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138
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 55 (5 self)
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1 Transverse measure for flows 4 2 Transverse measure for foliations 6
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 29 (10 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
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 26 (0 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.
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 21 (6 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.
The Hadamard condition for Dirac fields and adiabatic states on RobertsonWalker spacetimes
 Commun. Math. Phys
, 2001
"... We characterise the homogeneous and isotropic gauge invariant and quasifree states for free Dirac quantum fields on RobertsonWalker spacetimes in any even dimension. Using this characterisation, we construct adiabatic vacuum states of order n corresponding to some Cauchy surface. We then show that ..."
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Cited by 16 (2 self)
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We characterise the homogeneous and isotropic gauge invariant and quasifree states for free Dirac quantum fields on RobertsonWalker spacetimes in any even dimension. Using this characterisation, we construct adiabatic vacuum states of order n corresponding to some Cauchy surface. We then show that any two such states (of sufficiently high order) are locally quasiequivalent. We propose a microlocal version of the Hadamard condition for spinor fields on arbitrary spacetimes, which is shown to entail the usual short distance behaviour of the twopoint function. The polarisation set of these twopoint functions is determined from the Dencker connection of the spinorial KleinGordon operator which we show to be equals the (pullback) of the spin connection. Finally it is demonstrated that adiabatic states of infinite order are Hadamard, and that those of order n correspond, in some sense, to a truncated Hadamard series and will therefore allow for a point splitting renormalisation of the expected stressenergy tensor.
The spectral shift operator
 IN MATHEMATICAL RESULTS IN QUANTUM MECHANICS
, 1999
"... We introduce the concept of a spectral shift operator and use it to derive Krein’s spectral shift function for pairs of selfadjoint operators. Our principal tools are operatorvalued Herglotz functions and their logarithms. Applications to Krein’s trace formula and to the BirmanSolomyak spectral ..."
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Cited by 16 (7 self)
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We introduce the concept of a spectral shift operator and use it to derive Krein’s spectral shift function for pairs of selfadjoint operators. Our principal tools are operatorvalued Herglotz functions and their logarithms. Applications to Krein’s trace formula and to the BirmanSolomyak spectral averaging formula are discussed.
An analytic approach to spectral flow in von Neumann algebras
, 2005
"... The analytic approach to spectral flow is about ten years old. In that time it has evolved to cover an ever wider range of examples. The most critical extension was to replace Fredholm operators in the classical sense by BreuerFredholm operators in a semifinite von Neumann algebra. The latter have ..."
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Cited by 14 (7 self)
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The analytic approach to spectral flow is about ten years old. In that time it has evolved to cover an ever wider range of examples. The most critical extension was to replace Fredholm operators in the classical sense by BreuerFredholm operators in a semifinite von Neumann algebra. The latter have continuous spectrum so that the notion of spectral flow turns out to be rather more difficult to deal with. However quite remarkably there is a uniform approach in which the proofs do not depend on discreteness of the spectrum of the operators in question. The first part of this paper gives a brief account of this theory extending and refining earlier results. It is then applied in the latter parts of the paper to a series of examples. One of the most powerful tools is an integral formula for spectral flow first analysed in the classical setting by Getzler and extended to BreuerFredholm operators by some of the current authors. This integral formula was known for Dirac operators in a variety of forms ever since the fundamental papers of Atiyah, Patodi and Singer. One of the purposes of this exposition is to make contact with this early work so that one can understand the recent developments in a proper historical context. In addition we show how to derive these spectral flow formulae in the setting of Dirac operators on (noncompact) covering spaces of a compact spin manifold using the adiabatic method. This answers a question of Mathai connecting Atiyah’s L 2index theorem to our analytic spectral flow. Finally we relate our work to that of Coburn, Douglas, Schaeffer and Singer on Toeplitz operators with almost periodic symbol. We generalise their work to cover the case of matrix valued almost periodic symbols on R N using some ideas of Shubin. This provides us with an opportunity to describe the deepest part of the theory namely the semifinite local index theorem in noncommutative geometry. This theorem, which gives a formula for spectral flow was recently proved by some of the present authors. It provides a farreaching generalisation of the original 1995 result of Connes and Moscovici.
Khintchine type inequalities for reduced free products and applications,’ preprint
"... 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 13 (2 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. 1