scholarly journals E-theory for C[0, 1]-algebras with finitely many singular points

2014 ◽  
Vol 13 (2) ◽  
pp. 249-274 ◽  
Author(s):  
Marius Dadarlat ◽  
Prahlad Vaidyanathan
Keyword(s):  
Finite Rank ◽  
Unit Interval ◽  
Singular Points ◽  
Hausdorff Spaces ◽  
Classification Result ◽  
C Algebras ◽  
K Theory ◽  
Non Hausdorff Spaces

AbstractWe study the E-theory group E[0, 1](A, B) for a class of C*-algebras over the unit interval with finitely many singular points, called elementary C[0, 1]-algebras. We use results on E-theory over non-Hausdorff spaces to describe E[0, 1](A, B) where A is a sky-scraper algebra. Then we compute E[0, 1](A, B) for two elementary C[0, 1]-algebras in the case where the fibers A(x) and B(y) of A and B are such that E1 (A(x), B(y)) = 0 for all x, y ∈ [0, 1]. This result applies whenever the fibers satisfy the UCT, their K0-groups are free of finite rank and their K1-groups are zero. In that case we show that E[0, 1](A, B) is isomorphic to Hom(0(A), 0(B)), the group of morphisms of the K-theory sheaves of A and B. As an application, we give a streamlined partially new proof of a classification result due to the first author and Elliott.

An Introduction to K-Theory for C*-Algebras

2000 ◽  
Author(s):  
M. Rørdam ◽  
F. Larsen ◽  
N. Laustsen
Keyword(s):  
C Algebras ◽  
K Theory

scholarly journals Graded K-theory and K-homology of relative Cuntz–Pimsner algebras and graph C⁎-algebras

2021 ◽  
Vol 496 (2) ◽  
pp. 124822
Author(s):  
Quinn Patterson ◽  
Adam Sierakowski ◽  
Aidan Sims ◽  
Jonathan Taylor
Keyword(s):  
C Algebras ◽  
K Theory

scholarly journals K-theory for certain group C*-algebras

1983 ◽  
Vol 151 (0) ◽  
pp. 209-230 ◽  
Author(s):  
E. Christopher Lance
Keyword(s):  
C Algebras ◽  
K Theory

scholarly journals ON THE CLASSIFICATION OF NUCLEAR C*-ALGEBRAS

2002 ◽  
Vol 85 (1) ◽  
pp. 168-210 ◽  
Author(s):  
MARIUS DADARLAT ◽  
SØREN EILERS
Keyword(s):  
Subject Classification ◽  
Unified Approach ◽  
C Algebras ◽  
Starting Point ◽  
K Theory

We employ results from KK-theory, along with quasidiagonality techniques, to obtain wide-ranging classification results for nuclear C*-algebras. Using a new realization of the Cuntz picture of the Kasparov groups we show that two morphisms inducing equal KK-elements are approximately stably unitarily equivalent. Using K-theory with coefficients to associate a partial KK-element to an approximate morphism, our result is generalized to cover such maps. Conversely, we study the problem of lifting a (positive) partial KK-element to an approximate morphism. These results are employed to obtain classification results for certain classes of quasidiagonal C*-algebras introduced by H. Lin, and to reprove the classification of purely infinite simple nuclear C*-algebras of Kirchberg and Phillips. It is our hope that this work can be the starting point of a unified approach to the classification of nuclear C*-algebras.2000 Mathematical Subject Classification: primary 46L35; secondary 19K14, 19K35, 46L80.

scholarly journals Skew compact semigroups

2003 ◽  
Vol 4 (1) ◽  
pp. 133
Author(s):  
Ralph D. Kopperman ◽  
Desmond Robbie
Keyword(s):  
Continuous Semigroup ◽  
Minimal Ideal ◽  
Compact Semigroup ◽  
Closed Ideal ◽  
Hausdorff Spaces ◽  
Compact Spaces ◽  
Compact Semigroups ◽  
Compact Topology ◽  
Hausdorff Group ◽  
Non Hausdorff Spaces

<p>Skew compact spaces are the best behaving generalization of compact Hausdorff spaces to non-Hausdorff spaces. They are those (X ; τ ) such that there is another topology τ* on X for which τ V τ* is compact and (X; τ ; τ*) is pairwise Hausdorff; under these conditions, τ uniquely determines τ *, and (X; τ*) is also skew compact. Much of the theory of compact T<sub>2</sub> semigroups extends to this wider class. We show:</p> <p>A continuous skew compact semigroup is a semigroup with skew compact topology τ, such that the semigroup operation is continuous τ<sup>2</sup>→ τ. Each of these contains a unique minimal ideal which is an upper set with respect to the specialization order.</p> <p>A skew compact semigroup which is a continuous semigroup with respect to both topologies is called a de Groot semigroup. Given one of these, we show:</p> <p>It is a compact Hausdorff group if either the operation is cancellative, or there is a unique idempotent and S<sup>2</sup> = S.</p> <p>Its topology arises from its subinvariant quasimetrics.</p> <p>Each *-closed ideal ≠ S is contained in a proper open ideal.</p>

scholarly journals The convergence-theoretic approach to weakly first countable spaces and symmetrizable spaces

2019 ◽  
Vol 69 (1) ◽  
pp. 185-198
Author(s):  
Fadoua Chigr ◽  
Frédéric Mynard
Keyword(s):  
Space Structures ◽  
Theoretic Approach ◽  
Hausdorff Spaces ◽  
Sequential Space ◽  
Countable Space ◽  
The Stability ◽  
First Countable Space ◽  
Algebraic Proofs ◽  
Non Hausdorff Spaces

AbstractThis article fits in the context of the approach to topological problems in terms of the underlying convergence space structures, and serves as yet another illustration of the power of the method. More specifically, we spell out convergence-theoretic characterizations of the notions of weak base, weakly first-countable space, semi-metrizable space, and symmetrizable spaces. With the help of the already established similar characterizations of the notions of Frchet-Ursyohn, sequential, and accessibility spaces, we give a simple algebraic proof of a classical result regarding when a symmetrizable (respectively, weakly first-countable, respectively sequential) space is semi-metrizable (respectively first-countable, respectively Fréchet) that clarifies the situation for non-Hausdorff spaces. Using additionally known results on the commutation of the topologizer with product, we obtain simple algebraic proofs of various results of Y. Tanaka on the stability under product of symmetrizability and weak first-countability, and we obtain the same way a new characterization of spaces whose product with every metrizable topology is weakly first-countable, respectively symmetrizable.

scholarly journals Delocalized Eta Invariants, Algebraicity, and K-Theory of Group C*-Algebras

2019 ◽  
Author(s):  
Zhizhang Xie ◽  
Guoliang Yu
Keyword(s):  
Conjugacy Class ◽  
Discrete Group ◽  
Polynomial Growth ◽  
Algebraic Numbers ◽  
Eta Invariant ◽  
Trace Map ◽  
Novikov Conjecture ◽  
C Algebras ◽  
Eta Invariants ◽  
K Theory

Abstract In this paper, we establish a precise connection between higher rho invariants and delocalized eta invariants. Given an element in a discrete group, if its conjugacy class has polynomial growth, then there is a natural trace map on the $K_0$-group of its group $C^\ast$-algebra. For each such trace map, we construct a determinant map on secondary higher invariants. We show that, under the evaluation of this determinant map, the image of a higher rho invariant is precisely the corresponding delocalized eta invariant of Lott. As a consequence, we show that if the Baum–Connes conjecture holds for a group, then Lott’s delocalized eta invariants take values in algebraic numbers. We also generalize Lott’s delocalized eta invariant to the case where the corresponding conjugacy class does not have polynomial growth, provided that the strong Novikov conjecture holds for the group.

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