The Group of Unital C-extensions

Let A and B be separable C∗-algebras, A unital and B stable. It is shown that there is a natural six-terms exact sequence which relates the group which arises by considering all semi-split extensions of A by B to the group which arises by restricting the attention to unital semi-split extensions of A by B. The six-terms exact sequence is an unpublished result of G. Skandalis. Let A,B be separable C-algebras, B stable. As is well-known the C-algebra extensions of A by B can be identified with Hom(A.Q(B)), the set of ∗-homomorphisms A→ Q(B) where Q(B) = M(B)/B is the generalized Calkin algebra. Two extensions φ, ψ : A → Q(B) are unitarily equivalent when there is a unitary u ∈M(B) such that Ad q(u) ◦ ψ = φ, where q : M(B) → Q(B) is the quotient map. The unitary equivalence classes of extensions of A by B have an abelian semi-group structure thanks to the stability of B: Choose isometries V1, V2 ∈ M(B) such that V1V ∗ 1 + V2V ∗ 2 = 1, and define the sum φ⊕ ψ : A→ Q(B) of φ, ψ ∈ Hom(A,Q(B)) by (ψ ⊕ φ)(a) = Ad q(V1) ◦ ψ(a) + Ad q(V2) ◦ φ(a). (1) The isometries, V1 and V2, are fixed in the following. An extension φ : A→ Q(B) is split when there is a ∗-homomorphisms π : A → M(B) such that φ = q ◦ π. To trivialize the split extensions we declare two extensions φ, ψ : A→ Q(B) to be stably equivalent when there there is a split extension π such that ψ⊕π and φ⊕π are unitarily equivalent. This is an equivalence relation because the sum (1) of two split extensions is itself split. We denote by Ext(A,B) the semigroup of stable equivalence classes of extensions of A by B. It was proved in [Th], as a generalization of results of Kasparov, that there exists an absorbing split extension π0 : A → Q(B), i.e. a split extension with the property that π0 ⊕ π is unitarily equivalent to π0 for every split extension π. Thus two extensions φ, ψ are stably equivalent if and only if φ⊕π0 and ψ⊕π0 are unitarily equivalent. The classes of stably equivalent extensions of A by B is an abelian semigroup Ext(A,B) in which any split extension (like 0) represents the neutral element. As is well-documented the semi-group is generally not a group, and we denote by Ext(A,B) the abelian group of invertible elements in Ext(A,B). It is also well-known that this group is one way of describing the KK-groups of Kasparov. Specifically, Ext(A,B) = KK(SA,B) = KK(A, SB). Assume now that A is unital. It is then possible, and sometimes even advantageous, to restrict attention to unital extensions of A by B, i.e. to short exact sequences 0 // B // E // A // 0 (2) 1 2 VLADIMIR MANUILOV AND KLAUS THOMSEN of C-algebras with E is unital, or equivalently to ∗-homomorphisms A→ Q(B) that are unital. The preceding definitions are all amenable to such a restriction, if done consistently. Specifically, we say that a unital extension φ : A → Q(B) is unitally split when there is a unital ∗-homomorphism π : A→M(B) such that φ = q◦π. The sum (1) of two unital extensions is again unital, and we say that two unital extensions φ, ψ : A→ Q(B) are unitally stably equivalent when there there is a unital split extension π such that ψ ⊕ π and φ⊕ π are unitarily equivalent. It was proved in [Th] that there always exists a unitally absorbing split extension π0 : A → Q(B), i.e. a unitally split extension with the property that π0 ⊕ π is unitarily equivalent to π0 for every unitally split extension π. Thus two unital extensions φ, ψ are unitally stably equivalent if and only if φ ⊕ π0 and ψ ⊕ π0 are unitarily equivalent. The classes of unitally stably equivalent extensions of A by B is an abelian semi-group which we denote by Extunital(A,B). The unitally absorbing split extension π0, or any other unitally split extension, represents the neutral element of Extunital(A,B), and we denote by Ext unital(A,B) the abelian group of invertible elements in Extunital(A,B). As we shall see there is a difference between Ext unital(A,B) and Ext (A,B) arising from the fact that while the class in Ext(A,B) of a unital extension A→ Q(B) can not be changed by conjugating it with a unitary from Q(B), its class in Ext unital(A,B) can. In a sense the main result of this note is that this is the only way in which the two groups differ. Note that there is a group homomorphism Ext unital(A,B) → Ext (A,B), obtained by forgetting the word ’unital’. It will be shown that this forgetful map fits into a six-terms exact sequence K0(B) u0 // Ext unital(A,B) // Ext(A,B)