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Full-Text Articles in Science and Technology Studies
The K-Theory Of Heegaard Quantum Lens Spaces, Piotr M. Hajac, Adam Rennie, Bartosz Zielinski
The K-Theory Of Heegaard Quantum Lens Spaces, Piotr M. Hajac, Adam Rennie, Bartosz Zielinski
Associate Professor Adam Rennie
Representing Z/NZ as roots of unity, we restrict a natural U(1)-action on the Heegaard quantum sphere to Z/NZ, and call the quotient spaces Heegaard quantum lens spaces. Then we use this representation of Z/NZ to construct an associated complex loine bundle. This paper proves the stable non-triviality of these line bundles over any of the quantum lens spaces we consider. We use the pullback structure of the C*-algebra of the lens space to compute its K-theory via the Mayer-Vietoris sequence, and an explicit form of the Bass connecting homomorphism to prove the stable non-triviality of the bundles. On the algebraic …
Kk-Theory And Spectral Flow In Von Neumann Algebras, J Kaad, R Nest, Adam C. Rennie
Kk-Theory And Spectral Flow In Von Neumann Algebras, J Kaad, R Nest, Adam C. Rennie
Associate Professor Adam Rennie
We present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko(J).Given a semifinite spectral triple (A, H, D) relative to (N, t) with A separable, we construct a class [D] ? KK1(A, K(N)). For a unitary u ? A, the von Neumann spectral flow between D and u*Du is equal to the Kasparov product [u]A[D], and is simply related to the numerical spectral flow, and a refined C*-spectral flow.
Spectral Flow Invariants And Twisted Cyclic Theory For The Haar State On Suq(2), A L. Carey, A Rennie, K Tong
Spectral Flow Invariants And Twisted Cyclic Theory For The Haar State On Suq(2), A L. Carey, A Rennie, K Tong
Associate Professor Adam Rennie
In [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], we presented a K-theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely SUq.2/ and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes from the generator of …
Spectral Triples: Examples And Index Theory, Alan L. Carey, John Phillips, Adam C. Rennie
Spectral Triples: Examples And Index Theory, Alan L. Carey, John Phillips, Adam C. Rennie
Associate Professor Adam Rennie
The main objective of these notes is to give some intuition about spectral triples and the role they play in index theory. The notes are basically a road map, with much detail omitted. To give a complete account of all the topics covered would require at least a book, so we have opted for a sketch.
Twisted Cyclic Theory, Equivariant Kk-Theory And Kms States, Alan L. Carey, Sergey Neshveyev, R Nest, Adam Rennie
Twisted Cyclic Theory, Equivariant Kk-Theory And Kms States, Alan L. Carey, Sergey Neshveyev, R Nest, Adam Rennie
Associate Professor Adam Rennie
Given a C-algebra A with a KMS weight for a circle action, we construct and compute a secondary invariant on the equivariant K-theory of the mapping cone of AT ,! A, both in terms of equivariant KK-theory and in terms of a semifinite spectral flow. This in particular puts the previously considered examples of Cuntz algebras [10] and SUqð2Þ [14] in a general framework. As a new example we consider the Araki-Woods IIIl representations of the Fermion algebra.