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48
Inverse semigroups and combinatorial C*algebras
, 2008
"... We describe a special class of representations of an inverse semigroup S on Hilbert’s space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the sp ..."
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Cited by 30 (7 self)
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We describe a special class of representations of an inverse semigroup S on Hilbert’s space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the space of ultrafilters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*algebra of the groupoid of germs for the action of S on its tight spectrum. We then treat the case of certain inverse semigroups constructed from semigroupoids, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in onetoone correspondence with representations of the semigroupoid, and consequently the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case.
Integrable and proper actions on C ∗ algebras, and squareintegrable representations of groups
, 1999
"... Abstract. We propose a definition of what should be meant by a proper action of a locally compact group on a C ∗algebra. We show that when the C ∗algebra is commutative this definition exactly captures the usual notion of a proper action on a locally compact space. We then discuss how one might de ..."
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Abstract. We propose a definition of what should be meant by a proper action of a locally compact group on a C ∗algebra. We show that when the C ∗algebra is commutative this definition exactly captures the usual notion of a proper action on a locally compact space. We then discuss how one might define a generalized fixedpoint algebra. The goal is to show that the generalized fixedpoint algebra is strongly Morita equivalent to an ideal in the crossed product algebra, as happens in the commutative case. We show that one candidate gives the desired algebra when the C ∗algebra is commutative. But very recently Exel has shown that this candidate is too big in general. Finally, we consider in detail the application of these ideas to actions of a locally compact group on the algebra of compact operators (necessarily coming from unitary representations), and show that this gives an attractive view of the subject of squareintegrable representations. There is a variety of situations in which actions of locally compact groups on noncommutative C ∗algebras appear to be “proper ” in a way analogous to proper actions of
Spaces Which Are Not Negatively Curved
 Comm. in Anal. and Geom
, 1997
"... this paper will be 2dimensional. Definition of a lamination ..."
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Cited by 13 (6 self)
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this paper will be 2dimensional. Definition of a lamination
Index theory and NonCommutative Geometry I. Higher Families Index Theory
, 2005
"... We prove an index theorem for foliated manifolds. We do so by constructing a push forward map in cohomology for a Koriented map from an arbitrary manifold to the space of leaves of an oriented foliation, and by constructing a ChernConnes character from the Ktheory of the compactly supported smoo ..."
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Cited by 12 (7 self)
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We prove an index theorem for foliated manifolds. We do so by constructing a push forward map in cohomology for a Koriented map from an arbitrary manifold to the space of leaves of an oriented foliation, and by constructing a ChernConnes character from the Ktheory of the compactly supported smooth functions on the holonomy groupoid of the foliation to the Haefliger cohomology of the foliation. Combining these with the ConnesSkandalis topological index map and the classical Chern character gives a commutative diagram from which the index theorem follows immediately.
State spaces of operator algebras: Basic theory, orientations, and C*products, by
"... One of the most important auxiliary objects associated with an operator algebra is its state space. The two books under review describe the authors ’ solutions, obtained together with H. HancheOlsen and B. Iochum [1], [2], [9], to the problems: What data must be added to a state space so that the o ..."
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Cited by 10 (4 self)
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One of the most important auxiliary objects associated with an operator algebra is its state space. The two books under review describe the authors ’ solutions, obtained together with H. HancheOlsen and B. Iochum [1], [2], [9], to the problems: What data must be added to a state space so that the operator algebra can be recovered? and Which convex sets can arise as state spaces? As that work is now around twenty years old, they are able to present it here in a very finished form. Operator algebras come in two varieties, C*algebras and von Neumann algebras. (My friends who work in the non selfadjoint theory will forgive me for using the term in this way for the purposes of this review.) Concretely, a C*algebra is a linear subspace of B(H) (the space of bounded operators on a complex Hilbert space H) which is algebraically closed under operator products and adjoints and is topologically closed in norm. Concrete von Neumann algebras are defined similarly, now requiring closure in the weak * topology. There are abstract characterizations as well: C*algebras are complex Banach algebras equipped with an involution satisfying ‖x ∗ x ‖ = ‖x ‖ 2, and von Neumann algebras are C*algebras that have a
Bivariant Ktheory via correspondences
, 2008
"... Abstract. We use correspondences to define a purely topological equivariant bivariant Ktheory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. We replace smooth Koriented maps by a class of Koriented normal maps, which are ..."
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Abstract. We use correspondences to define a purely topological equivariant bivariant Ktheory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. We replace smooth Koriented maps by a class of Koriented normal maps, which are maps together with a certain factorisation. Our construction does not use any special features of equivariant Ktheory. To highlight this, we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories. We formulate necessary and sufficient conditions for certain duality isomorphisms in the geometric bivariant Ktheory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant Ktheory to Ktheory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, both bivariant Ktheories agree if there is such a duality isomorphism. 1.
GENERALIZED FIXED POINT ALGEBRAS AND SQUAREINTEGRABLE GROUP ACTIONS
, 2000
"... Abstract. We analzye Rieffel’s construction of generalized fixed point algebras in the setting of group actions on Hilbert modules. Let G be a locally compact group acting on a C ∗algebra B. We construct a Hilbert module F over the reduced crossed product of G and B, using a pair (E, R), where E is ..."
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Abstract. We analzye Rieffel’s construction of generalized fixed point algebras in the setting of group actions on Hilbert modules. Let G be a locally compact group acting on a C ∗algebra B. We construct a Hilbert module F over the reduced crossed product of G and B, using a pair (E, R), where E is an equivariant Hilbert module over B and R is a dense subspace of E with certain properties. The generalized fixed point algebra is the C ∗algebra of compact operators on F. Any Hilbert module over the reduced crossed product arises by this construction for a pair (E, R) that is unique up to isomorphism. A necessary condition for the existence of R is that E be squareintegrable. The consideration of squareintegrable representations of Abelian groups on Hilbert space shows that this condition is not sufficient and that different choices for R may yield different generalized fixed point algebras. If B is proper in Kasparov’s sense, there is a unique R with the required properties. Thus the generalized fixed point algebra only depends on E. 1.