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209
A survey of foliations and operator algebras
 Proc. Sympos. Pure
, 1982
"... 1 Transverse measure for flows 4 2 Transverse measure for foliations 6 ..."
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Cited by 82 (6 self)
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1 Transverse measure for flows 4 2 Transverse measure for foliations 6
On spectral theory for Schrödinger operators with strongly singular potentials
 Math. Nachr
, 2006
"... Abstract. We examine two kinds of spectral theoretic situations: First, we recall the case of selfadjoint halfline Schrödinger operators on [a, ∞), a ∈ R, with a regular finite end point a and the case of Schrödinger operators on the real line with locally integrable potentials, which naturally le ..."
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Cited by 64 (9 self)
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Abstract. We examine two kinds of spectral theoretic situations: First, we recall the case of selfadjoint halfline Schrödinger operators on [a, ∞), a ∈ R, with a regular finite end point a and the case of Schrödinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and 2 × 2 matrixvalued Herglotz functions representing the associated Weyl–Titchmarsh coefficients. Second, we contrast this with the case of selfadjoint halfline Schrödinger operators on (a, ∞) with a potential strongly singular at the end point a. We focus on situations where the potential is so singular that the associated maximally defined Schrödinger operator is selfadjoint (equivalently, the associated minimally defined Schrödinger operator is essentially selfadjoint) and hence no boundary condition is required at the finite end point a. For this case we show that the Weyl–Titchmarsh coefficient in this strongly singular context still determines the associated spectral function, but ceases to posses the Herglotz property. However, as will be shown, Herglotz function techniques continue to play a decisive role in the spectral theory for strongly singular Schrödinger operators. 1.
Embeddings of reduced free products of operator algebras
 Pacific J. Math. 199
, 2001
"... Abstract. Given reduced amalgamated free products of C ∗ –algebras (A, φ) = ∗ ι∈I (Aι, φι) and (D, ψ) = ∗ ι∈I (Dι, ψι), an embedding A ↩ → D is shown to exist assuming there are conditional expectation preserving embeddings Aι ↩ → Dι. This result is extended to show the existence of the reduced am ..."
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Cited by 48 (14 self)
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Abstract. Given reduced amalgamated free products of C ∗ –algebras (A, φ) = ∗ ι∈I (Aι, φι) and (D, ψ) = ∗ ι∈I (Dι, ψι), an embedding A ↩ → D is shown to exist assuming there are conditional expectation preserving embeddings Aι ↩ → Dι. This result is extended to show the existence of the reduced amalgamated free product of certain classes of unital completely positive maps. Finally, the reduced amalgamated free product of von Neumann algebras is defined in the general case and analogues of the above mentioned results are proved for von Neumann algebras. Introduction. We begin with some standard facts about freeness in groups, analogues of which we will consider in C ∗ –algebras. If H is a subgroup of a group G and if Gι is a subgroup of G containing H for every ι in some index set I, let us say that the family (Gι)ι∈I is free over H (or free with amalgamation over H) if g1g2 · · · gn / ∈ H whenever gj ∈ Gιj \H for some ιj ∈ I
On the spectral decomposition of affine Hecke algebras
 J. Inst. Math. Jussieu
"... Abstract. An affine Hecke algebra H contains a large abelian subalgebra A spanned by the BernsteinZelevinskiLusztig basis elements θx, where x runs over (an extension of) the root lattice. The center Z of H is the subalgebra of Weyl group invariant elements in A. The natural trace (“evaluation at ..."
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Cited by 39 (11 self)
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Abstract. An affine Hecke algebra H contains a large abelian subalgebra A spanned by the BernsteinZelevinskiLusztig basis elements θx, where x runs over (an extension of) the root lattice. The center Z of H is the subalgebra of Weyl group invariant elements in A. The natural trace (“evaluation at the identity”) of the affine Hecke algebra can be written as integral of a certain rational nform (with values in the linear dual of H) over a cycle in the algebraic torus T = spec(A). This cycle is homologous to a union of “local cycles”. We show that this gives rise to a decomposition of the trace as an integral of positive local traces against an explicit probability measure on the spectrum W0\T of Z. From this result we derive the Plancherel formula of the affine Hecke algebra.
Khintchine type inequalities for reduced free products and applications
, 2005
"... We prove Khintchine type inequalities for words of a fixed length in a reduced free product of C ∗algebras (or von Neumann algebras). These inequalities imply that the natural projection from a reduced free product onto the subspace generated by the words of a fixed length d is completely bounded w ..."
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Cited by 36 (3 self)
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We prove Khintchine type inequalities for words of a fixed length in a reduced free product of C ∗algebras (or von Neumann algebras). These inequalities imply that the natural projection from a reduced free product onto the subspace generated by the words of a fixed length d is completely bounded with norm depending linearly on d. We then apply these results to various approximation properties on reduced free products. As a first application, we give a quick proof of Dykema’s theorem on the stability of exactness under the reduced free product for C ∗algebras. We next study the stability of the completely contractive approximation property (CCAP) under reduced free product. Our first result in this direction is that a reduced free product of finite dimensional C ∗algebras has the CCAP. The second one asserts that a von Neumann reduced free product of injective von Neumann algebras has the weak ∗ CCAP. In the case of group C ∗algebras, we show that a free product of weakly amenable groups with constant 1 is weakly amenable.
TYPE II NON COMMUTATIVE GEOMETRY. I. Dixmier Trace In Von Neumann Algebras
, 2003
"... We define the notion of Connesvon Neumann spectral triple and consider the associated index problem. We compute the analytic ChernConnes character of such a generalized spectral triple and prove the corresponding local formula for its Hochschild class. This formula involves the Dixmier trace for ..."
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Cited by 35 (3 self)
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We define the notion of Connesvon Neumann spectral triple and consider the associated index problem. We compute the analytic ChernConnes character of such a generalized spectral triple and prove the corresponding local formula for its Hochschild class. This formula involves the Dixmier trace for II∞ von Neumann algebras. In the case of foliations, we identify this Dixmier trace with the corresponding measured Wodzicki residue.
Amenability for dual Banach algebras
 Run 2] [Sel] [Spr] [Woo 1] [Woo 2] V. Runde, Lectures on Amenability. Lecture Notes in Mathematics 1774
, 2002
"... We define a Banach algebra A to be dual if A = (A∗) ∗ for a closed submodule A ∗ of A ∗. The class of dual Banach algebras includes all W ∗algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Ban ..."
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Cited by 30 (8 self)
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We define a Banach algebra A to be dual if A = (A∗) ∗ for a closed submodule A ∗ of A ∗. The class of dual Banach algebras includes all W ∗algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions an amenable dual Banach algebra is already superamenable and thus finitedimensional. We then develop two notions of amenability — Connesamenability and strong Connesamenability — which take the w ∗topology on dual Banach algebras into account. We relate the amenability of an Arens regular Banach algebra A to the (strong) Connesamenability of A ∗ ∗ ; as an application, we show that there are reflexive Banach spaces with the approximation property such that L(E) is not Connesamenable. We characterize the amenability of inner amenable locally compact groups in terms of their algebras of pseudomeasures. Finally, we give a proof of the known fact that the amenable von Neumann algebras are the subhomogeneous ones which avoids the equivalence of amenability and nuclearity for C ∗algebras.
A C∗ algebraic framework for quantum groups
, 2003
"... We develop a general framework to deal with the unitary representations of quantum groups using the language of C ∗algebras. Using this framework, we prove that the duality holds in a general context. This extends the framework of the duality theorem using the language of von Neumann algebras previ ..."
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Cited by 29 (1 self)
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We develop a general framework to deal with the unitary representations of quantum groups using the language of C ∗algebras. Using this framework, we prove that the duality holds in a general context. This extends the framework of the duality theorem using the language of von Neumann algebras previously developed by Masuda and Nakagami.