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Keywords
(4)
Operator Algebra
Spectral Flow
Spectral Triple
Von Neumann Algebra
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KKtheory and Spectral Flow in von Neumann Algebras
KKtheory and Spectral Flow in von Neumann Algebras,Jens Kaad,Ryszard Nest,Adam Rennie
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KKtheory and Spectral Flow in von Neumann Algebras
(
Citations: 10
)
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Jens Kaad
,
Ryszard Nest
,
Adam Rennie
We present a definition of
spectral flow
relative to any norm closed ideal J in any
von Neumann algebra
N. Given a path D(t) of selfadjoint operators in N which are invertible in N/J, the
spectral flow
produces a class in K_0(J). In the case when N is semifinite, the numerical
spectral flow
of the path coincides with the value of trace on the associated Kclass. Given a semifinite
spectral triple
(A,H,D) relative to a semifinite
von Neumann algebra
N, we construct a class [D] in KK^1(A,N') such that, for a unitary u in A, the von Neumann
spectral flow
between D and u*Du is equal to the Kasparov product of [u] and [D].
Published in 2007.
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Citation Context
(6)
...from the fact that the construction in this paper can be applied to many index problems in seminite noncommutative geometry using [
9
] (which we plan to address elsewhere)...
...More detailed information about the KKtheory version of this can be found in [
9
]...
...If v is unitary over A, we recover the usual Kasparov pairing between K1(A) and KK1(A;F ), [
9
], [15, Appendix]...
A. L. Carey
,
et al.
Noncommutative AtiyahPatodiSinger boundary conditions and index pair...
...Remarks. (i) This is a special case of a result in [
KNR
] (other special cases appeared in [PR, PRS])...
...(ii) The seminite index is known to be related to pairings in KKtheory, [CPR1,
KNR
], but the modular index introduced here is still mysterious...
A. L. Carey
,
et al.
Spectral flow invariants and twisted cyclic theory from the Haar state...
...Just as ordinary B(H) spectral triples represent Khomology classes, [C, CPRS1], and seminite spectral triples represent KKclasses, [
KNR
], modular spectral triples provide analytic representatives of some Ktheoretic type data which we now describe...
...This follows in the same way as the seminite case, [
KNR
]...
A. L. Carey
,
et al.
Twisted cyclic theory and an index theory for the gauge invariant KMS ...
...yields the same kind of ‘semifinite Kasparov modules’ as are described in [
37
]...
...This is explained by a result of KaadNestRennie [
37
]...
A. L. Carey
,
et al.
SemiFinite Noncommutative Geometry and some Applications
...from the fact that the construction in this paper can be applied to many index problems in semifinite noncommutative geometry using [
9
] (which we plan to address elsewhere)...
...More detailed information about the KKtheory version of this can be found in [
9
]...
...• If v is unitary over A, we recover the usual Kasparov pairing between K1(A) and KK1(A, F), [
9
], [15, Appendix]...
A. L. Carey
,
et al.
A noncommutative AtiyahPatodiSinger index theorem in KKtheory
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Citations
(10)
Families of Type {\rm III KMS} States on a Class of $C^*$Algebras containing $O_n$ and $\mathcal{Q}_\N$
A. L. Carey
,
J. Phillips
,
I. F. Putnam
,
A. Rennie
Published in 2010.
Noncommutative AtiyahPatodiSinger boundary conditions and index pairings in KK theory
A. L. Carey
,
J. Phillips
,
A. Rennie
Journal:
Journal Fur Die Reine Und Angewandte Mathematik  J REINE ANGEW MATH
, vol. 2010, no. 643, pp. 59109, 2010
Modular Theory, NonCommutative Geometry and Quantum Gravity
Paolo Bertozzini
,
Roberto Conti
,
Wicharn Lewkeeratiyutkul
Journal:
Symmetry Integrability and Geometrymethods and Applications  SYMMETRY INTEGR GEOM
, 2010
Spectral Triples: Examples and Applications Notes for lectures given at the workshop on noncommutative geometry and physics
A. Rennie
Published in 2009.
Spectral flow invariants and twisted cyclic theory from the Haar state on SU_q(2)
(
Citations: 8
)
A. L. Carey
,
A. Rennie
,
K. Tong
Published in 2008.