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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 43 (0 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
A class of C∗algebras generalizing both graph algebras and homeomorphism C∗algebras I, . . .
, 2003
"... We introduce a new class of C∗algebras, which is a generalization of both graph algebras and homeomorphism C ∗algebras. This class is very large and also very tractable. We prove the socalled gaugeinvariant uniqueness theorem and the CuntzKrieger uniqueness theorem, and compute the Kgroups of ..."
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Cited by 37 (5 self)
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We introduce a new class of C∗algebras, which is a generalization of both graph algebras and homeomorphism C ∗algebras. This class is very large and also very tractable. We prove the socalled gaugeinvariant uniqueness theorem and the CuntzKrieger uniqueness theorem, and compute the Kgroups of our algebras.
Graphs, groupoids and CuntzKrieger algebras
, 1996
"... We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of onesided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions the ..."
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Cited by 33 (16 self)
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We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of onesided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions their C algebras are Morita equivalent; the groupoid C algebra C (G) is the CuntzKrieger algebra of an infinite f0; 1g matrix defined by G, and that the algebras C (G(?)) contain the C algebras used by Doplicher and Roberts in their duality theory for compact groups. We then analyse the ideal structure of these groupoid C algebras using the general theory of Renault, and calculate their Ktheory. 1 Introduction Over the past fifteen years many C algebras and classes of C algebras have been given groupoid models. Here we consider locally finite directed graphs, which may have infinitely many vertices, but only finitely many edges in and out of each vertex. We associate ...
On C∗algebras associated with C∗correspondences
, 2003
"... We study C∗algebras arising from C∗correspondences, which was introduced by the author. We prove the gaugeinvariant uniqueness theorem, and obtain conditions for our C∗algebras to be nuclear, exact, or satisfy the Universal Coefficient Theorem. We also obtain a 6term exact sequence of Kgroup ..."
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Cited by 21 (2 self)
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We study C∗algebras arising from C∗correspondences, which was introduced by the author. We prove the gaugeinvariant uniqueness theorem, and obtain conditions for our C∗algebras to be nuclear, exact, or satisfy the Universal Coefficient Theorem. We also obtain a 6term exact sequence of Kgroups involving the Kgroups of our C∗algebras.
IDEAL STRUCTURE OF C∗ALGEBRAS ASSOCIATED WITH C∗correspondences
, 2003
"... We study C∗algebras arising from C∗correspondences, which was introduced by the author. We prove the gaugeinvariant uniqueness theorem, and obtain conditions for our C∗algebras to be nuclear, exact, or satisfy the Universal Coefficient Theorem. We also obtain a 6term exact sequence of Kgroups ..."
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Cited by 11 (2 self)
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We study C∗algebras arising from C∗correspondences, which was introduced by the author. We prove the gaugeinvariant uniqueness theorem, and obtain conditions for our C∗algebras to be nuclear, exact, or satisfy the Universal Coefficient Theorem. We also obtain a 6term exact sequence of Kgroups involving the Kgroups of our C∗algebras.
The C∗algebra of a Hilbert bimodule
 BOLLETTINO UMI. SERIE VIII 1 B
, 1998
"... We regard a right Hilbert C∗ –module X over a C ∗ –algebra A endowed with an isometric ∗–homomorphism φ: A → LA(X) as an object XA of the C*–category of right Hilbert A–modules. Following [11], we associate to it a C ∗ –algebra OXA containing X as a “Hilbert A–bimodule in OXA ”. If X is full and fin ..."
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Cited by 10 (1 self)
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We regard a right Hilbert C∗ –module X over a C ∗ –algebra A endowed with an isometric ∗–homomorphism φ: A → LA(X) as an object XA of the C*–category of right Hilbert A–modules. Following [11], we associate to it a C ∗ –algebra OXA containing X as a “Hilbert A–bimodule in OXA ”. If X is full and finite projective OXA is the C ∗ –algebra C ∗ (X) , the generalization of the Cuntz–Krieger algebras introduced by Pimsner [27]. More generally, C ∗ (X) is canonically embedded in OXA as the C ∗ –subalgebra generated by X. Conversely, if X is full OXA is canonically embedded in C∗ (X) ∗ ∗. Moreover, regarding X as an object AXA of the C ∗ –category of Hilbert A–bimodules, we associate to it a C ∗ –subalgebra OAXA of OXA commuting with A, on which X induces a canonical endomorphism ρ. We discuss conditions under which A and O are the AXA relative commutant of each other and X is precisely the subspace of intertwiners in OXA between the identity and ρ on O AXA. We also discuss conditions which imply the simplicity of C ∗ (X) or of OXA; in particular, if X is finite projective and full, C ∗ (X) will be simple if A is X–simple and the “Connes spectrum ” of X is T.
Almost multiplicative morphisms and almost commuting matrices
 J. Operator Theory
, 1998
"... Abstract. We prove that a contractive positive linear map which is approximately multiplicative and approximately injective from C(X) into certain unital simple C∗algebras of real rank zero and stable rank one is close to a homomorphism (with finite dimensional range) if a necessary Ktheoretical ..."
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Cited by 7 (7 self)
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Abstract. We prove that a contractive positive linear map which is approximately multiplicative and approximately injective from C(X) into certain unital simple C∗algebras of real rank zero and stable rank one is close to a homomorphism (with finite dimensional range) if a necessary Ktheoretical obstruction vanishes and dimension of X is no more than two. We also show that the above is false it the dimension of X is greater than 2, in general.
QUANTUM LENS SPACES AND GRAPH ALGEBRAS
 PACIFIC JOURNAL OF MATHEMATICS
, 2003
"... We construct the C∗algebra C(Lq(p; m1,...,mn)) of continuous functions on the quantum lens space as the fixed point algebra for a suitable action of Zp on the algebra C(S 2n−1 q corresponding to the quantum odd dimensional sphere. We show that C(Lq(p; m1,...,mn)) is isomorphic to the graph algebra ..."
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Cited by 7 (1 self)
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We construct the C∗algebra C(Lq(p; m1,...,mn)) of continuous functions on the quantum lens space as the fixed point algebra for a suitable action of Zp on the algebra C(S 2n−1 q corresponding to the quantum odd dimensional sphere. We show that C(Lq(p; m1,...,mn)) is isomorphic to the graph algebra C∗ ( L (p;m1,...,mn) 2n−1. This allows us to determine the ideal structure and, at least in principle, calculate the Kgroups of C(Lq(p; m1,...,mn)). Passing to the limit with natural imbeddings of the quantum lens spaces we construct the quantum infinite lens space, or the quantum EilenbergMacLane space of type (Zp, 1).
Strong Morita equivalence of higherdimensional noncommutative tori
, 2005
"... Abstract. We show that two C ∗algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K0groups and centers, extending N. C. Phillips’s result in the case that the algebras are simple. This is also generalized to the twisted group C ∗algebras of arb ..."
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Cited by 5 (2 self)
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Abstract. We show that two C ∗algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K0groups and centers, extending N. C. Phillips’s result in the case that the algebras are simple. This is also generalized to the twisted group C ∗algebras of arbitrary finitely generated abelian groups. 1.
C ∗STRUCTURE AND KTHEORY OF BOUTET DE MONVEL’S ALGEBRA
, 2001
"... Abstract. We consider the norm closure A of the algebra of all operators of order and class zero in Boutet de Monvel’s calculus on a manifold X with boundary ∂X. We first describe the image and the kernel of the continuous extension of the boundary principal symbol homomorphism to A. If X is connect ..."
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Cited by 5 (4 self)
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Abstract. We consider the norm closure A of the algebra of all operators of order and class zero in Boutet de Monvel’s calculus on a manifold X with boundary ∂X. We first describe the image and the kernel of the continuous extension of the boundary principal symbol homomorphism to A. If X is connected and ∂X is not empty, we then show that the Kgroups of A are topologically determined. In case the manifold, its boundary, and the cotangent space of its interior have torsion free Ktheory, we get Ki(A/K) ≃ Ki(C(X))⊕K1−i(C0(T ∗ ˙ X)), i = 0,1, with K denoting the compact ideal, and T ∗ ˙ X denoting the cotangent bundle of the interior. Using Boutet de Monvel’s index theorem, we also prove that the above formula holds for i = 1 even without this torsionfree hypothesis. For the case of orientable, twodimensional X, K0(A) ≃ Z 2g+m and K1(A) ≃ Z 2g+m−1, where g is the genus of X and m is the number of connected components of ∂X. We also obtain a composition sequence 0 ⊂ K ⊂ G ⊂ A, with A/G commutative and G/K isomorphic to the algebra of all continuous functions on the cosphere bundle of ∂X with values in compact operators on L 2 (R+). 1.