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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 algebraCategories:47L10, 43A20, 43A30, 46H15, 46H25, 47L25

2. CJM Online first

Bourhim, Abdellatif; Costara, Constantin
 Linear maps preserving matrices of local spectral radius zero at a fixed vector In this paper, we characterize linear maps on matrix spaces which preserve matrices of local spectral radius zero at some fixed nonzero vector. Keywords:linear preserver, local spectrum, local spectral radius, matrixCategories:47B49, 47A10, 47A11

3. CJM Online first

Bickerton, Robert T.; Kakariadis, Evgenios T.A.
 Free Multivariate w*-Semicrossed Products: Reflexivity and the Bicommutant Property We study w*-semicrossed products over actions of the free semigroup and the free abelian semigroup on (possibly non-selfadjoint) w*-closed algebras. We show that they are reflexive when the dynamics are implemented by uniformly bounded families of invertible row operators. Combining with results of Helmer we derive that w*-semicrossed products of factors (on a separable Hilbert space) are reflexive. Furthermore we show that w*-semicrossed products of automorphic actions on maximal abelian selfadjoint algebras are reflexive. In all cases we prove that the w*-semicrossed products have the bicommutant property if and only if the ambient algebra of the dynamics does also. Keywords:reflexivity, semicrossed productCategories:47A15, 47L65, 47L75, 47L80

4. 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, projectiveCategories:46L05, 47A67

5. CJM 2017 (vol 70 pp. 515)

Chen, Yanni; Hadwin, Don; Liu, Zhe; Nordgren, Eric
 A Beurling Theorem for Generalized Hardy Spaces on a Multiply Connected Domain The object of this paper is to prove a version of the Beurling-Helson-Lowdenslager invariant subspace theorem for operators on certain Banach spaces of functions on a multiply connected domain in $\mathbb{C}$. The norms for these spaces are either the usual Lebesgue and Hardy space norms or certain continuous gauge norms. In the Hardy space case the expected corollaries include the characterization of the cyclic vectors as the outer functions in this context, a demonstration that the set of analytic multiplication operators is maximal abelian and reflexive, and a determination of the closed operators that commute with all analytic multiplication operators. Keywords:Beurling theorem, invariant subspace, generalized Hardy space, gauge norm, multiply connected domain, Forelli projection, inner-outer factorization, affiliated operatorCategories:47L10, 30H10

6. CJM 2017 (vol 70 pp. 97)

Ghaani Farashahi, Arash
 A Class of Abstract Linear Representations for Convolution Function Algebras over Homogeneous Spaces of Compact Groups This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $\mu$ be the normalized $G$-invariant measure over the compact homogeneous space $G/H$ associated to the Weil's formula and $1\le p\lt \infty$. We then present a structured class of abstract linear representations of the Banach convolution function algebras $L^p(G/H,\mu)$. Keywords:homogeneous space, linear representation, continuous unitary representation, convolution function algebra, compact group, convolution, involutionCategories:43A85, 47A67, 20G05

7. CJM 2016 (vol 69 pp. 1422)

Šemrl, Peter
 Order and Spectrum Preserving Maps on Positive Operators We describe the general form of surjective maps on the cone of all positive operators which preserve order and spectrum. The result is optimal as shown by counterexamples. As an easy consequence we characterize surjective order and spectrum preserving maps on the set of all self-adjoint operators. Keywords:spectrum preserver, order preserver, positive operatorCategory:47B49

8. CJM 2016 (vol 69 pp. 650)

 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 preservingCategories:47B38, 46B42

9. CJM 2016 (vol 69 pp. 434)

Lee, Hun Hee; Youn, Sang-gyun
 New Deformations of Convolution Algebras and Fourier Algebras on Locally Compact Groups In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some information of the underlying groups by examining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similarly, we investigate complete representability as an operator algebra of deformed Fourier algebras on some finitely generated discrete groups with connection to the growth rate of the group. Keywords:Fourier algebra, convolution algebra, operator algebra, Beurling algebraCategories:43A20, 43A30, 47L30, 47L25

10. 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 combinationsCategories:46L05, 46L35, 46L45, 47C15

11. 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 problemCategories:47L30, 46E22, 47A13

12. CJM 2016 (vol 68 pp. 1257)

Cascante, Carme; Fàbrega, Joan; Ortega, Joaquín M.
 Sharp Norm Estimates for the Bergman Operator from Weighted Mixed-norm Spaces to Weighted Hardy Spaces In this paper we give sharp norm estimates for the Bergman operator acting from weighted mixed-norm spaces to weighted Hardy spaces in the ball, endowed with natural norms. Keywords:weighted Hardy space, Bergman operator, sharp norm estimateCategories:47B38, 32A35, 42B25, 32A37

13. CJM 2016 (vol 68 pp. 816)

Guo, Xiaoli; Hu, Guoen
 On the Commutators of Singular Integral Operators with Rough Convolution Kernels Let $T_{\Omega}$ be the singular integral operator with kernel $\frac{\Omega(x)}{|x|^n}$, where $\Omega$ is homogeneous of degree zero, has mean value zero and belongs to $L^q(S^{n-1})$ for some $q\in (1,\,\infty]$. In this paper, the authors establish the compactness on weighted $L^p$ spaces, and the Morrey spaces, for the commutator generated by $\operatorname{CMO}(\mathbb{R}^n)$ function and $T_{\Omega}$. The associated maximal operator and the discrete maximal operator are also considered. Keywords:commutator, singular integral operator, compact operator, completely continuous operator, maximal operator, Morrey spaceCategories:42B20, 47B07

14. CJM 2016 (vol 68 pp. 309)

Daws, Matthew
 Categorical Aspects of Quantum Groups: Multipliers and Intrinsic Groups We show that the assignment of the (left) completely bounded multiplier algebra $M_{cb}^l(L^1(\mathbb G))$ to a locally compact quantum group $\mathbb G$, and the assignment of the intrinsic group, form functors between appropriate categories. Morphisms of locally compact quantum groups can be described by Hopf $*$-homomorphisms between universal $C^*$-algebras, by bicharacters, or by special sorts of coactions. We show that the whole theory of completely bounded multipliers can be lifted to the universal $C^*$-algebra level, and that then the different pictures of both multipliers (reduced, universal, and as centralisers) and morphisms interact in extremely natural ways. The intrinsic group of a quantum group can be realised as a class of multipliers, and so our techniques immediately apply. We also show how to think of the intrinsic group using the universal $C^*$-algebra picture, and then, again, show how the differing views on the intrinsic group interact naturally with morphisms. We show that the intrinsic group is the maximal classical'' quantum subgroup of a locally compact quantum group, show that it is even closed in the strong Vaes sense, and that the intrinsic group functor is an adjoint to the inclusion functor from locally compact groups to quantum groups. Keywords:locally compact quantum group, morphism, intrinsic group, multiplier, centraliserCategories:20G42, 22D25, 43A22, 43A35, 43A95, 46L52, 46L89, 47L25

15. CJM 2015 (vol 69 pp. 408)

Klep, Igor; Špenko, Špela
 Free Function Theory Through Matrix Invariants This paper concerns free function theory. Free maps are free analogs of analytic functions in several complex variables, and are defined in terms of freely noncommuting variables. A function of $g$ noncommuting variables is a function on $g$-tuples of square matrices of all sizes that respects direct sums and simultaneous conjugation. Examples of such maps include noncommutative polynomials, noncommutative rational functions and convergent noncommutative power series. In sharp contrast to the existing literature in free analysis, this article investigates free maps \emph{with involution} -- free analogs of real analytic functions. To get a grip on these, techniques and tools from invariant theory are developed and applied to free analysis. Here is a sample of the results obtained. A characterization of polynomial free maps via properties of their finite-dimensional slices is presented and then used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are series of generalized polynomials for which an invariant-theoretic characterization is given. Furthermore, an inverse and implicit function theorem for free maps with involution is obtained. Finally, with a selection of carefully chosen examples it is shown that free maps with involution do not exhibit strong rigidity properties enjoyed by their involution-free counterparts. Keywords:free algebra, free analysis, invariant theory, polynomial identities, trace identities, concomitants, analytic maps, inverse function theorem, generalized polynomialsCategories:16R30, 32A05, 46L52, 15A24, 47A56, 15A24, 46G20

16. CJM 2015 (vol 67 pp. 1384)

Graczyk, Piotr; Kemp, Todd; Loeb, Jean-Jacques
 Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic (LSH) functions. We introduce a new large class of measures, Euclidean regular and exponential type, in addition to all compactly-supported measures, for which this equivalence holds. We prove a Sobolev density theorem through LSH functions and use it to prove the equivalence of strong hypercontractivity and the strong logarithmic Sobolev inequality for such log-subharmonic functions. Keywords:logarithmic Sobolev inequalitiesCategory:47D06

17. CJM 2014 (vol 66 pp. 1110)

Li, Dong; Xu, Guixiang; Zhang, Xiaoyi
 On the Dispersive Estimate for the Dirichlet SchrÃ¶dinger Propagator and Applications to Energy Critical NLS We consider the obstacle problem for the SchrÃ¶dinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under the radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet SchrÃ¶dinger propagator $e^{it\Delta_D}$ and give a robust algorithm to prove sharp $L^1 \rightarrow L^{\infty}$ dispersive estimates. We showcase the analysis in dimensions $n=5,7$. As an application, we obtain global well-posedness and scattering for defocusing energy-critical NLS on $\Omega=\mathbb{R}^n\backslash \overline{B(0,1)}$ with Dirichlet boundary condition and radial data in these dimensions. Keywords:Dirichlet SchrÃ¶dinger propagator, dispersive estimate, Dirichlet boundary condition, scattering theory, energy criticalCategories:35P25, 35Q55, 47J35

18. CJM 2013 (vol 67 pp. 132)

Clouâtre, Raphaël
 Unitary Equivalence and Similarity to Jordan Models for Weak Contractions of Class $C_0$ We obtain results on the unitary equivalence of weak contractions of class $C_0$ to their Jordan models under an assumption on their commutants. In particular, our work addresses the case of arbitrary finite multiplicity. The main tool is the theory of boundary representations due to Arveson. We also generalize and improve previously known results concerning unitary equivalence and similarity to Jordan models when the minimal function is a Blaschke product. Keywords:weak contractions, operators of class $C_0$, Jordan model, unitary equivalenceCategories:47A45, 47L55

19. CJM 2013 (vol 66 pp. 1143)

Plevnik, Lucijan; Šemrl, Peter
 Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional separable Hilbert spaces and ${\rm Lat}\,\mathcal{H}$ the lattice of all closed subspaces oh $\mathcal{H}$. We describe the general form of pairs of bijective maps $\phi , \psi : {\rm Lat}\,\mathcal{H} \to {\rm Lat}\,\mathcal{K}$ having the property that for every pair $U,V \in {\rm Lat}\,\mathcal{H}$ we have $\mathcal{H} = U \oplus V \iff \mathcal{K} = \phi (U) \oplus \psi (V)$. Then we reformulate this theorem as a description of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known structural results for maps on idempotents are easy consequences. Keywords:Hilbert space, lattice of closed subspaces, complemented subspaces, adjacent subspaces, idempotentsCategories:46B20, 47B49

20. CJM 2013 (vol 65 pp. 1005)

Forrest, Brian; Miao, Tianxuan
 Uniformly Continuous Functionals and M-Weakly Amenable Groups Let $G$ be a locally compact group. Let $A_{M}(G)$ ($A_{0}(G)$)denote the closure of $A(G)$, the Fourier algebra of $G$ in the space of bounded (completely bounded) multipliers of $A(G)$. We call a locally compact group M-weakly amenable if $A_M(G)$ has a bounded approximate identity. We will show that when $G$ is M-weakly amenable, the algebras $A_{M}(G)$ and $A_{0}(G)$ have properties that are characteristic of the Fourier algebra of an amenable group. Along the way we show that the sets of tolopolically invariant means associated with these algebras have the same cardinality as those of the Fourier algebra. Keywords:Fourier algebra, multipliers, weakly amenable, uniformly continuous functionalsCategories:43A07, 43A22, 46J10, 47L25

21. CJM 2013 (vol 66 pp. 387)

Mashreghi, J.; Shabankhah, M.
 Composition of Inner Functions We study the image of the model subspace $K_\theta$ under the composition operator $C_\varphi$, where $\varphi$ and $\theta$ are inner functions, and find the smallest model subspace which contains the linear manifold $C_\varphi K_\theta$. Then we characterize the case when $C_\varphi$ maps $K_\theta$ into itself. This case leads to the study of the inner functions $\varphi$ and $\psi$ such that the composition $\psi\circ\varphi$ is a divisor of $\psi$ in the family of inner functions. Keywords:composition operators, inner functions, Blaschke products, model subspacesCategories:30D55, 30D05, 47B33

22. CJM 2013 (vol 65 pp. 783)

Garcés, Jorge J.; Peralta, Antonio M.
 Generalised Triple Homomorphisms and Derivations We introduce generalised triple homomorphism between Jordan Banach triple systems as a concept which extends the notion of generalised homomorphism between Banach algebras given by K. Jarosz and B.E. Johnson in 1985 and 1987, respectively. We prove that every generalised triple homomorphism between JB$^*$-triples is automatically continuous. When particularised to C$^*$-algebras, we rediscover one of the main theorems established by B.E. Johnson. We shall also consider generalised triple derivations from a Jordan Banach triple $E$ into a Jordan Banach triple $E$-module, proving that every generalised triple derivation from a JB$^*$-triple $E$ into itself or into $E^*$ is automatically continuous. Keywords:generalised homomorphism, generalised triple homomorphism, generalised triple derivation, Banach algebra, Jordan Banach triple, C$^*$-algebra, JB$^*$-tripleCategories:46L05, 46L70, 47B48, 17C65, 46K70, 46L40, 47B47, 47B49

23. CJM 2013 (vol 66 pp. 641)

Grigor'yan, Alexander; Hu, Jiaxin
 Heat Kernels and Green Functions on Metric Measure Spaces We prove that, in a setting of local Dirichlet forms on metric measure spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent to the conjunction of the volume doubling propety, the elliptic Harnack inequality and a certain estimate of the capacity between concentric balls. The main technical tool is the equivalence between the capacity estimate and the estimate of a mean exit time in a ball, that uses two-sided estimates of a Green function in a ball. Keywords:Dirichlet form, heat kernel, Green function, capacityCategories:35K08, 28A80, 31B05, 35J08, 46E35, 47D07

24. CJM 2012 (vol 65 pp. 768)

 Nonself-adjoint Semicrossed Products by Abelian Semigroups Let $\mathcal{S}$ be the semigroup $\mathcal{S}=\sum^{\oplus k}_{i=1}\mathcal{S}_i$, where for each $i\in I$, $\mathcal{S}_i$ is a countable subsemigroup of the additive semigroup $\mathbb{R}_+$ containing $0$. We consider representations of $\mathcal{S}$ as contractions $\{T_s\}_{s\in\mathcal{S}}$ on a Hilbert space with the Nica-covariance property: $T_s^*T_t=T_tT_s^*$ whenever $t\wedge s=0$. We show that all such representations have a unique minimal isometric Nica-covariant dilation. This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant representations of the action of $\mathcal{S}$ on an operator algebra $\mathcal{A}$ by completely contractive endomorphisms. We conclude by calculating the $C^*$-envelope of the isometric nonself-adjoint semicrossed product algebra (in the sense of Kakariadis and Katsoulis). Keywords:semicrossed product, crossed product, C*-envelope, dilationsCategories:47L55, 47A20, 47L65

25. CJM 2011 (vol 64 pp. 1329)

Izuchi, Kei Ji; Nguyen, Quang Dieu; Ohno, Shûichi
 Composition Operators Induced by Analytic Maps to the Polydisk We study properties of composition operators induced by symbols acting from the unit disk to the polydisk. This result will be involved in the investigation of weighted composition operators on the Hardy space on the unit disk and moreover be concerned with composition operators acting from the Bergman space to the Hardy space on the unit disk. Keywords:composition operators, Hardy spaces, polydiskCategories:47B33, 32A35, 30H10
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