scholarly journals The noncommutative geometry of k-graph C*-algebras

2007 ◽  
Vol 1 (2) ◽  
pp. 259-304 ◽  
Author(s):  
David Pask ◽  
Adam Rennie ◽  
Aidan Sims
Keyword(s):  
Fixed Point ◽  
Isomorphism Class ◽  
Lower Semicontinuous ◽  
Index Theorem ◽  
Graph Algebras ◽  
Gauge Invariant ◽  
Local Index ◽  
C Algebras ◽  
Local Index Theorem ◽  
Fixed Point Algebra

AbstractThis paper is comprised of two related parts. First we discuss which k-graph algebras have faithful traces. We characterise the existence of a faithful semifinite lower-semicontinuous gauge-invariant trace on C* (Λ) in terms of the existence of a faithful graph trace on Λ.Second, for k-graphs with faithful gauge invariant trace, we construct a smooth (k,∞)-summable semifinite spectral triple. We use the semifinite local index theorem to compute the pairing with K-theory. This numerical pairing can be obtained by applying the trace to a KK-pairing with values in the K-theory of the fixed point algebra of the Tk action. As with graph algebras, the index pairing is an invariant for a finer structure than the isomorphism class of the algebra.

scholarly journals Applications of the Gauge-invariant uniqueness theorem for graph algebras

2002 ◽  
Vol 66 (1) ◽  
pp. 57-67 ◽  
Author(s):  
Teresa Bates
Keyword(s):  
Fixed Point ◽  
Uniqueness Theorem ◽  
Directed Graphs ◽  
Cartesian Product ◽  
Graph Algebras ◽  
Gauge Invariant ◽  
Shift Equivalence ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Fixed Point Algebra

We give applications of the gauge-invariant uniqueness theorem, which states that the Cuntz-Krieger algebras of directed graphs are characterised by the existence of a canonical action of . We classify the C*-algebras of higher order graphs, identify the C*-algebras of cartesian product graphs with a certain fixed point algebra and investigate a relation called elementary shift equivalence on graphs and its effect on the associated graph C*-algebras.

A local index theorem for non K�hler manifolds

1989 ◽  
Vol 284 (4) ◽  
pp. 681-699 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Index Theorem ◽  
Local Index ◽  
Local Index Theorem

scholarly journals Local index theorem for families.

1988 ◽  
Vol 35 (1) ◽  
pp. 11-20 ◽  
Author(s):  
Harold Donnelly
Keyword(s):  
Index Theorem ◽  
Local Index ◽  
Local Index Theorem

NONUNITAL SPECTRAL TRIPLES ASSOCIATED TO DEGENERATE METRICS

2004 ◽  
Vol 16 (01) ◽  
pp. 125-146
Author(s):  
A. RENNIE
Keyword(s):  
Finite Dimension ◽  
Index Theorem ◽  
Smooth Manifolds ◽  
Spectral Triples ◽  
Local Index ◽  
Dimension Spectrum ◽  
Spectrum Hypothesis ◽  
Local Index Theorem ◽  
K Theory

We show that one can define (p,∞)-summable spectral triples using degenerate metrics on smooth manifolds. Furthermore, these triples satisfy Connes–Moscovici's discrete and finite dimension spectrum hypothesis, allowing one to use the Local Index Theorem [1] to compute the pairing with K-theory. We demonstrate this with a concrete example.

scholarly journals Superselection Structures for C*-Algebras with Nontrivial Center

1997 ◽  
Vol 09 (07) ◽  
pp. 785-819 ◽  
Author(s):  
Hellmut Baumgärtel ◽  
Fernando Lledó
Keyword(s):  
Fixed Point ◽  
Compact Group ◽  
C Algebras ◽  
Nontrivial Center ◽  
Relative Commutant ◽  
Fixed Point Algebra

We present and prove some results within the framework of Hilbert C*-systems [Formula: see text] with a compact group [Formula: see text]. We assume that the fixed point algebra [Formula: see text] of [Formula: see text] has a nontrivial center [Formula: see text] and its relative commutant w.r.t. ℱ coincides with [Formula: see text], i.e. we have [Formula: see text]. In this context we propose a generalization of the notion of an irreducible endomorphism and study the behaviour of such irreducibles w.r.t. [Formula: see text]. Finally, we give several characterizations of the stabilizer of [Formula: see text].

scholarly journals A Short Note on the Continuous Rokhlin Property and the Universal Coefficient Theorem in E-theory

2015 ◽  
Vol 58 (2) ◽  
pp. 374-380 ◽  
Author(s):  
Gábor Szabó
Keyword(s):  
Fixed Point ◽  
Compact Group ◽  
Short Note ◽  
Circle Actions ◽  
Continuous Action ◽  
C Algebra ◽  
C Algebras ◽  
Group A ◽  
Fixed Point Algebra ◽  
Universal Coefficient Theorem

AbstractLet G be a metrizable compact group, A a separable C*-algebra, and α:G → Aut(A) a strongly continuous action. Provided that α satisfies the continuous Rokhlin property, we show that the property of satisfying the UCT in E-theory passes from Ato the crossed product C*-algebra A⋊α G and the ûxed point algebra Aα. This extends a similar result by Gardella for KK-theory in the case of unital C*-algebras but with a shorter and less technical proof. For circle actions on separable unital C*-algebras with the continuous Rokhlin property, we establish a connection between the Etheory equivalence class of A and that of its fixed point algebra Aα.

scholarly journals Local index theorem for orbifold Riemann surfaces

2018 ◽  
Vol 109 (5) ◽  
pp. 1119-1143 ◽  
Author(s):  
Leon A. Takhtajan ◽  
Peter Zograf
Keyword(s):  
Riemann Surfaces ◽  
Index Theorem ◽  
Local Index ◽  
Local Index Theorem
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