \rightarrow P$ has the tracial Rokhlin property.

Keywords:Jiang-Su absorption, inclusion of C*-algebra, strict comparison
Categories:46L55, 46L35

13. CJM Online first

Courtney, Kristin; Shulman, Tatiana
Elements of $C^*$-algebras attaining their norm in a finite-dimensional representation
We characterize the class of RFD $C^*$-algebras as those containing a dense subset of elements that attain their norm under a finite-dimensional representation. We show further that this subset is the whole space precisely when every irreducible representation of the $C^*$-algebra is finite-dimensional, which is equivalent to the $C^*$-algebra having no simple infinite-dimensional AF subquotient. We apply techniques from this proof to show the existence of elements in more general classes of $C^*$-algebras whose norms in finite-dimensional representations fit certain prescribed properties.

Keywords:AF-telescope, RFD, projective
Categories:46L05, 47A67

14. CJM Online first

Li, Hui; Yang, Dilian
Boundary quotient C*-algebras of products of odometers
In this paper, we study the boundary quotient C*-algebras associated to products of odometers. One of our main results shows that the boundary quotient C*-algebra of the standard product of $k$ odometers over $n_i$-letter alphabets ($1\le i\le k$) is always nuclear, and that it is a UCT Kirchberg algebra if and only if $\{\ln n_i: 1\le i\le k\}$ is rationally independent, if and only if the associated single-vertex $k$-graph C*-algebra is simple. To achieve this, one of our main steps is to construct a topological $k$-graph such that its associated Cuntz-Pimsner C*-algebra is isomorphic to the boundary quotient C*-algebra. Some relations between the boundary quotient C*-algebra and the C*-algebra $\mathrm{Q}_\mathbb{N}$ introduced by Cuntz are also investigated.

Keywords:C*-algebra; semigroup; odometer; topological $k$-graph; product system; Zappa-Szép product
Category:46L05

15. CJM 2017 (vol 70 pp. 3)

Benaych-Georges, Florent; Cébron, Guillaume; Rochet, Jean
Fluctuation of matrix entries and application to outliers of elliptic matrices
For any family of $N\times N$ random matrices $(\mathbf{A}_k)_{k\in K}$ which is invariant, in law, under unitary conjugation, we give general sufficient conditions for central limit theorems for random variables of the type $\operatorname{Tr}(\mathbf{A}_k \mathbf{M})$, where the matrix $\mathbf{M}$ is deterministic (such random variables include for example the normalized matrix entries of the $\mathbf{A}_k$'s). A consequence is the asymptotic independence of the projection of the matrices $\mathbf{A}_k$ onto the subspace of null trace matrices from their projections onto the orthogonal of this subspace. These results are used to study the asymptotic behavior of the outliers of a spiked elliptic random matrix. More precisely, we show that the fluctuations of these outliers around their limits can have various rates of convergence, depending on the Jordan Canonical Form of the additive perturbation. Also, some correlations can arise between outliers at a macroscopic distance from each other. These phenomena have already been observed with random matrices from the Single Ring Theorem.

Keywords:random matrix, Gaussian fluctuation, spiked model, elliptic random matrix, Weingarten calculus, Haar measure
Categories:60B20, 15B52, 60F05, 46L54

16. CJM Online first

Handelman, David
Nearly approximate transitivity (AT) for circulant matrices
By previous work of Giordano and the author, ergodic actions of $\mathbf Z$ (and other discrete groups) are completely classified measure-theoretically by their dimension space, a construction analogous to the dimension group used in C*-algebras and topological dynamics. Here we investigate how far from AT (approximately transitive) can actions be which derive from circulant (and related) matrices. It turns out not very: although non-AT actions can arise from this method of construction, under very modest additional conditions, ATness arises; in addition, if we drop the positivity requirement in the isomorphism of dimension spaces, then all these ergodic actions satisfy an analogue of AT. Many examples are provided.

Keywords:approximately transitive, ergodic transformation, circulant matrix, hemicirculant matrix, dimension space, matrix-valued random walk
Categories:37A05, 06F25, 28D05, 46B40, 60G50

17. CJM 2017 (vol 70 pp. 294)

Eilers, Søren; Restorff, Gunnar; Ruiz, Efren; Sørensen, Adam P. W.
Geometric classification of graph C*-algebras over finite graphs
We address the classification problem for graph $C^*$-algebras of finite graphs (finitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that the graphs satisfy the standard condition (K), so that the graph $C^*$-algebras may come with uncountably many ideals. We find that in this generality, stable isomorphism of graph $C^*$-algebras does not coincide with the geometric notion of Cuntz move equivalence. However, adding a modest condition on the graphs, the two notions are proved to be mutually equivalent and equivalent to the $C^*$-algebras having isomorphic $K$-theories. This proves in turn that under this condition, the graph $C^*$-algebras are in fact classifiable by $K$-theory, providing in particular complete classification when the $C^*$-algebras in question are either of real rank zero or type I/postliminal. The key ingredient in obtaining these results is a characterization of Cuntz move equivalence using the adjacency matrices of the graphs. Our results are applied to discuss the classification problem for the quantum lens spaces defined by Hong and Szymański, and to complete the classification of graph $C^*$-algebras associated to all simple graphs with four vertices or less.

Keywords:graph $C^*$-algebra, geometric classification, $K$-theory, flow equivalence
Categories:46L35, 46L80, 46L55, 37B10

18. CJM 2017 (vol 69 pp. 1385)

Pasnicu, Cornel; Phillips, N. Christopher
The Weak Ideal Property and Topological Dimension Zero
Following up on previous work, we prove a number of results for C*-algebras with the weak ideal property or topological dimension zero, and some results for C*-algebras with related properties. Some of the more important results include: $\bullet$ The weak ideal property implies topological dimension zero. $\bullet$ For a separable C*-algebra~$A$, topological dimension zero is equivalent to ${\operatorname{RR}} ({\mathcal{O}}_2 \otimes A) = 0$, to $D \otimes A$ having the ideal property for some (or any) Kirchberg algebra~$D$, and to $A$ being residually hereditarily in the class of all C*-algebras $B$ such that ${\mathcal{O}}_{\infty} \otimes B$ contains a nonzero projection. $\bullet$ Extending the known result for ${\mathbb{Z}}_2$, the classes of C*-algebras with residual (SP), which are residually hereditarily (properly) infinite, or which are purely infinite and have the ideal property, are closed under crossed products by arbitrary actions of abelian $2$-groups. $\bullet$ If $A$ and $B$ are separable, one of them is exact, $A$ has the ideal property, and $B$ has the weak ideal property, then $A \otimes_{\mathrm{min}} B$ has the weak ideal property. $\bullet$ If $X$ is a totally disconnected locally compact Hausdorff space and $A$ is a $C_0 (X)$-algebra all of whose fibers have one of the weak ideal property, topological dimension zero, residual (SP), or the combination of pure infiniteness and the ideal property, then $A$ also has the corresponding property (for topological dimension zero, provided $A$ is separable). $\bullet$ Topological dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable C*-algebras including all separable locally AH~algebras. $\bullet$ The weak ideal property does not imply the ideal property for separable $Z$-stable C*-algebras. We give other related results, as well as counterexamples to several other statements one might hope for.

Keywords:ideal property, weak ideal property, topological dimension zero, $C_0 (X)$-algebra, purely infinite C*-algebra
Category:46L05

19. CJM 2017 (vol 70 pp. 53)

Dantas, Sheldon; García, Domingo; Maestre, Manuel; Martín, Miguel
The Bishop-Phelps-Bollobás property for compact operators
We study the Bishop-Phelps-Bollobás property (BPBp for short) for compact operators. We present some abstract techniques which allows to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$ has the BPBp for compact operators, then so do $(C_0(L),Y)$ for every locally compact Hausdorff topological space $L$ and $(X,Y)$ whenever $X^*$ is isometrically isomorphic to $\ell_1$. If $X^*$ has the Radon-Nikodým property and $(\ell_1(X),Y)$ has the BPBp for compact operators, then so does $(L_1(\mu,X),Y)$ for every positive measure $\mu$; as a consequence, $(L_1(\mu,X),Y)$ has the the BPBp for compact operators when $X$ and $Y$ are finite-dimensional or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(\nu)$ for any positive measure $\nu$ and $1\lt p\lt \infty$. For $1\leq p \lt \infty$, if $(X,\ell_p(Y))$ has the BPBp for compact operators, then so does $(X,L_p(\mu,Y))$ for every positive measure $\mu$ such that $L_1(\mu)$ is infinite-dimensional. If $(X,Y)$ has the BPBp for compact operators, then so do $(X,L_\infty(\mu,Y))$ for every $\sigma$-finite positive measure $\mu$ and $(X,C(K,Y))$ for every compact Hausdorff topological space $K$.

Keywords:Bishop-Phelps theorem, Bishop-Phelps-Bollobás property, norm attaining operator, compact operator
Categories:46B04, 46B20, 46B28, 46B25, 46E40

20. CJM 2017 (vol 70 pp. 26)

Bosa, Joan; Petzka, Henning
Comparison Properties of the Cuntz semigroup and applications to C*-algebras
We study comparison properties in the category $\mathrm{Cu}$ aiming to lift results to the C*-algebraic setting. We introduce a new comparison property and relate it to both the CFP and $\omega$-comparison. We show differences of all properties by providing examples, which suggest that the corona factorization for C*-algebras might allow for both finite and infinite projections. In addition, we show that R{\o}rdam's simple, nuclear C*-algebra with a finite and an infinite projection does not have the CFP.

Keywords:classification of C*-algebras, cuntz semigroup
Categories:46L35, 06F05, 46L05, 19K14

21. CJM 2017 (vol 70 pp. 721)

Bao, Guanlong; Göğüş, Nihat Gökhan; Pouliasis, Stamatis
On Dirichlet Spaces with a Class of Superharmonic Weights
In this paper, we investigate Dirichlet spaces $\mathcal{D}_\mu$ with superharmonic weights induced by positive Borel measures $\mu$ on the open unit disk. We establish the Alexander-Taylor-Ullman inequality for $\mathcal{D}_\mu$ spaces and we characterize the cases where equality occurs. We define a class of weighted Hardy spaces $H_{\mu}^{2}$ via the balayage of the measure $\mu$. We show that $\mathcal{D}_\mu$ is equal to $H_{\mu}^{2}$ if and only if $\mu$ is a Carleson measure for $\mathcal{D}_\mu$. As an application, we obtain the reproducing kernel of $\mathcal{D}_\mu$ when $\mu$ is an infinite sum of point mass measures. We consider the boundary behavior and inner-outer factorization of functions in $\mathcal{D}_\mu$. We also characterize the boundedness and compactness of composition operators on $\mathcal{D}_\mu$.

Keywords:Dirichlet space, Hardy space, superharmonic weight
Categories:30H10, 31C25, 46E15

22. CJM 2017 (vol 69 pp. 1312)

Fricain, Emmanuel; Rupam, Rishika
On Asymptotically Orthonormal Sequences
An asymptotically orthonormal sequence is a sequence which is "nearly" orthonormal in the sense that it satisfies the Parseval equality up to two constants close to one. In this paper, we explore such sequences formed by normalized reproducing kernels for model spaces and de Branges-Rovnyak spaces.

Keywords:function space, de Branges-Rovnyak and model space, reproducing kernel, asymptotically orthonormal sequence
Categories:30J05, 30H10, 46E22

23. CJM 2017 (vol 70 pp. 804)

Giannopoulos, Apostolos; Koldobsky, Alexander; Valettas, Petros
Inequalities for the Surface Area of Projections of Convex Bodies
We provide general inequalities that compare the surface area $S(K)$ of a convex body $K$ in ${\mathbb R}^n$ to the minimal, average or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of $K$. We examine separately the dependence of the constants on the dimension in the case where $K$ is in some of the classical positions or $K$ is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

Keywords:surface area, convex body, projection
Categories:52A20, 46B05

24. CJM 2017 (vol 69 pp. 1109)

Ng, P. W.; Skoufranis, P.
Closed Convex Hulls of Unitary Orbits in Certain Simple Real Rank Zero C$^*$-algebras
In this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple C$^*$-algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of projections with respect to tracial states. In addition, an upper bound for the number of unitary conjugates in a convex combination needed to approximate a self-adjoint are obtained.

Keywords:convex hull of unitary orbits, real rank zero C*-algebras simple, eigenvalue function, majorization
Category:46L05

25. CJM 2017 (vol 69 pp. 1064)

Crann, Jason
Amenability and Covariant Injectivity of Locally Compact Quantum Groups II
Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group $\mathbb{G}$ and 1-injectivity of $L^{\infty}(\widehat{\mathbb{G}})$ as an operator $L^1(\widehat{\mathbb{G}})$-module. In particular, a locally compact group $G$ is amenable if and only if its group von Neumann algebra $VN(G)$ is 1-injective as an operator module over the Fourier algebra $A(G)$. As an application, we provide a decomposability result for completely bounded $L^1(\widehat{\mathbb{G}})$-module maps on $L^{\infty}(\widehat{\mathbb{G}})$, and give a simplified proof that amenable discrete quantum groups have co-amenable compact duals which avoids the use of modular theory and the Powers--Størmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.

Keywords:locally compact quantum group, amenability, injective module
Categories:22D35, 46M10, 46L89
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