Coincidence and common fixed point theorems for b-quasi contractive mappings on metric spaces endowed with binary relations and involving suitable comparison functions are presented. Our results generalize, improve, and extend several recent results. As an application, we study the existence of solutions for some class of integral equations.
We shall consider separable C *-dynamical systems (A, G, α) for which the induced action of the group G on the primitive ideal space Prim(A) of the C *-algebra A is free. We shall discuss how the representation theory of the associated crossed product C *-algebra A α G depends on the representation theory of A and the properties of the action of G on Prim(A) and the spectrumÂ. After surveying some earlier results, we shall describe some recent joint work with Astrid an Huef. The main tools are the notion of strength of convergence in orbit spaces and the notions of upper and lower multiplicities for irreducible representations. We apply these ideas to give necessary and sufficient conditions, in terms of A and the action of G, for A α G to be (i) a continuous trace C *-algebra, (ii) a Fell algebra and (iii) a bounded trace C *-algebra. For the case of amenable G, we can apply a result of Leung and Ng to determine when A α G is (iv) a liminal C *-algebra and (v) a Type I C *-algebra. The results in (i) and (iii)-(v) extend some earlier special cases in which the C *-algebra A was assumed to have the corresponding property.
Let H be a closed connected subgroup of a connected, simply connected exponential solvable Lie group G. We consider the deformation space [Formula: see text] of a discontinuous subgroup Γ of G for the homogeneous space G/H. When H contains [G, G], we exhibit a description of the space [Formula: see text] which appears to involve GLk(ℝ) as a direct product factor, where k designates the rank of Γ. The moduli space [Formula: see text] is also described. Consequently, we prove in such a setup that the local rigidity property fails to hold globally on [Formula: see text] and that every element of the parameters space is topologically stable.
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