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1. CJM Online first

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 operatorCategories:46L55, 19K56, 46L80

2. CJM Online first

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

3. CJM Online first

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

4. CJM Online first

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

5. CJM Online first

Kamgarpour, Masoud
 On the notion of conductor in the local geometric Langlands correspondence Under the local Langlands correspondence, the conductor of an irreducible representation of $\operatorname{Gl}_n(F)$ is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection. Keywords:local geometric Langlands, connections, cyclic vectors, opers, conductors, Segal-Sugawara operators, Chervov-Molev operators, critical level, smooth representations, affine Kac-Moody algebra, categorical representationsCategories:17B67, 17B69, 22E50, 20G25

6. CJM 2015 (vol 67 pp. 1290)

Charlesworth, Ian; Nelson, Brent; Skoufranis, Paul
 On Two-faced Families of Non-commutative Random Variables We demonstrate that the notions of bi-free independence and combinatorial-bi-free independence of two-faced families are equivalent using a diagrammatic view of bi-non-crossing partitions. These diagrams produce an operator model on a Fock space suitable for representing any two-faced family of non-commutative random variables. Furthermore, using a Kreweras complement on bi-non-crossing partitions we establish the expected formulas for the multiplicative convolution of a bi-free pair of two-faced families. Keywords:free probability, operator algebras, bi-freeCategory:46L54

7. CJM 2015 (vol 67 pp. 1161)

Zhang, Junqiang; Cao, Jun; Jiang, Renjin; Yang, Dachun
 Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators Let $w$ be either in the Muckenhoupt class of $A_2(\mathbb{R}^n)$ weights or in the class of $QC(\mathbb{R}^n)$ weights, and $L_w:=-w^{-1}\mathop{\mathrm{div}}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$, $n\ge 2$. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$ for $p\in (0,1]$ and, when $p\in (\frac{n}{n+1},1]$ and $w\in A_{q_0}(\mathbb{R}^n)$ with $q_0\in[1,\frac{p(n+1)}n)$, the authors prove that the associated Riesz transform $\nabla L_w^{-1/2}$ is bounded from $H_{L_w}^p(\mathbb{R}^n)$ to the weighted classical Hardy space $H_w^p(\mathbb{R}^n)$. Keywords:degenerate elliptic operator, Hardy space, square function, maximal function, molecule, Riesz transformCategories:42B30, 42B35, 35J70

8. CJM 2014 (vol 67 pp. 573)

Chen, Fulin; Gao, Yun; Jing, Naihuan; Tan, Shaobin
 Twisted Vertex Operators and Unitary Lie Algebras A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral $\mathbb Z_2$-lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac-Moody Lie algebra of type $A_n^{(2)}$ are recovered by the new method. Keywords:Lie algebra, vertex operator, representation theoryCategories:17B60, 17B69

9. 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

10. CJM 2013 (vol 66 pp. 1382)

Wu, Xinfeng
 Weighted Carleson Measure Spaces Associated with Different Homogeneities In this paper, we introduce weighted Carleson measure spaces associated with different homogeneities and prove that these spaces are the dual spaces of weighted Hardy spaces studied in a forthcoming paper. As an application, we establish the boundedness of composition of two CalderÃ³n-Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong-Stein. Keywords:composition of operators, weighted Carleson measure spaces, dualityCategories:42B20, 42B35

11. 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

12. CJM 2013 (vol 65 pp. 1217)

Cruz, Victor; Mateu, Joan; Orobitg, Joan
 Beltrami Equation with Coefficient in Sobolev and Besov Spaces Our goal in this work is to present some function spaces on the complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f (z)$, have first derivatives locally in $X(\mathbb C)$, provided that the Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$. Keywords:quasiregular mappings, Beltrami equation, Sobolev spaces, CalderÃ³n-Zygmund operatorsCategories:30C62, 35J99, 42B20

13. CJM 2012 (vol 65 pp. 989)

Chu, C-H.; Velasco, M. V.
 Automatic Continuity of Homomorphisms in Non-associative Banach Algebras We introduce the concept of a rare element in a non-associative normed algebra and show that the existence of such element is the only obstruction to continuity of a surjective homomorphism from a non-associative Banach algebra to a unital normed algebra with simple completion. Unital associative algebras do not admit any rare element and hence automatic continuity holds. Keywords:automatic continuity, non-associative algebra, spectrum, rare operator, rare elementCategories:46H40, 46H70

14. CJM 2011 (vol 64 pp. 1036)

Koh, Doowon; Shen, Chun-Yen
 Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields In this paper we study the extension problem, the averaging problem, and the generalized ErdÅs-Falconer distance problem associated with arbitrary homogeneous varieties in three dimensional vector spaces over finite fields. In the case when the varieties do not contain any plane passing through the origin, we obtain the best possible results on the aforementioned three problems. In particular, our result on the extension problem modestly generalizes the result by Mockenhaupt and Tao who studied the particular conical extension problem. In addition, investigating the Fourier decay on homogeneous varieties enables us to give complete mapping properties of averaging operators. Moreover, we improve the size condition on a set such that the cardinality of its distance set is nontrivial. Keywords:extension problems, averaging operator, finite fields, ErdÅs-Falconer distance problems, homogeneous polynomialCategories:42B05, 11T24, 52C17

15. 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

16. CJM 2011 (vol 64 pp. 669)

Pantano, Alessandra; Paul, Annegret; Salamanca-Riba, Susana A.
 The Genuine Omega-regular Unitary Dual of the Metaplectic Group We classify all genuine unitary representations of the metaplectic group whose infinitesimal character is real and at least as regular as that of the oscillator representation. In a previous paper we exhibited a certain family of representations satisfying these conditions, obtained by cohomological induction from the tensor product of a one-dimensional representation and an oscillator representation. Our main theorem asserts that this family exhausts the genuine omega-regular unitary dual of the metaplectic group. Keywords:Metaplectic group, oscillator representation, bottom layer map, cohomological induction, Parthasarathy's Dirac Operator Inequality, pseudospherical principal seriesCategory:22E46

17. CJM 2011 (vol 64 pp. 805)

Chapon, François; Defosseux, Manon
 Quantum Random Walks and Minors of Hermitian Brownian Motion Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam, and van Moerbeke that the process of eigenvalues of two consecutive minors of a Hermitian Brownian motion is a Markov process; whereas, if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion. Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor processCategories:46L53, 60B20, 14L24

18. CJM 2011 (vol 64 pp. 892)

Hytönen, Tuomas; Liu, Suile; Yang, Dachun; Yang, Dongyong
 Boundedness of CalderÃ³n-Zygmund Operators on Non-homogeneous Metric Measure Spaces Let $({\mathcal X}, d, \mu)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$. In this paper, we show that the boundedness of a CalderÃ³n-Zygmund operator $T$ on $L^2(\mu)$ is equivalent to that of $T$ on $L^p(\mu)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(\mu)$ to $L^{1,\,\infty}(\mu).$ As an application, we prove that if $T$ is a CalderÃ³n-Zygmund operator bounded on $L^2(\mu)$, then its maximal operator is bounded on $L^p(\mu)$ for all $p\in (1, \infty)$ and from the space of all complex-valued Borel measures on ${\mathcal X}$ to $L^{1,\,\infty}(\mu)$. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition. Keywords:upper doubling, geometrical doubling, dominating function, weak type $(1,1)$ estimate, CalderÃ³n-Zygmund operator, maximal operatorCategories:42B20, 42B25, 30L99

19. CJM 2011 (vol 64 pp. 183)

 Negative Powers of Laguerre Operators We study negative powers of Laguerre differential operators in $\mathbb{R}^d$, $d\ge1$. For these operators we prove two-weight $L^p-L^q$ estimates with ranges of $q$ depending on $p$. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution type is less straightforward. For half-integer type indices $\alpha$ we transfer the desired results from the Hermite setting and then apply an interpolation argument based on a device we call the convexity principle to cover the continuous range of $\alpha\in[-1/2,\infty)^d$. Finally, we investigate negative powers of the Dunkl harmonic oscillator in the context of a finite reflection group acting on $\mathbb{R}^d$ and isomorphic to $\mathbb Z^d_2$. The two weight $L^p-L^q$ estimates we obtain in this setting are essentially consequences of those for Laguerre function expansions of convolution type. Keywords:potential operator, fractional integral, Riesz potential, negative power, harmonic oscillator, Laguerre operator, Dunkl harmonic oscillatorCategories:47G40, 31C15, 26A33

20. CJM 2011 (vol 63 pp. 862)

Hosokawa, Takuya; Nieminen, Pekka J.; Ohno, Shûichi
 Linear Combinations of Composition Operators on the Bloch Spaces We characterize the compactness of linear combinations of analytic composition operators on the Bloch space. We also study their boundedness and compactness on the little Bloch space. Keywords: composition operator, compactness, Bloch spaceCategories:47B33, 30D45, 47B07

21. CJM 2010 (vol 63 pp. 181)

Ismail, Mourad E. H.; Obermaier, Josef
 Characterizations of Continuous and Discrete $q$-Ultraspherical Polynomials We characterize the continuous $q$-ultraspherical polynomials in terms of the special form of the coefficients in the expansion $\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$ being the Askey--Wilson divided difference operator. The polynomials are assumed to be symmetric, and the connection coefficients are multiples of the reciprocal of the square of the $L^2$ norm of the polynomials. A similar characterization is given for the discrete $q$-ultraspherical polynomials. A new proof of the evaluation of the connection coefficients for big $q$-Jacobi polynomials is given. Keywords:continuous $q$-ultraspherical polynomials, big $q$-Jacobi polynomials, discrete $q$-ultra\-spherical polynomials, Askey--Wilson operator, $q$-difference operator, recursion coefficientsCategories:33D45, 42C05

22. CJM 2010 (vol 62 pp. 1419)

Yang, Dachun; Yang, Dongyong
 BMO-Estimates for Maximal Operators via Approximations of the Identity with Non-Doubling Measures Let $\mu$ be a nonnegative Radon measure on $\mathbb{R}^d$ that satisfies the growth condition that there exist constants $C_0>0$ and $n\in(0,d]$ such that for all $x\in\mathbb{R}^d$ and $r>0$, ${\mu(B(x,\,r))\le C_0r^n}$, where $B(x,r)$ is the open ball centered at $x$ and having radius $r$. In this paper, the authors prove that if $f$ belongs to the $\textrm {BMO}$-type space $\textrm{RBMO}(\mu)$ of Tolsa, then the homogeneous maximal function $\dot{\mathcal{M}}_S(f)$ (when $\mathbb{R}^d$ is not an initial cube) and the inhomogeneous maximal function $\mathcal{M}_S(f)$ (when $\mathbb{R}^d$ is an initial cube) associated with a given approximation of the identity $S$ of Tolsa are either infinite everywhere or finite almost everywhere, and in the latter case, $\dot{\mathcal{M}}_S$ and $\mathcal{M}_S$ are bounded from $\textrm{RBMO}(\mu)$ to the $\textrm {BLO}$-type space $\textrm{RBLO}(\mu)$. The authors also prove that the inhomogeneous maximal operator $\mathcal{M}_S$ is bounded from the local $\textrm {BMO}$-type space $\textrm{rbmo}(\mu)$ to the local $\textrm {BLO}$-type space $\textrm{rblo}(\mu)$. Keywords:Non-doubling measure, maximal operator, approximation of the identity, RBMO(mu), RBLO(mu), rbmo(mu), rblo(mu)Categories:42B25, 42B30, 47A30, 43A99

23. CJM 2010 (vol 62 pp. 1037)

Calviño-Louzao, E.; García-Río, E.; Vázquez-Lorenzo, R.
 Riemann Extensions of Torsion-Free Connections with Degenerate Ricci Tensor {Correspondence} between torsion-free connections with {nilpotent skew-symmetric curvature operator} and IP Riemann extensions is shown. Some consequences are derived in the study of four-dimensional IP metrics and locally homogeneous affine surfaces. Keywords:Walker metric, Riemann extension, curvature operator, projectively flat and recurrent affine connectionCategories:53B30, 53C50

24. CJM 2009 (vol 62 pp. 218)

Xing, Yang
 The General Definition of the Complex Monge--AmpÃ¨re Operator on Compact KÃ¤hler Manifolds We introduce a wide subclass ${\mathcal F}(X,\omega)$ of quasi-plurisubharmonic functions in a compact KÃ¤hler manifold, on which the complex Monge-AmpÃ¨re operator is well defined and the convergence theorem is valid. We also prove that ${\mathcal F}(X,\omega)$ is a convex cone and includes all quasi-plurisubharmonic functions that are in the Cegrell class. Keywords:complex Monge--AmpÃ¨re operator, compact KÃ¤hler manifoldCategories:32W20, 32Q15

25. CJM 2009 (vol 62 pp. 202)

Tang, Lin
 Interior $h^1$ Estimates for Parabolic Equations with $\operatorname{LMO}$ Coefficients In this paper we establish \emph{a priori} $h^1$-estimates in a bounded domain for parabolic equations with vanishing $\operatorname{LMO}$ coefficients. Keywords:parabolic operator, Hardy space, parabolic, singular integrals and commutatorsCategories:35K20, 35B65, 35R05
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