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1. 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

2. CJM Online first

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

3. 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

4. CJM Online first

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

5. 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

6. CJM Online first

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

7. 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

8. CJM Online first

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

9. CJM Online first

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

10. CJM Online first

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

11. CJM Online first

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

12. CJM Online first

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

13. CJM Online first

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

14. CJM Online first

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

15. 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

16. 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

17. CJM 2016 (vol 69 pp. 650)

Oikhberg, Timur; Tradacete, Pedro
Almost Disjointness Preservers
We study the stability of disjointness preservers on Banach lattices. In many cases, we prove that an "almost disjointness preserving" operator is well approximable by a disjointness preserving one. However, this approximation is not always possible, as our examples show.

Keywords:Banach lattice, disjointness preserving
Categories:47B38, 46B42

18. CJM 2016 (vol 69 pp. 548)

Hartglass, Michael
Free Product C*-algebras Associated with Graphs, Free Differentials, and Laws of Loops
We study a canonical C$^*$-algebra, $\mathcal{S}(\Gamma, \mu)$, that arises from a weighted graph $(\Gamma, \mu)$, specific cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting which ensure simplicity and uniqueness of trace of $\mathcal{S}(\Gamma, \mu)$, and study the structure of its positive cone. We then study the $*$-algebra, $\mathcal{A}$, generated by the generators of $\mathcal{S}(\Gamma, \mu)$, and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko, as well as Mai, Speicher, and Weber to show that certain ``loop" elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements $x \in M_{n}(\mathcal{A})$ have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials in Wishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

Keywords:free probability, C*-algebra

19. CJM 2016 (vol 68 pp. 999)

Izumi, Masaki; Morrison, Scott; Penneys, David
Quotients of $A_2 * T_2$
We study unitary quotients of the free product unitary pivotal category $A_2*T_2$. We show that such quotients are parametrized by an integer $n\geq 1$ and an $2n$-th root of unity $\omega$. We show that for $n=1,2,3$, there is exactly one quotient and $\omega=1$. For $4\leq n\leq 10$, we show that there are no such quotients. Our methods also apply to quotients of $T_2*T_2$, where we have a similar result. The essence of our method is a consistency check on jellyfish relations. While we only treat the specific cases of $A_2 * T_2$ and $T_2 * T_2$, we anticipate that our technique can be extended to a general method for proving nonexistence of planar algebras with a specified principal graph. During the preparation of this manuscript, we learnt of Liu's independent result on composites of $A_3$ and $A_4$ subfactor planar algebras (arxiv:1308.5691). In 1994, Bisch-Haagerup showed that the principal graph of a composite of $A_3$ and $A_4$ must fit into a certain family, and Liu has classified all such subfactor planar algebras. We explain the connection between the quotient categories and the corresponding composite subfactor planar algebras. As a corollary of Liu's result, there are no such quotient categories for $n\geq 4$. This is an abridged version of arxiv:1308.5723.

Keywords:pivotal category, free product, quotient, subfactor, intermediate subfactor

20. CJM 2016 (vol 69 pp. 373)

Kaftal, Victor; Ng, Ping Wong; Zhang, Shuang
Strict Comparison of Positive Elements in Multiplier Algebras
Main result: If a C*-algebra $\mathcal{A}$ is simple, $\sigma$-unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier algebra $\operatorname{\mathcal{M}}(\mathcal{A})$ also has strict comparison of positive elements by traces. The same results holds if ``finitely many extremal traces" is replaced by ``quasicontinuous scale". A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary $\sigma$-unital C*-algebra can be approximated by a bi-diagonal series. An application of strict comparison: If $\mathcal{A}$ is a simple separable stable C*-algebra with real rank zero, stable rank one, and strict comparison of positive elements by traces, then whether a positive element is a positive linear combination of projections is determined by the trace values of its range projection.

Keywords:strict comparison, bi-diagonal form, positive combinations
Categories:46L05, 46L35, 46L45, 47C15

21. CJM 2016 (vol 68 pp. 1023)

Phillips, John; Raeburn, Iain
Centre-valued Index for Toeplitz Operators with Noncommuting Symbols
We formulate and prove a ``winding number'' index theorem for certain ``Toeplitz'' operators in the same spirit as Gohberg-Krein, Lesch and others. The ``number'' is replaced by a self-adjoint operator in a subalgebra $Z\subseteq Z(A)$ of a unital $C^*$-algebra, $A$. We assume a faithful $Z$-valued trace $\tau$ on $A$ left invariant under an action $\alpha:{\mathbf R}\to Aut(A)$ leaving $Z$ pointwise fixed.If $\delta$ is the infinitesimal generator of $\alpha$ and $u$ is invertible in $\operatorname{dom}(\delta)$ then the ``winding operator'' of $u$ is $\frac{1}{2\pi i}\tau(\delta(u)u^{-1})\in Z_{sa}.$ By a careful choice of representations we extend $(A,Z,\tau,\alpha)$ to a von Neumann setting $(\mathfrak{A},\mathfrak{Z},\bar\tau,\bar\alpha)$ where $\mathfrak{A}=A^{\prime\prime}$ and $\mathfrak{Z}=Z^{\prime\prime}.$ Then $A\subset\mathfrak{A}\subset \mathfrak{A}\rtimes{\bf R}$, the von Neumann crossed product, and there is a faithful, dual $\mathfrak{Z}$-trace on $\mathfrak{A}\rtimes{\bf R}$. If $P$ is the projection in $\mathfrak{A}\rtimes{\bf R}$ corresponding to the non-negative spectrum of the generator of $\mathbf R$ inside $\mathfrak{A}\rtimes{\mathbf R}$ and $\tilde\pi:A\to\mathfrak{A}\rtimes{\mathbf R}$ is the embedding then we define for $u\in A^{-1}$, $T_u=P\tilde\pi(u) P$ and show it is Fredholm in an appropriate sense and the $\mathfrak{Z}$-valued index of $T_u$ is the negative of the winding operator. In outline the proof follows the proof of the scalar case done previously by the authors. The main difficulty is making sense of the constructions with the scalars replaced by $\mathfrak{Z}$ in the von Neumann setting. The construction of the dual $\mathfrak{Z}$-trace on $\mathfrak{A}\rtimes{\mathbf R}$ required the nontrivial development of a $\mathfrak{Z}$-Hilbert Algebra theory. We show that certain of these Fredholm operators fiber as a ``section'' of Fredholm operators with scalar-valued index and the centre-valued index fibers as a section of the scalar-valued indices.

Keywords:index ,Toeplitz operator
Categories:46L55, 19K56, 46L80

22. CJM 2016 (vol 69 pp. 54)

Hartz, Michael
On the Isomorphism Problem for Multiplier Algebras of Nevanlinna-Pick Spaces
We continue the investigation of the isomorphism problem for multiplier algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick property. In contrast to previous work in this area, we do not study these spaces by identifying them with restrictions of a universal space, namely the Drury-Arveson space. Instead, we work directly with the Hilbert spaces and their reproducing kernels. In particular, we show that two multiplier algebras of Nevanlinna-Pick spaces on the same set are equal if and only if the Hilbert spaces are equal. Most of the article is devoted to the study of a special class of complete Nevanlinna-Pick spaces on homogeneous varieties. We provide a complete answer to the question of when two multiplier algebras of spaces of this type are algebraically or isometrically isomorphic. This generalizes results of Davidson, Ramsey, Shalit, and the author.

Keywords:non-selfadjoint operator algebras, reproducing kernel Hilbert spaces, multiplier algebra, Nevanlinna-Pick kernels, isomorphism problem
Categories:47L30, 46E22, 47A13

23. CJM 2016 (vol 68 pp. 1067)

Runde, Volker; Viselter, Ami
On Positive Definiteness over Locally Compact Quantum Groups
The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on "square roots" of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.

Keywords:bicrossed product, locally compact quantum group, non-commutative $L^p$-space, positive-definite function, positive-definite measure, separation property
Categories:20G42, 22D25, 43A35, 46L51, 46L52, 46L89

24. CJM 2016 (vol 68 pp. 698)

Skalski, Adam; Sołtan, Piotr
Quantum Families of Invertible Maps and Related Problems
The notion of families of quantum invertible maps (C$^*$-algebra homomorphisms satisfying Podleś' condition) is employed to strengthen and reinterpret several results concerning universal quantum groups acting on finite quantum spaces. In particular Wang's quantum automorphism groups are shown to be universal with respect to quantum families of invertible maps. Further the construction of the Hopf image of Banica and Bichon is phrased in the purely analytic language and employed to define the quantum subgroup generated by a family of quantum subgroups or more generally a family of quantum invertible maps.

Keywords:quantum families of invertible maps, Hopf image, universal quantum group
Categories:46L89, 46L65

25. CJM 2016 (vol 69 pp. 321)

De Bernardi, Carlo Alberto; Veselý, Libor
Tilings of Normed Spaces
By a tiling of a topological linear space $X$ we mean a covering of $X$ by at least two closed convex sets, called tiles, whose nonempty interiors are pairwise disjoint. Study of tilings of infinite-dimensional spaces initiated in the 1980's with pioneer papers by V. Klee. We prove some general properties of tilings of locally convex spaces, and then apply these results to study existence of tilings of normed and Banach spaces by tiles possessing certain smoothness or rotundity properties. For a Banach space $X$, our main results are the following. 1. $X$ admits no tiling by Fréchet smooth bounded tiles. 2. If $X$ is locally uniformly rotund (LUR), it does not admit any tiling by balls. 3. On the other hand, some $\ell_1(\Gamma)$ spaces, $\Gamma$ uncountable, do admit a tiling by pairwise disjoint LUR bounded tiles.

Keywords:tiling of normed space, Fréchet smooth body, locally uniformly rotund body, $\ell_1(\Gamma)$-space
Categories:46B20, 52A99, 46A45
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