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On Groupoid C∗Algebras, Persistent Homology and TimeFrequency Analysis
"... We study some topological aspects in timefrequency analysis in the context of dimensionality reduction using C ∗algebras and noncommutative topology. Our main objective is to propose and analyze new conceptual and algorithmic strategies for computing topological features of datasets arising in tim ..."
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Cited by 44 (1 self)
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We study some topological aspects in timefrequency analysis in the context of dimensionality reduction using C ∗algebras and noncommutative topology. Our main objective is to propose and analyze new conceptual and algorithmic strategies for computing topological features of datasets arising in timefrequency analysis. The main result of our work is to illustrate how noncommutative C ∗algebras and the concept of Morita equivalence can be applied as a new type of analysis layer in signal processing. From a conceptual point of view, we use groupoid C∗algebras constructed with timefrequency data in order to study a given signal. From a computational point of view, we consider persistent homology as an algorithmic tool for estimating topological properties in timefrequency analysis. The usage of C∗algebras in our environment, together with the problem of designing computational algorithms, naturally leads to our proposal of using AFalgebras in the persistent homology setting. Finally, a computational toy example is presented, illustrating some elementary aspects of our framework. Due to the interdisciplinary nature
STRONG MORITA EQUIVALENCE OF INVERSE SEMIGROUPS
, 2009
"... We introduce strong Morita equivalence for inverse semigroups. This notion encompasses Mark Lawson’s concept of enlargement. Strongly Morita equivalent inverse semigroups have Morita equivalent universal groupoids in the sense of Paterson and hence strongly Morita equivalent universal and reduced C ..."
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Cited by 13 (4 self)
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We introduce strong Morita equivalence for inverse semigroups. This notion encompasses Mark Lawson’s concept of enlargement. Strongly Morita equivalent inverse semigroups have Morita equivalent universal groupoids in the sense of Paterson and hence strongly Morita equivalent universal and reduced C ∗algebras. As a consequence we obtain a new proof of a result of Khoshkam and Skandalis showing that the C ∗algebra of an Finverse semigroup is strongly Morita equivalent to a cross product of a commutative C ∗algebra by a group.
Nuclearity of semigroup C*algebras and the connection to amenability
 Advances in Math. 244
, 2013
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FIXEDPOINT ALGEBRAS FOR PROPER ACTIONS AND CROSSED PRODUCTS BY HOMOGENEOUS SPACES
, 2009
"... We consider a fixed free and proper action of a locally compact group G on a space T, and actions α: G → Aut A on C∗algebras for which there is an equivariant embedding of (C0(T), rt) in (M(A), α). A recent theorem of Rieffel implies that α is proper and saturated with respect to the subalgebra C ..."
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Cited by 3 (2 self)
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We consider a fixed free and proper action of a locally compact group G on a space T, and actions α: G → Aut A on C∗algebras for which there is an equivariant embedding of (C0(T), rt) in (M(A), α). A recent theorem of Rieffel implies that α is proper and saturated with respect to the subalgebra C0(T)AC0(T) of A, so that his general theory of proper actions gives a Morita equivalence between A ⋊α,r G and a generalised fixedpoint algebra A α. Here we investigate the functor (A, α) ↦ → A α and the naturality of Rieffel’s Morita equivalence, focusing in particular on the relationship between the different functors associated to subgroups and quotients. We then use the results to study induced representations for crossed products by coactions of homogeneous spaces G/H of G, which were previously shown by an Huef and Raeburn to be fixedpoint algebras for the dual action of H on the crossed product by G.
THE MACKEY MACHINE FOR CROSSED PRODUCTS BY REGULAR GROUPOIDS. I
, 2009
"... We first describe a Rieffel induction system for groupoid crossed products. We then use this induction system to show that, given a regular groupoid G and a dynamical system (A, G, α), every irreducible representation of A⋊G is induced from a representation of the group crossed product A(u)⋊Su wher ..."
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Cited by 2 (1 self)
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We first describe a Rieffel induction system for groupoid crossed products. We then use this induction system to show that, given a regular groupoid G and a dynamical system (A, G, α), every irreducible representation of A⋊G is induced from a representation of the group crossed product A(u)⋊Su where u ∈ G (0) , A(u) is a fibre of A, and Su is a stabilizer subgroup of G.
MATH. SCAND. 117 (2015), 5–30 TOPOLOGICAL AMENABILITY IS A BOREL PROPERTY
"... We establish that a σcompact locally compact groupoid possessing a continuous Haar system is topologically amenable if and only if it is Borel amenable. We give some examples and applications. 1. ..."
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We establish that a σcompact locally compact groupoid possessing a continuous Haar system is topologically amenable if and only if it is Borel amenable. We give some examples and applications. 1.