On the $K$-theory of $C^*$-algebras associated to substitution tilings

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.

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$K$-Theory and Ext-theory for Rectangular Unitary $C^*$-Algebras

Rocky Mountain Journal of Mathematics ◽

10.1216/rmjm/1181072541 ◽

1993 ◽

Vol 23 (3) ◽

pp. 1063-1080 ◽

Cited By ~ 6

Author(s):

Kevin McClanahan

Keyword(s):

C Algebras ◽

K Theory

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Ideal-related $K$-theory for Leavitt path algebras and graph $C^*$-algebras

Indiana University Mathematics Journal ◽

10.1512/iumj.2013.62.5123 ◽

2013 ◽

Vol 62 (5) ◽

pp. 1587-1620 ◽

Cited By ~ 5

Author(s):

Efren Ruiz ◽

Mark Tomforde

Keyword(s):

C Algebras ◽

Path Algebras ◽

K Theory ◽

Leavitt Path Algebras

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

International Mathematics Research Notices ◽

10.1093/imrn/rnz170 ◽

2019 ◽

Cited By ~ 2

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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