Judith A. Packer

Twisted crossed products of C *-algebras

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.

On the structure of twisted group C*-algebras

We first give general structural results for the twisted group algebras C*(G, σ) of a locally compact group G with large abelian subgroups. In particular, we use a theorem of Williams to realise C*(G, σ) 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 Γ is a discrete subgroup of a solvable Lie group G, the K-groups K * (C*(Γ, σ)) are isomorphic to certain twisted K-groups K*(G/Γ, δ(σ)) of the homogeneous space G/Γ, and we discuss how the twisting class δ(σ) ∈ H 3 (G/Γ, Z) depends on the cocycle σ

Construction of Parseval wavelets from redundant filter systems

We consider wavelets in L2(Rd) which have generalized multiresolutions. This means that the initial resolution subspace V0 in L2(Rd) is not singly generated. As a result, the representation of the integer lattice Zd restricted to V0 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 Rd can be constructed directly from the generalized wavelet filters.

A direct integral decomposition of the wavelet representation

In this paper we use the concept of wavelet sets, as introduced by X. Dai and D. Larson, to decompose the wavelet representation of the discrete group associated to an arbitrary n×n integer dilation matrix as a direct integral of irreducible monomial representations. In so doing we generalize a result of F. Martin and A. Valette in which they show that the wavelet representation is weakly equivalent to the regular representation for the Baumslag-Solitar groups.

K-theoretic invariants for C∗-algebras associated to transformations and induced flows

Abstract We discuss a technique of studying the K-theory of a unital C∗-algebra associated to a homomorphism on a compact metric space (Y Z ) by examining the non-unital C∗-algebra associated to the induced topological flow (Ind Z R (Y), R ). The Thom isomorphism of Connes and the Schwartzman asymptotic cycle are used to calculate the range of the trace corresponding to an invariant measure on the K0 group of C∗(X, R ) for a continuous flow on a compact metric space (X, R ). Under certain conditions projections in C∗(X, R ) with trace r corresponding to cross sections to the flow can be constructed for every positive real number r in this range, again by combining techniques of Connes and Schwartzman. Applications to the calculation of the tracial range of K0(C∗(X, R )) are discussed. In particular, this invariant is calculated for minimal affine actions of Z on n-tori which have quasi-discrete spectrum, and for minimal actions of Z on compact abelian groups with topologically discrete spectrum. In both cases this tracial invariant is shown to be the preimage of the eigenvalues for (Y, Z ) under the natural projection o: R → R / Z =S1.

Wavelets and Graph C∗-Algebras

Here we give an overview on the connection between wavelet theory and representation theory for graph C∗-algebras, including the higher-rank graph C∗-algebras of A. Kumjian and D. Pask. Many authors have studied different aspects of this connection over the last 20 years, and we begin this paper with a survey of the known results. We then discuss several new ways to generalize these results and obtain wavelets associated to representations of higher-rank graphs. In Farsi et al. (J Math Anal Appl 425:241–270, 2015), we introduced the “cubical wavelets” associated to a higher-rank graph. Here, we generalize this construction to build wavelets of arbitrary shapes. We also present a different but related construction of wavelets associated to a higher-rank graph, which we anticipate will have applications to traffic analysis on networks. Finally, we generalize the spectral graph wavelets of Hammond et al. (Appl Comput Harmon Anal 30:129–150, 2011) to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs.

Flow Equivalence for Dynamical Systems and the Corresponding C*-Algebras

The purpose of this paper is to describe a method of using flow equivalence for homeomorphisms to construct certain projections in the corresponding transformation group C*-algebras, by following a program which was outlined in our recent paper [15]. Using methods stemming from work of Connes [3] and Rieffel and Green [21] we will explicitly construct the strong Morita equivalence bimodule relating two C -algebras associated to flow equivalent transformations, and the corresponding projections; these projections will be related to the isomorphism of Connes between KO(C*(Y,Z)) and K1(M(Y)) where (Y,Z) is a dynamical system, C*(Y,Z) the associated C*-algebra, and M(Y) is the mapping torus for (Y.Z).

Probability measures on solenoids corresponding to fractal wavelets

The measure on generalized solenoids constructed using filters by Dutkay and Jorgensen in (12) is analyzed further by writing the solenoid as the product of a torus and a Cantor set. Using this decomposition, key differences are revealed between solenoid measures associated with classical filters in R d and those associated with filters on inflated fractal sets. In particular, it is shown that the classical case produces atomic fiber measures, and as a result supports both suitably defined solenoid MSF wavelets and systems of imprimitivity for the corresponding wavelet representation of the generalized Baumslag-Solitar group. In contrast, the fiber measures for filters on inflated fractal spaces cannot be atomic, and thus can support neither MSF wavelets nor systems of imprimitivity.

A Survey of Projective Multiresolution Analyses and a Projective Multiresolution Analysis Corresponding to the Quincunx Lattice

We give a survey of the concept of projective multiresolution analyses as introduced by M. Rieffel and studied further by M. Rieffel and the author. We give examples of projective multiresolution analyses corresponding to the nondiagonal 2 × 2 integer dilation matrix \(\left( {\begin{array}{*{20}c} 0&1\\ 2&0\\ \end{array}} \right)\) that has determinant -2, and also to the nondiagonal 2 × 2 matrix \(\left( {\begin{array}{*{20}c} 1&1\\ {-1}&1\\ \end{array}} \right)\) having determinant 2 related to the quincunx lattice. The method of construction follows that given by Rieffel and the author in their earlier work but also poses new problems. In both examples given here, the one-dimensional initial C(T2)-modules are not free, but the in the quincunx case, the one-dimensional wavelet module is free, whereas in the case corresponding to the dilation matrix whose determinant is negative, the one-dimensional wavelet module is not free either.

On the normalizer of certain subalgebras of group-measure factors

We study operator algebras constructed from ergodic actions of discrete groups on compact Lebesgue spaces. We show that under appropriate conditions, the normalizer of a subalgebra corresponding to a quotient action depends on the relative elementary spectrum of the action over the quotient action. Using these results and methods of groupoid cohomology we prove the existence of an uncountable family of Cartan subalgebras of the hyperfinite IIj factor, no two of which are inner conjugate. 1. Introductory remarks. Let the countable discrete group G act freely and ergodically on the compact Lebesgue space (X, ix) so as to leave the finite measure li quasi-invariant. By the von Neumann-Murray group-measure construction we then may form the factor F(X, G). Let (Y, v, G) be a quotient action of (X, /i, G); by this we mean that there exists a surjective Borel map <p : X —• Y satisfying <p*(p) = v and v(xg) = v(x)g Qx a. e.) for every g in G; we also call (X, ix, G) an extension of (Yt v, G). Throughout the following we will assume that in the decomposition of ix over the fibers of </ > as ix = fixydv, ixyg = g*Qxy). This implies that there exists a G-invariant conditional expectation of L°°(X) onto L°°(Y). The map y provides a homomorphism of the ergodic measure groupoid X x G, onto the groupoid Y x G, and one thus obtains an injection of F(Y, G) into F(Xt G), which we shall denote by </?*. Since G acts ergodically on (X, ix) it follows that G acts ergodically on (Y, v), and if G acts freely on (Y, v) as well, F(Y, G) injects naturally as a subfactor of F(Xt G). THEOREM 1. Let (Y, v, G) be a free quotient action of(X, ix> G) where X and Y are compact Lebesgue spaces and ix (hence v) is finite and G-invariant, and where the countable discrete group G acts freely and ergodically on (X, LX). Then there is a canonical correspondence between the intermediate subalgebras ofF(Yt G) and F(Y, G) and the quotient actions (Z, G) of(X, G) which are extension actions of (Y, G). REMARK 1. Since (Z, G) will inherit ergodicity from (X, G) and freeness from (Y, G), it follows immediately from the theorem that any intermediate subalgebra, which will be of the form F(Z, G) for some (Z, G), will be a subfactor. Received by the editors April 1, 1982 and, in revised form, June 7, 1982 1980 Mathematical Subject Classification. Primary 46L40, 46L10, 28D99; Secondary 47C15, 47D99.