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Fedor Sukochev - One of the best experts on this subject based on the ideXlab platform.
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m embedded symmetric operator spaces and the derivation problem
Mathematical Proceedings of the Cambridge Philosophical Society, 2020Co-Authors: Jinghao Huang, Galina Levitina, Fedor SukochevAbstract:Let ℳ be a Semifinite von Neumann algebra with a faithful Semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary Semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 < p < ∞, has a positive answer.
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positive linear isometries in symmetric operator spaces
Integral Equations and Operator Theory, 2018Co-Authors: Fedor Sukochev, Aleksandr VekslerAbstract:Let $$(\mathcal {M},\tau )$$ and $$(\mathcal {N},\nu )$$ be Semifinite von Neumann algebras equipped with faithful normal Semifinite traces and let $$E(\mathcal {M},\tau )$$ and $$F(\mathcal {N},\nu )$$ be symmetric operator spaces associated with these algebras. We provide a sufficient condition on the norm of the space $$F(\mathcal {N},\nu )$$ guaranteeing that every positive linear isometry $$T:E(\mathcal {M},\tau ){\mathop {\longrightarrow }\limits ^{into}} F(\mathcal {N},\nu )$$ is “disjointness preserving” in the sense that $$T(x)T(y)=0$$ provided that $$xy=0$$ , $$0\le x,y\in E(\mathcal {M},\tau )$$ . This fact, in turn, allows us to describe the general form of such isometries.
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sums and intersections of symmetric operator spaces
Journal of Mathematical Analysis and Applications, 2014Co-Authors: E M Semenov, Fedor SukochevAbstract:Abstract Let L ( H ) be the algebra of all bounded operators on a separable Hilbert space H. We completely describe symmetric operator ideals E in L ( H ) , which can be represented as a sum (or, an intersection) of two other (distinct from E ) symmetric operator ideals in L ( H ) . We also present a version of our results for symmetric operator spaces affiliated with a Semifinite atomless von Neumann algebra M .
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completeness of quasi normed symmetric operator spaces
Indagationes Mathematicae, 2014Co-Authors: Fedor SukochevAbstract:Abstract We show that (generalized) Calkin correspondence between quasi-normed symmetric sequence spaces and symmetrically quasi-normed ideals of compact operators on an infinite-dimensional Hilbert space preserves completeness. We also establish a Semifinite version of this result.
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The local index formula in Semifinite von Neumann algebras. II. The even case
2012Co-Authors: Alan L. Carey, Adam Rennie, John Phillips, Fedor SukochevAbstract:All authors were supported by grants from ARC (Australia) and NSERC (Canada), in addition the third named author acknowledges a University of Newcastle early career researcher grant. 1 Address for correspondence 1 2 THE LOCAL INDEX FORMULA IN Semifinite VON NEUMANN ALGEBRAS II: THE EVEN CASE We generalise the even local index formula of Connes and Moscovici to the case of spectral triples for a ∗-subalgebra A of a general Semifinite von Neumann algebra. The proof is a variant of that for the odd case which appears in Part I. To allow for algebras with a non-trivial centre we have to establish a theory of unbounded Fredholm operators in a general Semifinite von Neumann algebra and in particular prove a generalised McKean-Singer formula.
Alan L. Carey - One of the best experts on this subject based on the ideXlab platform.
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Twisted cyclic theory, equivariant KK-theory and KMS states
2013Co-Authors: Alan L. Carey, Sergey Neshveyev, Ryszard Nest, Adam RennieAbstract:Recently, examples of an index theory for KMS states of circle actions were discovered, [9, 13]. We show that these examples are not isolated. Rather there is a general framework in which we use KMS states for circle actions on a C ∗-algebra A to construct Kasparov modules and Semifinite spectral triples. By using a residue construction analogous to that used in the Semifinite local index formula we associate to these triples a twisted cyclic cocycle on a dense subalgebra of A. This cocycle pairs with the equivariant KK-theory of the mapping cone algebra for the inclusion of the fixed point algebra of the circle action in A. The pairing is expressed in terms of spectral flow between a pair of unbounded self adjoint operators that are Fredholm in the Semifinite sense. A novel aspect of our work is the discovery of an eta cocycle that forms a part of our twisted residue cocycle. To illustrate our theorems we observe firstly that they incorporate the results in [9, 13] as special cases. Next we use the Araki-Woods IIIλ representations of the Fermion algebra to show that there are examples which are not Cuntz-Krieger systems. 1
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The local index formula in Semifinite von Neumann algebras. II. The even case
2012Co-Authors: Alan L. Carey, Adam Rennie, John Phillips, Fedor SukochevAbstract:All authors were supported by grants from ARC (Australia) and NSERC (Canada), in addition the third named author acknowledges a University of Newcastle early career researcher grant. 1 Address for correspondence 1 2 THE LOCAL INDEX FORMULA IN Semifinite VON NEUMANN ALGEBRAS II: THE EVEN CASE We generalise the even local index formula of Connes and Moscovici to the case of spectral triples for a ∗-subalgebra A of a general Semifinite von Neumann algebra. The proof is a variant of that for the odd case which appears in Part I. To allow for algebras with a non-trivial centre we have to establish a theory of unbounded Fredholm operators in a general Semifinite von Neumann algebra and in particular prove a generalised McKean-Singer formula.
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Operator Integrals, Spectral Shift, and Spectral Flow
Canadian Journal of Mathematics, 2009Co-Authors: Nurulla Azamov, Alan L. Carey, Peter G. Dodds, Fedor SukochevAbstract:We present a new and simple approach to the theory of multiple operator integrals that ap- plies to unbounded operators affiliated with general von Neumann algebras. For Semifinite von Neu- mann algebras we give applications to the Frechet differentiation of operator functions that sharpen existing results, and establish the Birman-Solomyak representation of the spectral shift function of M.G.Krein in terms of an average of spectral measuresin the type II setting. We also exhibit a surpris- ing connection between the spectral shift function and spectral flow.
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The spectral shift function and spectral flow
Communications in Mathematical Physics, 2007Co-Authors: Nurulla Azamov, Alan L. Carey, Fedor SukochevAbstract:At the 1974 International Congress, I. M. Singer proposed that eta invariants and hence spectral flow should be thought of as the integral of a one form. In the intervening years this idea has lead to many interesting developments in the study of both eta invariants and spectral flow. Using ideas of [24] Singer’s proposal was brought to an advanced level in [16] where a very general formula for spectral flow as the integral of a one form was produced in the framework of noncommutative geometry. This formula can be used for computing spectral flow in a general Semifinite von Neumann algebra as described and reviewed in [5]. In the present paper we take the analytic approach to spectral flow much further by giving a large family of formulae for spectral flow between a pair of unbounded self-adjoint operators D and D + V with D having compact resolvent belonging to a general Semifinite von Neumann algebra \({\mathcal{N}}\) and the perturbation \(V \in {\mathcal{N}}\) . In noncommutative geometry terms we remove summability hypotheses. This level of generality is made possible by introducing a new idea from [3]. There it was observed that M. G. Krein’s spectral shift function (in certain restricted cases with V trace class) computes spectral flow. The present paper extends Krein’s theory to the setting of Semifinite spectral triples where D has compact resolvent belonging to \({\mathcal{N}}\) and V is any bounded self-adjoint operator in \({\mathcal{N}}\) . We give a definition of the spectral shift function under these hypotheses and show that it computes spectral flow. This is made possible by the understanding discovered in the present paper of the interplay between spectral shift function theory and the analytic theory of spectral flow. It is this interplay that enables us to take Singer’s idea much further to create a large class of one forms whose integrals calculate spectral flow. These advances depend critically on a new approach to the calculus of functions of non-commuting operators discovered in [3] which generalizes the double operator integral formalism of [8–10]. One surprising conclusion that follows from our results is that the Krein spectral shift function is computed, in certain circumstances, by the Atiyah-Patodi-Singer index theorem [2].
Sukochev, Fedor A - One of the best experts on this subject based on the ideXlab platform.
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The Chern character of Semifinite spectral triples
'European Mathematical Society Publishing House', 2016Co-Authors: Carey Alan, Phillips John, Rennie, Adam Charles, Sukochev, Fedor AAbstract:In previouswork we generalised both the odd and even local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra A of a general Semifinite von Neumann algebra. Our proofs are novel even in the setting of the original th
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Integration on locally compact noncommutative spaces
'Elsevier BV', 2016Co-Authors: Carey Alan, Rennie Adam, Sukochev, Fedor AAbstract:We present an ab initio approach to integration theory for nonunital spectral triples. This is done without reference to local units and in the full generality of Semifinite noncommutative geometry. The main result is an equality between the Dixmier trace and generalised residue of the zeta function and heat kernel of suitable operators. We also examine definitions for integrable bounded elements of a spectral triple based on zeta function, heat kernel and Dixmier trace techniques. We show that zeta functions and heat kernels yield equivalent notions of integrability, which imply Dixmier traceability
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The local index formula in Semifinite von Neumann algebras II: the even case
'Elsevier BV', 2015Co-Authors: Carey Alan, Phillips John, Rennie, Adam Charles, Sukochev, Fedor AAbstract:We generalise the even local index formula of Connes and Moscovici to the case of spectral triples for a*-subalgebra A of a general Semifinite von Neumann algebra. The proof is a variant of that for the odd case which appears in Part I. To allow for algebras with a non-trivial centre we have to establish a theory of unbounded Fredholm operators in a general Semifinite von Neumann algebra and in particular prove a generalised McKean-Singer formula
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Operator Integrals, Spectral Shift, and Spectral Flow
'Canadian Mathematical Society', 2015Co-Authors: Azamov N A, Carey Alan, Dodds Peter, Sukochev, Fedor AAbstract:We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For Semifinite von Neumann algebras we give applications to the Fréchet differentiation o
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The Hochschild class of the Chern character for Semifinite spectral triples
Academic Press, 2015Co-Authors: Carey Alan, Phillips John, Rennie, Adam Charles, Sukochev, Fedor AAbstract:We provide a proof of Connes' formula for a representative of the Hochschild class of the Chern character for (p,∞)-summable spectral triples. Our proof is valid for all Semifinite von Neumann algebras, and all integral p≥1. We employ the minimum pos
Nurulla Azamov - One of the best experts on this subject based on the ideXlab platform.
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Operator Integrals, Spectral Shift, and Spectral Flow
Canadian Journal of Mathematics, 2009Co-Authors: Nurulla Azamov, Alan L. Carey, Peter G. Dodds, Fedor SukochevAbstract:We present a new and simple approach to the theory of multiple operator integrals that ap- plies to unbounded operators affiliated with general von Neumann algebras. For Semifinite von Neu- mann algebras we give applications to the Frechet differentiation of operator functions that sharpen existing results, and establish the Birman-Solomyak representation of the spectral shift function of M.G.Krein in terms of an average of spectral measuresin the type II setting. We also exhibit a surpris- ing connection between the spectral shift function and spectral flow.
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The spectral shift function and spectral flow
Communications in Mathematical Physics, 2007Co-Authors: Nurulla Azamov, Alan L. Carey, Fedor SukochevAbstract:At the 1974 International Congress, I. M. Singer proposed that eta invariants and hence spectral flow should be thought of as the integral of a one form. In the intervening years this idea has lead to many interesting developments in the study of both eta invariants and spectral flow. Using ideas of [24] Singer’s proposal was brought to an advanced level in [16] where a very general formula for spectral flow as the integral of a one form was produced in the framework of noncommutative geometry. This formula can be used for computing spectral flow in a general Semifinite von Neumann algebra as described and reviewed in [5]. In the present paper we take the analytic approach to spectral flow much further by giving a large family of formulae for spectral flow between a pair of unbounded self-adjoint operators D and D + V with D having compact resolvent belonging to a general Semifinite von Neumann algebra \({\mathcal{N}}\) and the perturbation \(V \in {\mathcal{N}}\) . In noncommutative geometry terms we remove summability hypotheses. This level of generality is made possible by introducing a new idea from [3]. There it was observed that M. G. Krein’s spectral shift function (in certain restricted cases with V trace class) computes spectral flow. The present paper extends Krein’s theory to the setting of Semifinite spectral triples where D has compact resolvent belonging to \({\mathcal{N}}\) and V is any bounded self-adjoint operator in \({\mathcal{N}}\) . We give a definition of the spectral shift function under these hypotheses and show that it computes spectral flow. This is made possible by the understanding discovered in the present paper of the interplay between spectral shift function theory and the analytic theory of spectral flow. It is this interplay that enables us to take Singer’s idea much further to create a large class of one forms whose integrals calculate spectral flow. These advances depend critically on a new approach to the calculus of functions of non-commuting operators discovered in [3] which generalizes the double operator integral formalism of [8–10]. One surprising conclusion that follows from our results is that the Krein spectral shift function is computed, in certain circumstances, by the Atiyah-Patodi-Singer index theorem [2].
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the krein spectral shift function in Semifinite von neumann algebras
Integral Equations and Operator Theory, 2006Co-Authors: Nurulla Azamov, P G Dodds, Fedor SukochevAbstract:We show the existence of a spectral shift function in the sense of Krein for bounded trace class perturbations of a self-adjoint operator affiliated with a Semifinite von Neumann algebra
Carey Alan - One of the best experts on this subject based on the ideXlab platform.
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The Chern character of Semifinite spectral triples
'European Mathematical Society Publishing House', 2016Co-Authors: Carey Alan, Phillips John, Rennie, Adam Charles, Sukochev, Fedor AAbstract:In previouswork we generalised both the odd and even local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra A of a general Semifinite von Neumann algebra. Our proofs are novel even in the setting of the original th
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Integration on locally compact noncommutative spaces
'Elsevier BV', 2016Co-Authors: Carey Alan, Rennie Adam, Sukochev, Fedor AAbstract:We present an ab initio approach to integration theory for nonunital spectral triples. This is done without reference to local units and in the full generality of Semifinite noncommutative geometry. The main result is an equality between the Dixmier trace and generalised residue of the zeta function and heat kernel of suitable operators. We also examine definitions for integrable bounded elements of a spectral triple based on zeta function, heat kernel and Dixmier trace techniques. We show that zeta functions and heat kernels yield equivalent notions of integrability, which imply Dixmier traceability
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The local index formula in Semifinite von Neumann algebras II: the even case
'Elsevier BV', 2015Co-Authors: Carey Alan, Phillips John, Rennie, Adam Charles, Sukochev, Fedor AAbstract:We generalise the even local index formula of Connes and Moscovici to the case of spectral triples for a*-subalgebra A of a general Semifinite von Neumann algebra. The proof is a variant of that for the odd case which appears in Part I. To allow for algebras with a non-trivial centre we have to establish a theory of unbounded Fredholm operators in a general Semifinite von Neumann algebra and in particular prove a generalised McKean-Singer formula
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Operator Integrals, Spectral Shift, and Spectral Flow
'Canadian Mathematical Society', 2015Co-Authors: Azamov N A, Carey Alan, Dodds Peter, Sukochev, Fedor AAbstract:We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For Semifinite von Neumann algebras we give applications to the Fréchet differentiation o
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The Hochschild class of the Chern character for Semifinite spectral triples
Academic Press, 2015Co-Authors: Carey Alan, Phillips John, Rennie, Adam Charles, Sukochev, Fedor AAbstract:We provide a proof of Connes' formula for a representative of the Hochschild class of the Chern character for (p,∞)-summable spectral triples. Our proof is valid for all Semifinite von Neumann algebras, and all integral p≥1. We employ the minimum pos
