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1. CJM Online first

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, involution
Categories:43A85, 47A67, 20G05

2. CJM 2016 (vol 68 pp. 841)

Gupta, Sanjiv Kumar; Hare, Kathryn
Characterizing the Absolute Continuity of the Convolution of Orbital Measures in a Classical Lie Algebra
Let $\mathfrak{g}$ be a compact, simple Lie algebra of dimension $d$. It is a classical result that the convolution of any $d$ non-trivial, $G$-invariant, orbital measures is absolutely continuous with respect to Lebesgue measure on $\mathfrak{g}$ and the sum of any $d$ non-trivial orbits has non-empty interior. The number $d$ was later reduced to the rank of the Lie algebra (or rank $+1$ in the case of type $A_{n}$). More recently, the minimal integer $k=k(X)$ such that the $k$-fold convolution of the orbital measure supported on the orbit generated by $X$ is an absolutely continuous measure was calculated for each $X\in \mathfrak{g}$. In this paper $\mathfrak{g}$ is any of the classical, compact, simple Lie algebras. We characterize the tuples $(X_{1},\dots,X_{L})$, with $X_{i}\in \mathfrak{g},$ which have the property that the convolution of the $L$-orbital measures supported on the orbits generated by the $X_{i}$ is absolutely continuous and, equivalently, the sum of their orbits has non-empty interior. The characterization depends on the Lie type of $\mathfrak{g}$ and the structure of the annihilating roots of the $X_{i}$. Such a characterization was previously known only for type $A_{n}$.

Keywords:compact Lie algebra, orbital measure, absolutely continuous measure
Categories:43A80, 17B45, 58C35

3. 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 space
Categories:42B20, 47B07

4. CJM 2015 (vol 67 pp. 1091)

Mine, Kotaro; Yamashita, Atsushi
Metric Compactifications and Coarse Structures
Let $\mathbf{TB}$ be the category of totally bounded, locally compact metric spaces with the $C_0$ coarse structures. We show that if $X$ and $Y$ are in $\mathbf{TB}$ then $X$ and $Y$ are coarsely equivalent if and only if their Higson coronas are homeomorphic. In fact, the Higson corona functor gives an equivalence of categories $\mathbf{TB}\to\mathbf{K}$, where $\mathbf{K}$ is the category of compact metrizable spaces. We use this fact to show that the continuously controlled coarse structure on a locally compact space $X$ induced by some metrizable compactification $\tilde{X}$ is determined only by the topology of the remainder $\tilde{X}\setminus X$.

Keywords:coarse geometry, Higson corona, continuously controlled coarse structure, uniform continuity, boundary at infinity
Categories:18B30, 51F99, 53C23, 54C20

5. CJM 2015 (vol 67 pp. 481)

an Huef, Astrid; Archbold, Robert John
The C*-algebras of Compact Transformation Groups
We investigate the representation theory of the crossed-product $C^*$-algebra associated to a compact group $G$ acting on a locally compact space $X$ when the stability subgroups vary discontinuously. Our main result applies when $G$ has a principal stability subgroup or $X$ is locally of finite $G$-orbit type. Then the upper multiplicity of the representation of the crossed product induced from an irreducible representation $V$ of a stability subgroup is obtained by restricting $V$ to a certain closed subgroup of the stability subgroup and taking the maximum of the multiplicities of the irreducible summands occurring in the restriction of $V$. As a corollary we obtain that when the trivial subgroup is a principal stability subgroup, the crossed product is a Fell algebra if and only if every stability subgroup is abelian. A second corollary is that the $C^*$-algebra of the motion group $\mathbb{R}^n\rtimes \operatorname{SO}(n)$ is a Fell algebra. This uses the classical branching theorem for the special orthogonal group $\operatorname{SO}(n)$ with respect to $\operatorname{SO}(n-1)$. Since proper transformation groups are locally induced from the actions of compact groups, we describe how some of our results can be extended to transformation groups that are locally proper.

Keywords:compact transformation group, proper action, spectrum of a C*-algebra, multiplicity of a representation, crossed-product C*-algebra, continuous-trace C*-algebra, Fell algebra
Categories:46L05, 46L55

6. CJM 2014 (vol 67 pp. 330)

Bernardes, Nilson C.; Vermersch, Rômulo M.
Hyperspace Dynamics of Generic Maps of the Cantor Space
We study the hyperspace dynamics induced from generic continuous maps and from generic homeomorphisms of the Cantor space, with emphasis on the notions of Li-Yorke chaos, distributional chaos, topological entropy, chain continuity, shadowing and recurrence.

Keywords:cantor space, continuous maps, homeomorphisms, hyperspace, dynamics
Categories:37B99, 54H20, 54E52

7. 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 functionals
Categories:43A07, 43A22, 46J10, 47L25

8. CJM 2011 (vol 64 pp. 755)

Brown, Lawrence G.; Lee, Hyun Ho
Homotopy Classification of Projections in the Corona Algebra of a Non-simple $C^*$-algebra
We study projections in the corona algebra of $C(X)\otimes K$, where K is the $C^*$-algebra of compact operators on a separable infinite dimensional Hilbert space and $X=[0,1],[0,\infty),(-\infty,\infty)$, or $[0,1]/\{ 0,1 \}$. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in $K_0$, Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct.

Keywords:essential codimension, continuous field of Hilbert spaces, Corona algebra
Categories:46L05, 46L80

9. CJM 2011 (vol 63 pp. 1188)

Śliwa, Wiesław; Ziemkowska, Agnieszka
On Complemented Subspaces of Non-Archimedean Power Series Spaces
The non-archimedean power series spaces, $A_1(a)$ and $A_\infty(b)$, are the best known and most important examples of non-archimedean nuclear Fréchet spaces. We prove that the range of every continuous linear map from $A_p(a)$ to $A_q(b)$ has a Schauder basis if either $p=1$ or $p=\infty$ and the set $M_{b,a}$ of all bounded limit points of the double sequence $(b_i/a_j)_{i,j\in\mathbb{N}}$ is bounded. It follows that every complemented subspace of a power series space $A_p(a)$ has a Schauder basis if either $p=1$ or $p=\infty$ and the set $M_{a,a}$ is bounded.

Keywords:non-archimedean Köthe space, range of a continuous linear map, Schauder basis
Categories:46S10, 47S10, 46A35

10. CJM 2011 (vol 63 pp. 500)

Dadarlat, Marius; Elliott, George A.; Niu, Zhuang
One-Parameter Continuous Fields of Kirchberg Algebras. II
Parallel to the first two authors' earlier classification of separable, unita one-parameter, continuous fields of Kirchberg algebras with torsion free $\mathrm{K}$-groups supported in one dimension, one-parameter, separable, uni continuous fields of AF-algebras are classified by their ordered $\mathrm{K}_0$-sheaves. Effros-Handelman-Shen type theorems are pr separable unital one-parameter continuous fields of AF-algebras and Kirchberg algebras.

Keywords:continuous fields of C$^*$-algebras, $\mathrm{K}_0$-presheaves, Effros--Handeen type theorem

11. 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 coefficients
Categories:33D45, 42C05

12. CJM 2007 (vol 59 pp. 1135)

Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari
Sobolev Extensions of Hölder Continuous and Characteristic Functions on Metric Spaces
We study when characteristic and H\"older continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and H\"older continuous functions into globally defined Sobolev functions.

Keywords:characteristic function, Newtonian function, metric space, resolutivity, Hölder continuous, Perron solution, $p$-harmonic, Sobolev extension, Whitney covering
Categories:46E35, 31C45

13. CJM 2007 (vol 59 pp. 3)

Biller, Harald
Holomorphic Generation of Continuous Inverse Algebras
We study complex commutative Banach algebras (and, more generally, continuous inverse algebras) in which the holomorphic functions of a fixed $n$-tuple of elements are dense. In particular, we characterize the compact subsets of~$\C^n$ which appear as joint spectra of such $n$-tuples. The characterization is compared with several established notions of holomorphic convexity by means of approximation conditions.

Keywords:holomorphic functional calculus, commutative continuous inverse algebra, holomorphic convexity, Stein compacta, meromorphic convexity, holomorphic approximation
Categories:46H30, 32A38, 32E30, 41A20, 46J15

14. CJM 2005 (vol 57 pp. 471)

Ciesielski, Krzysztof; Pawlikowski, Janusz
Small Coverings with Smooth Functions under the Covering Property Axiom
In the paper we formulate a Covering Property Axiom, \psmP, which holds in the iterated perfect set model, and show that it implies the following facts, of which (a) and (b) are the generalizations of results of J. Stepr\={a}ns. \begin{compactenum}[\rm(a)~~] \item There exists a family $\F$ of less than continuum many $\C^1$ functions from $\real$ to $\real$ such that $\real^2$ is covered by functions from $\F$, in the sense that for every $\la x,y\ra\in\real^2$ there exists an $f\in\F$ such that either $f(x)=y$ or $f(y)=x$. \item For every Borel function $f\colon\real\to\real$ there exists a family $\F$ of less than continuum many ``$\C^1$'' functions ({\em i.e.,} differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of $f$. \item For every $n>0$ and a $D^n$ function $f\colon\real\to\real$ there exists a family $\F$ of less than continuum many $\C^n$ functions whose graphs cover the graph of $f$. \end{compactenum} We also provide the examples showing that in the above properties the smoothness conditions are the best possible. Parts (b), (c), and the examples are closely related to work of A. Olevski\v{\i}.

Keywords:continuous, smooth, covering
Categories:26A24, 03E35

15. CJM 2002 (vol 54 pp. 709)

Ismail, Mourad E. H.; Stanton, Dennis
$q$-Integral and Moment Representations for $q$-Orthogonal Polynomials
We develop a method for deriving integral representations of certain orthogonal polynomials as moments. These moment representations are applied to find linear and multilinear generating functions for $q$-orthogonal polynomials. As a byproduct we establish new transformation formulas for combinations of basic hypergeometric functions, including a new representation of the $q$-exponential function $\mathcal{E}_q$.

Keywords:$q$-integral, $q$-orthogonal polynomials, associated polynomials, $q$-difference equations, generating functions, Al-Salam-Chihara polynomials, continuous $q$-ultraspherical polynomials
Categories:33D45, 33D20, 33C45, 30E05

16. CJM 1997 (vol 49 pp. 1089)

Burke, Maxim R.; Ciesielski, Krzysztof
Sets on which measurable functions are determined by their range
We study sets on which measurable real-valued functions on a measurable space with negligibles are determined by their range.

Keywords:measurable function, measurable space with negligibles, continuous image, set of range uniqueness (SRU)
Categories:28A20, 28A05, 54C05, 26A30, 03E35, 03E50

17. CJM 1997 (vol 49 pp. 520)

Ismail, Mourad E. H.; Stanton, Dennis
Classical orthogonal polynomials as moments
We show that the Meixner, Pollaczek, Meixner-Pollaczek, the continuous $q$-ultraspherical polynomials and Al-Salam-Chihara polynomials, in certain normalization, are moments of probability measures. We use this fact to derive bilinear and multilinear generating functions for some of these polynomials. We also comment on the corresponding formulas for the Charlier, Hermite and Laguerre polynomials.

Keywords:Classical orthogonal polynomials, \ACP, continuous, $q$-ultraspherical polynomials, generating functions, multilinear, generating functions, transformation formulas, umbral calculus
Categories:33D45, 33D20, 33C45, 30E05

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