18 research outputs found
Composition Operators and Endomorphisms
If is an inner function, then composition with induces an
endomorphism, , of that leaves
invariant. We investigate the structure of the
endomorphisms of and that implement
through the representations of and
in terms of multiplication operators on
and . Our analysis, which is based on work
of R. Rochberg and J. McDonald, will wind its way through the theory of
composition operators on spaces of analytic functions to recent work on Cuntz
families of isometries and Hilbert -modules
Wavelets and graph -algebras
Here we give an overview on the connection between wavelet theory and
representation theory for graph -algebras, including the higher-rank
graph -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 \cite{FGKP}, 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 \cite{hammond} to higher-rank graphs, giving a
third family of wavelets associated to higher-rank graphs
\'Etale groupoids and Steinberg algebras, a concise introduction
We give a concise introduction to (discrete) algebras arising from \'etale
groupoids, (aka Steinberg algebras) and describe their close relationship with
groupoid C*-algebras. Their connection to partial group rings via inverse
semigroups also explored
The groupoid approach to Leavitt path algebras
When the theory of Leavitt path algebras was already quite advanced, it was discovered that some of the more difficult questions were susceptible to a new approach using topological groupoids. The main result that makes this possible is that the Leavitt path algebra of a graph is graded isomorphic to the Steinberg algebra of the graphâs boundary path groupoid. This expository paper has three parts: Part 1 is on the Steinberg algebra of a groupoid, Part 2 is on the path space and boundary path groupoid of a graph, and Part 3 is on the Leavitt path algebra of a graph. It is a self-contained reference on these topics, intended to be useful to beginners and experts alike. While revisiting the fundamentals, we prove some results in greater generality than can be found elsewhere, including the uniqueness theorems for Leavitt path algebras
Subquotients of Hecke C*-algebras
We realize the Hecke C*-algebra C-Q of Bost and Connes as a direct limit of Hecke C*-algebras which are semigroup crossed products by N-F, for F a finite set of primes. For each approximating Hecke C*-algebra we describe a composition series of ideals. In all cases there is a large type I ideal and a commutative quotient, and the intermediate subquotients are direct sums of simple C*-algebras. We can describe the simple summands as ordinary crossed products by actions of Z(S) for S a finite set of primes. When vertical bar S vertical bar = 1, these actions are odometers and the crossed products are Bunce-Deddens algebras; when vertical bar S vertical bar > 1, the actions are an apparently new class of higher-rank odometer actions, and the crossed products are an apparently new class of classifiable AT-algebras
Co-universal C*-algebras associated to generalised graphs
We introduce P-graphs, which are generalisations of directed graphs in which paths have a degree in a semigroup P rather than a length in â. We focus on semigroups P arising as part of a quasi-lattice ordered group (G, P) in the sense of Nica, and on P-graphs which are finitely aligned in the sense of Raeburn and Sims. We show that each finitely aligned P-graph admits a C*-algebra C*min (Î) which is co-universal for partialisometric representations of Î which admit a coaction of G compatible with the P-valued length function. We also characterise when a homomorphism induced by the co-universal property is injective. Our results combined with those of Spielberg show that every Kirchberg algebra is Morita equivalent to Cmin* (Î) for some (â2* â)-graph Î. © 2012 Hebrew University Magnes Press