Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

D. Kucerovsky / Journal of Functional Analysis 236 (2006) 395–408 407
and that the equivalence relation on Ext exactly corresponds to the stabilized cp equivalence
relation in KK 1 , meaning equivalence modulo perturbation by Jφ, unitary equivalence, and addition
of stabilized degenerate cycles [2, Section 17.6.4] which may be taken to be unital if the
original cycle is unital [11]. Addition of stabilized degenerate cycles corresponds to the addition
of trivial extensions in Ext, so we see that for absorbing cycles (φ1,F1) and (φ2,F2), wehave
that (φ1,F1) ∼cp (φ2,F2) in KK 1 if and only if π(F1φ1F ∗ 1 ) ∼u π(F2φ2F ∗ 2 ).
We now give an example showing that a cycle can be absorbing even if (1 − F)φ(·)(1 − F)
is small.
Example 7.1. Let φ be the Kasparov GNS representation of A on B: thus, φ : A → M(B ⊗ K)
is given by 1 ⊗ π where π is the usual universal representation of A on B(H) ∼ = M(K). Then it
follows from Kasparov’s results [11] that (M(B), φ, 1) ∈ KK 1 (A, B) is an absorbing triple. The
complement (M(B), φ, 0) is zero. We can elaborate this example somewhat, by letting F = e11
in M2(M(B ⊗ K)), and then letting φ be the direct sum of Kasparov’s GNS representation in
the e11 corner and any arbitrary homomorphism in the e22 corner.
It would be desirable, of course, to give elegant necessary and sufficient condition for a
specific triple to be absorbing. This is a hard problem, however, we point out one very pretty
sufficient condition. An operator F ∈ M(B) is said to be quasi-Fredholm if it is quasi-invertible
in the corona, or equivalently, if there are x,y ∈ M(B) such that xFy = 1 + b for some b ∈ B.
Theorem 7.2. Let B be stable and separable. A Fredholm triple (M(B), φ, F ) ∈ KK 1 (A, B) is
absorbing if F φ(a)F ∗ is a quasi-Fredholm operator whenever it is positive.
Proof. Let C be the extension algebra of the associated Busby map a ↦→ F φ(a)F ∗ . A positive
element of C not in B is of the form F φ(a)F ∗ + b for some a ∈ A and b ∈ B, where F is the
given essential projection from the Fredholm triple.
The hypothesis implies that there is r1 and r2 in the corona such that d := F φ(a)F ∗ satisfies
r1dr2 = 1Q. But then, 1 r1d 2 r ∗ 1 r2r ∗ 2 , so that there is an r′ with r ′ d 2 r ′∗ = 1. Lifting r ′ to
˜r ′ in the multipliers, we have that the original element d + b satisfies ˜r ′ (d + b) 2 ˜r ′∗ = 1 + b0
for some b0 in the canonical ideal, and we can cut down by a isometry obtained from stability,
obtaining v ∗ ˜r ′ (d + b) 2 ˜r ′∗ v = 1 + v ∗ b0v with the norm of v ∗ b0v small. Thus, this expression is
invertible, and r ′′ (d + b) 2 r ′′∗ = 1forsomer ′′ in the multipliers. We now have an isometry V :=
(d + b)r ′′∗ that implements an equivalence of VBV ∗ ⊂ (d + b)B(d + b) and B, showing that
(d + b)B(d + b) contains a stable (full) hereditary subalgebra. Thus, the absorption criterion of
Theorem 3.1 holds. Actually, the conclusion of that theorem implicitly has two cases, according
to whether the given extension is unital or not, however, in both cases the hypothesis just involves
existence of stable full hereditary subalgebras as shown above. ✷
References
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