Group algebras and crossed products have always played an important role in the theory of C*-algebras, and there has also been considerable interest in various twisted analogues, where the multiplication is twisted by a two-cocycle. Here we shall discuss a very general family of twisted actions of locally compact groups on C*-algebras, and the corresponding twisted crossed product C*-algebras. We shall then establish some of the basic properties of these algebras, motivated by the requirements of some applications we have in mind [2, 9, 10]. Some of our results will be known to others, at least in principle, but we feel that a coherent account might be useful.
Let Λ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of C * (Λ) on certain separable Hilbert spaces of the form L 2 (X, µ), by introducing the notion of a Λ-semibranching function system (a generalization of the semibranching function systems studied by Marcolli and Paolucci). In particular, if Λ is aperiodic, we obtain a faithful representation ofwhere M is the Perron-Frobenius probability measure on the infinite path space Λ ∞ recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a Λ-semibranching function system gives rise to KMS states for C * (Λ). For the higher-rank graphs of Robertson and Steger, we also obtain a representation of, where X is a fractal subspace of [0, 1] by embedding Λ ∞ into [0, 1] as a fractal subset X of [0,1]. In this latter case we additionally show that there exists a KMS state for C * (Λ) whose inverse temperature is equal to the Hausdorff dimension of X. Finally, we construct a wavelet system for L 2 (Λ ∞ , M ) by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.2010 Mathematics Subject Classification: 46L05.
We define the notion of "projective" multiresolution analyses, for which, by definition, the initial space corresponds to a finitely generated projective module over the algebra C(T n ) of continuous complex-valued functions on an n-torus. The case of ordinary multi-wavelets is that in which the projective module is actually free. We discuss the properties of projective multiresolution analyses, including the frames which they provide for L 2 (R n ). Then we show how to construct examples for the case of any diagonal 2 × 2 dilation matrix with integer entries, with initial module specified to be any fixed finitely generated projective C(T 2 )-module. We compute the isomorphism classes of the corresponding wavelet modules.In classical wavelet theory one uses multi-resolution analyses to construct (multi-) wavelets and their corresponding orthonormal bases or frames for L 2 (R n ). In almost all applications the scaling functions and wavelets have continuous Fourier transforms. This continuity is a significant and interesting condition. If one requires just a bit more, then one finds that in the frequency domain one is dealing with what are called projective modules over C(T n ) (or equivalently, with the spaces of continuous cross-sections of complex vector bundles over T n ). The case of a single scaling function or wavelet corresponds to the free module of rank 1, whereas the case of several orthogonal scaling functions or wavelets corresponds to free modules of higher rank. This leads one to ask whether wavelet theory carries over to the case of general projective modules over C(T n ). It is the purpose of this article to show that the answer is affirmative.For a given dilation matrix A we define a projective multiresolution analysis for L 2 (R n ) to be an increasing sequence {V j } of subspaces having the usual properties, with the one exception that instead of V 0 being the linear span of the integer translates of one or Math Subject Classifications. Primary: 46L99; secondary: 42C15, 46H25, 47A05.
We consider wavelets in L^2(R^d) which have generalized multiresolutions. This means that the initial resolution subspace V_0 in L^2(R^d) is not singly generated. As a result, the representation of the integer lattice Z^d restricted to V_0 has a nontrivial multiplicity function. We show how the corresponding analysis and synthesis for these wavelets can be understood in terms of unitary-matrix-valued functions on a torus acting on a certain vector bundle. Specifically, we show how the wavelet functions on R^d can be constructed directly from the generalized wavelet filters.Comment: 34 pages, AMS-LaTeX ("amsproc" document class) v2 changes minor typos in Sections 1 and 4, v3 adds a number of references on GMRA theory and wavelet multiplicity analysis; v4 adds material on pages 2, 3, 5 and 10, and two more reference
We first give general structural results for the twisted group algebras C*iG, a) of a locally compact group G with large abelian subgroups. In particular, we use a theorem of Williams to realise C*(G, a) as the sections of a C*-bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras. Next we use a device of Rosenberg to show that, when T is a discrete subgroup of a solvable Lie group G,the #-groups Km{C*(T, a)) are isomorphic to certain twisted Kgroups K*{G/r, ô{a)) of the homogeneous space G/V, and we discuss how the twisting class <5((j) £ //3(<7/T, Z) depends on the cocycle a. For many particular groups, such as Z" or the integer Heisenberg group,
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