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26. CJM 2009 (vol 62 pp. 34)

Campbell, Peter S.; Nevins, Monica
 Branching Rules for Ramified Principal Series Representations of $\mathrm{GL}(3)$ over a $p$-adic Field We decompose the restriction of ramified principal series representations of the $p$-adic group $\mathrm{GL}(3,\mathrm{k})$ to its maximal compact subgroup $K=\mathrm{GL}(3,R)$. Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in $K$. We establish several irreducibility results and illustrate the decomposition with some examples. Keywords:principal series representations, branching rules, maximal compact subgroups, representations of $p$-adic groupsCategories:20G25, 20G05

27. CJM 2009 (vol 61 pp. 1300)

Hubard, Isabel; Orbani\'c, Alen; Weiss, Asia Ivi\'c
 Monodromy Groups and Self-Invariance For every polytope $\mathcal{P}$ there is the universal regular polytope of the same rank as $\mathcal{P}$ corresponding to the Coxeter group $\mathcal{C} =[\infty, \dots, \infty]$. For a given automorphism $d$ of $\mathcal{C}$, using monodromy groups, we construct a combinatorial structure $\mathcal{P}^d$. When $\mathcal{P}^d$ is a polytope isomorphic to $\mathcal{P}$ we say that $\mathcal{P}$ is self-invariant with respect to $d$, or $d$-invariant. We develop algebraic tools for investigating these operations on polytopes, and in particular give a criterion on the existence of a $d$\nobreakdash-auto\-morphism of a given order. As an application, we analyze properties of self-dual edge-transitive polyhedra and polyhedra with two flag-orbits. We investigate properties of medials of such polyhedra. Furthermore, we give an example of a self-dual equivelar polyhedron which contains no polarity (duality of order 2). We also extend the concept of Petrie dual to higher dimensions, and we show how it can be dealt with using self-invariance. Keywords:maps, abstract polytopes, self-duality, monodromy groups, medials of polyhedraCategories:51M20, 05C25, 05C10, 05C30, 52B70

28. CJM 2009 (vol 61 pp. 382)

Miao, Tianxuan
 Unit Elements in the Double Dual of a Subalgebra of the Fourier Algebra $A(G)$ Let $\mathcal{A}$ be a Banach algebra with a bounded right approximate identity and let $\mathcal B$ be a closed ideal of $\mathcal A$. We study the relationship between the right identities of the double duals ${\mathcal B}^{**}$ and ${\mathcal A}^{**}$ under the Arens product. We show that every right identity of ${\mathcal B}^{**}$ can be extended to a right identity of ${\mathcal A}^{**}$ in some sense. As a consequence, we answer a question of Lau and \"Ulger, showing that for the Fourier algebra $A(G)$ of a locally compact group $G$, an element $\phi \in A(G)^{**}$ is in $A(G)$ if and only if $A(G) \phi \subseteq A(G)$ and $E \phi = \phi$ for all right identities $E$ of $A(G)^{**}$. We also prove some results about the topological centers of ${\mathcal B}^{**}$ and ${\mathcal A}^{**}$. Keywords:Locally compact groups, amenable groups, Fourier algebra, identity, Arens product, topological centerCategory:43A07

29. CJM 2009 (vol 61 pp. 124)

Dijkstra, Jan J.; Mill, Jan van
 Characterizing Complete Erd\H os Space The space now known as {\em complete Erd\H os space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the closed subspace of the Hilbert space $\ell^2$ consisting of all vectors such that every coordinate is in the convergent sequence $\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G. Oversteegen we present simple and useful topological characterizations of $\cerdos$. As an application we determine the class of factors of $\cerdos$. In another application we determine precisely which of the spaces that can be constructed in the Banach spaces $\ell^p$ according to the Erd\H os method' are homeomorphic to $\cerdos$. A novel application states that if $I$ is a Polishable $F_\sigma$-ideal on $\omega$, then $I$ with the Polish topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$, $\Z\times2^\omega$, or $\cerdos$. This last result answers a question that was asked by Stevo Todor{\v{c}}evi{\'c}. Keywords:Complete Erd\H os space, Lelek fan, almost zero-dimensional, nowhere zero-dimensional, Polishable ideals, submeasures on $\omega$, $\R$-trees, line-free groups in Banach spacesCategories:28C10, 46B20, 54F65

30. CJM 2008 (vol 60 pp. 1010)

Galé, José E.; Miana, Pedro J.
 $H^\infty$ Functional Calculus and Mikhlin-Type Multiplier Conditions Let $T$ be a sectorial operator. It is known that the existence of a bounded (suitably scaled) $H^\infty$ calculus for $T$, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra $\Lambda_{\infty,1}^\alpha(\R^+)$. Such an algebra includes functions defined by Mikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for $T$. In this paper, we use fractional derivation to analyse in detail the relationship between $\Lambda_{\infty,1}^\alpha$ and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence. Keywords:functional calculus, fractional calculus, Mikhlin multipliers, analytic semigroups, unbounded operators, quasimultipliersCategories:47A60, 47D03, 46J15, 26A33, 47L60, 47B48, 43A22

31. CJM 2008 (vol 60 pp. 1001)

Cornulier, Yves de; Tessera, Romain; Valette, Alain
 Isometric Group Actions on Hilbert Spaces: Structure of Orbits Our main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group. Keywords:affine actions, Hilbert spaces, minimal actions, nilpotent groupsCategories:22D10, 43A35, 20F69

32. CJM 2007 (vol 59 pp. 1301)

Furioli, Giulia; Melzi, Camillo; Veneruso, Alessandro
 Strichartz Inequalities for the Wave Equation with the Full Laplacian on the Heisenberg Group We prove dispersive and Strichartz inequalities for the solution of the wave equation related to the full Laplacian on the Heisenberg group, by means of Besov spaces defined by a Littlewood--Paley decomposition related to the spectral resolution of the full Laplacian. This requires a careful analysis due also to the non-homogeneous nature of the full Laplacian. This result has to be compared to a previous one by Bahouri, G\'erard and Xu concerning the solution of the wave equation related to the Kohn Laplacian. Keywords:nilpotent and solvable Lie groups, smoothness and regularity of solutions of PDEsCategories:22E25, 35B65

33. CJM 2006 (vol 58 pp. 897)

Courtès, François
 Distributions invariantes sur les groupes rÃ©ductifs quasi-dÃ©ployÃ©s Soit $F$ un corps local non archim\'edien, et $G$ le groupe des $F$-points d'un groupe r\'eductif connexe quasi-d\'eploy\'e d\'efini sur $F$. Dans cet article, on s'int\'eresse aux distributions sur $G$ invariantes par conjugaison, et \a l'espace de leurs restrictions \a l'alg\ebre de Hecke $\mathcal{H}$ des fonctions sur $G$ \a support compact biinvariantes par un sous-groupe d'Iwahori $I$ donn\'e. On montre tout d'abord que les valeurs d'une telle distribution sur $\mathcal{H}$ sont enti\erement d\'etermin\'ees par sa restriction au sous-espace de dimension finie des \'el\'ements de $\mathcal{H}$ \`a support dans la r\'eunion des sous-groupes parahoriques de $G$ contenant $I$. On utilise ensuite cette propri\'et\'e pour montrer, moyennant certaines conditions sur $G$, que cet espace est engendr\'e d'une part par certaines int\'egrales orbitales semi-simples, d'autre part par les int\'egrales orbitales unipotentes, en montrant tout d'abord des r\'esultats analogues sur les groupes finis. Keywords:reductive $p$-adic groups, orbital integrals, invariant distributionsCategories:22E35, 20G40

34. CJM 2006 (vol 58 pp. 23)

Dabbaghian-Abdoly, Vahid
 Constructing Representations of Finite Simple Groups and Covers Let $G$ be a finite group and $\chi$ be an irreducible character of $G$. An efficient and simple method to construct representations of finite groups is applicable whenever $G$ has a subgroup $H$ such that $\chi_H$ has a linear constituent with multiplicity $1$. In this paper we show (with a few exceptions) that if $G$ is a simple group or a covering group of a simple group and $\chi$ is an irreducible character of $G$ of degree less than 32, then there exists a subgroup $H$ (often a Sylow subgroup) of $G$ such that $\chi_H$ has a linear constituent with multiplicity $1$. Keywords:group representations, simple groups, central covers, irreducible representationsCategories:20C40, 20C15

35. CJM 2005 (vol 57 pp. 648)

Nevins, Monica
 Branching Rules for Principal Series Representations of $SL(2)$ over a $p$-adic Field We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL(2,k)$, for $k$ a $p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of $SL(2,k)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup. Keywords:representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-typesCategories:20G25, 22E35, 20H25

36. CJM 2004 (vol 56 pp. 344)

Miao, Tianxuan
 Predual of the Multiplier Algebra of $A_p(G)$ and Amenability For a locally compact group $G$ and $1 Keywords:Locally compact groups, amenable groups, multiplier algebra, Herz algebraCategory:43A07 37. CJM 2003 (vol 55 pp. 1000) Graczyk, P.; Sawyer, P.  Some Convexity Results for the Cartan Decomposition In this paper, we consider the set$\mathcal{S} = a(e^X K e^Y)$where$a(g)$is the abelian part in the Cartan decomposition of$g$. This is exactly the support of the measure intervening in the product formula for the spherical functions on symmetric spaces of noncompact type. We give a simple description of that support in the case of$\SL(3,\mathbf{F})$where$\mathbf{F} = \mathbf{R}$,$\mathbf{C}$or$\mathbf{H}$. In particular, we show that$\mathcal{S}$is convex. We also give an application of our result to the description of singular values of a product of two arbitrary matrices with prescribed singular values. Keywords:convexity theorems, Cartan decomposition, spherical functions, product formula, semisimple Lie groups, singular valuesCategories:43A90, 53C35, 15A18 38. CJM 2002 (vol 54 pp. 795) Möller, Rögnvaldur G.  Structure Theory of Totally Disconnected Locally Compact Groups via Graphs and Permutations Willis's structure theory of totally disconnected locally compact groups is investigated in the context of permutation actions. This leads to new interpretations of the basic concepts in the theory and also to new proofs of the fundamental theorems and to several new results. The treatment of Willis's theory is self-contained and full proofs are given of all the fundamental results. Keywords:totally disconnected locally compact groups, scale function, permutation groups, groups acting on graphsCategories:22D05, 20B07, 20B27, 05C25 39. CJM 2000 (vol 52 pp. 633) Walters, Samuel G.  Chern Characters of Fourier Modules Let$A_\theta$denote the rotation algebra---the universal$C^\ast$-algebra generated by unitaries$U,V$satisfying$VU=e^{2\pi i\theta}UV$, where$\theta$is a fixed real number. Let$\sigma$denote the Fourier automorphism of$A_\theta$defined by$U\mapsto V$,$V\mapsto U^{-1}$, and let$B_\theta = A_\theta \rtimes_\sigma \mathbb{Z}/4\mathbb{Z}$denote the associated$C^\ast$-crossed product. It is shown that there is a canonical inclusion$\mathbb{Z}^9 \hookrightarrow K_0(B_\theta)$for each$\theta$given by nine canonical modules. The unbounded trace functionals of$B_\theta$(yielding the Chern characters here) are calculated to obtain the cyclic cohomology group of order zero$\HC^0(B_\theta)$when$\theta$is irrational. The Chern characters of the nine modules---and more importantly, the Fourier module---are computed and shown to involve techniques from the theory of Jacobi's theta functions. Also derived are explicit equations connecting unbounded traces across strong Morita equivalence, which turn out to be non-commutative extensions of certain theta function equations. These results provide the basis for showing that for a dense$G_\delta$set of values of$\theta$one has$K_0(B_\theta)\cong\mathbb{Z}^9$and is generated by the nine classes constructed here. Keywords:$C^\ast$-algebras, unbounded traces, Chern characters, irrational rotation algebras,$K$-groupsCategories:46L80, 46L40 40. CJM 1999 (vol 51 pp. 96) Rösler, Margit; Voit, Michael  Partial Characters and Signed Quotient Hypergroups If$G$is a closed subgroup of a commutative hypergroup$K$, then the coset space$K/G$carries a quotient hypergroup structure. In this paper, we study related convolution structures on$K/G$coming from deformations of the quotient hypergroup structure by certain functions on$K$which we call partial characters with respect to$G$. They are usually not probability-preserving, but lead to so-called signed hypergroups on$K/G$. A first example is provided by the Laguerre convolution on$\left[ 0,\infty \right[$, which is interpreted as a signed quotient hypergroup convolution derived from the Heisenberg group. Moreover, signed hypergroups associated with the Gelfand pair$\bigl( U(n,1), U(n) \bigr)\$ are discussed. Keywords:quotient hypergroups, signed hypergroups, Laguerre convolution, Jacobi functionsCategories:43A62, 33C25, 43A20, 43A90

41. CJM 1998 (vol 50 pp. 342)

Giraldo, Antonio
 Shape fibrations, multivalued maps and shape groups The notion of shape fibration with the near lifting of near multivalued paths property is studied. The relation of these maps---which agree with shape fibrations having totally disconnected fibers---with Hurewicz fibrations with the unique path lifting property is completely settled. Some results concerning homotopy and shape groups are presented for shape fibrations with the near lifting of near multivalued paths property. It is shown that for this class of shape fibrations the existence of liftings of a fine multivalued map, is equivalent to an algebraic problem relative to the homotopy, shape or strong shape groups associated. Keywords:Shape fibration, multivalued map, homotopy groups, shape, groups, strong shape groupsCategories:54C56, 55P55, 55Q05, 55Q07, 55R05
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