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26. 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

27. 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

28. 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

29. 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

30. 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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