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1. CJM Online first
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 |
2. CJM 2014 (vol 67 pp. 330)
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 |
3. CJM 2013 (vol 65 pp. 1005)
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 |
4. CJM 2011 (vol 64 pp. 755)
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 |
5. CJM 2011 (vol 63 pp. 1188)
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 FrÃ©chet 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 KÃ¶the space, range of a continuous linear map, Schauder basis Categories:46S10, 47S10, 46A35 |
6. CJM 2011 (vol 63 pp. 500)
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 Category:46L35 |
7. CJM 2010 (vol 63 pp. 181)
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 |
8. CJM 2007 (vol 59 pp. 1135)
Sobolev Extensions of HÃ¶lder 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, HÃ¶lder continuous, Perron solution, $p$-harmonic, Sobolev extension, Whitney covering Categories:46E35, 31C45 |
9. CJM 2007 (vol 59 pp. 3)
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 |
10. CJM 2005 (vol 57 pp. 471)
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 |
11. CJM 2002 (vol 54 pp. 709)
$q$-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 |
12. CJM 1997 (vol 49 pp. 1089)
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 |
13. CJM 1997 (vol 49 pp. 520)
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 |