# Topological and Bivariant K-Theory by Joachim Cuntz

By Joachim Cuntz

Topological K-theory is among the most vital invariants for noncommutative algebras outfitted with an appropriate topology or bornology. Bott periodicity, homotopy invariance, and diverse lengthy special sequences distinguish it from algebraic K-theory.

We describe a bivariant K-theory for bornological algebras, which gives an unlimited generalization of topological K-theory. additionally, we speak about different methods to bivariant K-theories for operator algebras. As purposes, we research K-theory of crossed items, the Baum-Connes meeting map, twisted K-theory with a few of its purposes, and a few editions of the Atiyah-Singer Index Theorem.

Similar abstract books

Intégration: Chapitres 7 et 8

Intégration, Chapitres 7 et 8Les Éléments de mathématique de Nicolas BOURBAKI ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce quantity du Livre d’Intégration, sixième Livre du traité, traite de l’intégration sur les groupes localement compacts et de ses functions.

Extra info for Topological and Bivariant K-Theory

Sample text

Therefore, homotopy, equivalence, and similarity all provide the same equivalence relation on Idem M∞ (A). Proof. Let e ∈ Idem C([0, 1], A) be a homotopy between e0 and e1 . We have e ∈ Idem C([0, 1], AS ) for some Banach subalgebra AS ⊆ A. We can ﬁnd 0 = t0 ≤ t1 ≤ · · · ≤ tk+1 = 1 such that e(tj ) − e(tj+1 ) AS < 1 − 2e(tj ) −1 AS for j = 0, . . , k; thus xj := e(tj )e(tj+1 ) + 1 − e(tj ) 1 − e(tj+1 ) satisﬁes (xj ) < 1, so that ln(xj ) is deﬁned. Moreover, xj e(tj+1 )x−1 = e(tj ). j Now we construct u out of the paths of invertible elements exp t · ln(xj ) .

It is a local Banach algebra by deﬁnition. 20. Let A be a unital local Banach algebra over C. The spectrum of x ∈ A is the set ΣA (x) of all λ ∈ C for which x − λ · 1A is not invertible in A. 2. 21. Let A be a local Banach algebra. The spectral radius of a bounded subset S is deﬁned by A (S) := inf {r > 0 | r−1 S is power-bounded}. The spectral radius of an element x ∈ A is deﬁned by A (x) := A ({x}). These deﬁnitions still work for general bornological algebras, but the spectral radius may become ∞ and the spectrum need not have particularly nice properties.

If v ∗ v or vv ∗ is idempotent, then v is a partial isometry. 46. Let A be a unital C ∗ -algebra. (1) The set of projections in A is a deformation retract of the set of idempotents in A; thus any idempotent is homotopic to a projection, and two projections that are homotopic among idempotents are homotopic among projections. (2) Two projections p, q in A are similar if and only if they are unitarily equivalent, that is, upu−1 = q for some unitary u ∈ A. (3) Two projections p, q in A are equivalent if and only if they are Murray–vonNeumann equivalent, that is, there is a partial isometry v with v ∗ v = p and vv ∗ = q.