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26. CMB 2005 (vol 48 pp. 607)

Park, Efton
 Toeplitz Algebras and Extensions of\\Irrational Rotation Algebras For a given irrational number $\theta$, we define Toeplitz operators with symbols in the irrational rotation algebra ${\mathcal A}_\theta$, and we show that the $C^*$-algebra $\mathcal T({\mathcal A}_\theta)$ generated by these Toeplitz operators is an extension of ${\mathcal A}_\theta$ by the algebra of compact operators. We then use these extensions to explicitly exhibit generators of the group $KK^1({\mathcal A}_\theta,\mathbb C)$. We also prove an index theorem for $\mathcal T({\mathcal A}_\theta)$ that generalizes the standard index theorem for Toeplitz operators on the circle. Keywords:Toeplitz operators, irrational rotation algebras, index theoryCategories:47B35, 46L80

27. CMB 2005 (vol 48 pp. 175)

Borwein, David; Kratz, Werner
 Weighted Convolution Operators on $\ell_p$ The main results deal with conditions for the validity of the weighted convolution inequality $\sum_{n\in\mathbb Z}\left|b_n\sum_{k\in\mathbb Z} a_{n-k}x_k\right|^p\le C^p\sum_{k\in\mathbb Z} |x_k|^p$ when $p\ge1$. Keywords:convolution operators on $\ell_p$Categories:40G10;, 40E05

28. CMB 2004 (vol 47 pp. 615)

Randrianantoanina, Narcisse
 $C^*$-Algebras and Factorization Through Diagonal Operators Let $\cal A$ be a $C^*$-algebra and $E$ be a Banach space with the Radon-Nikodym property. We prove that if $j$ is an embedding of $E$ into an injective Banach space then for every absolutely summing operator $T:\mathcal{A}\longrightarrow E$, the composition $j \circ T$ factors through a diagonal operator from $l^{2}$ into $l^{1}$. In particular, $T$ factors through a Banach space with the Schur property. Similarly, we prove that for $2 Keywords:$C^*$-algebras, summing operators, diagonal operators,, Radon-Nikodym propertyCategories:46L50, 47D15 29. CMB 2001 (vol 44 pp. 385) Ballantine, Cristina M.  A Hypergraph with Commuting Partial Laplacians Let$F$be a totally real number field and let$\GL_{n}$be the general linear group of rank$n$over$F$. Let$\mathfrak{p}$be a prime ideal of$F$and$F_{\mathfrak{p}}$the completion of$F$with respect to the valuation induced by$\mathfrak{p}$. We will consider a finite quotient of the affine building of the group$\GL_{n}$over the field$F_{\mathfrak{p}}$. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph. Keywords:Hecke operators, buildingsCategories:11F25, 20F32 30. CMB 2000 (vol 43 pp. 406) Borwein, David  Weighted Mean Operators on$l_p$The weighted mean matrix$M_a$is the triangular matrix$\{a_k/A_n\}$, where$a_n > 0$and$A_n := a_1 + a_2 + \cdots + a_n$. It is proved that, subject to$n^c a_n$being eventually monotonic for each constant$c$and to the existence of$\alpha := \lim \frac{A_n}{na_n}$,$M_a \in B(l_p)$for$1 < p < \infty$if and only if$\alpha < p$. Keywords:weighted means, operators on$l_p$, norm estimatesCategories:47B37, 47A30, 40G05 31. CMB 2000 (vol 43 pp. 496) Xu, Yuan  Harmonic Polynomials Associated With Reflection Groups We extend Maxwell's representation of harmonic polynomials to$h$-harmonics associated to a reflection invariant weight function$h_k$. Let$\CD_i$,$1\le i \le d$, be Dunkl's operators associated with a reflection group. For any homogeneous polynomial$P$of degree$n$, we prove the polynomial$|\xb|^{2 \gamma +d-2+2n}P(\CD)\{1/|\xb|^{2 \gamma +d-2}\}$is a$h$-harmonic polynomial of degree$n$, where$\gamma = \sum k_i$and$\CD=(\CD_1,\ldots,\CD_d)$. The construction yields a basis for$h$-harmonics. We also discuss self-adjoint operators acting on the space of$h$-harmonics. Keywords:$h$-harmonics, reflection group, Dunkl's operatorsCategories:33C50, 33C45 32. CMB 1999 (vol 42 pp. 129) Baker, Andrew  Hecke Operations and the Adams$E_2$-Term Based on Elliptic Cohomology Hecke operators are used to investigate part of the$\E_2$-term of the Adams spectral sequence based on elliptic homology. The main result is a derivation of$\Ext^1$which combines use of classical Hecke operators and$p$-adic Hecke operators due to Serre. Keywords:Adams spectral sequence, elliptic cohomology, Hecke operatorsCategories:55N20, 55N22, 55T15, 11F11, 11F25 33. CMB 1998 (vol 41 pp. 129) Lee, Young Joo  Pluriharmonic symbols of commuting Toeplitz type operators on the weighted Bergman spaces A class of Toeplitz type operators acting on the weighted Bergman spaces of the unit ball in the$n$-dimensional complex space is considered and two pluriharmonic symbols of commuting Toeplitz type operators are completely characterized. Keywords:Pluriharmonic functions, Weighted Bergman spaces, Toeplitz type operators.Categories:47B38, 32A37 34. CMB 1998 (vol 41 pp. 10) Borwein, David  Simple conditions for matrices to be bounded operators on$l_p$The two theorems proved yield simple yet reasonably general conditions for triangular matrices to be bounded operators on$l_p$. The theorems are applied to N\"orlund and weighted mean matrices. Keywords:Triangular matrices, NÃ¶rlund matrices, weighted means, operators, on$l_p\$.Categories:47B37, 47A30, 40G05
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