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.

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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.