The KK-theory of amalgamated free products. (English) Zbl 1455.46074
This article continues the study of the K-theory and KK-theory of
amalgamated free products of C*-algebras. The setup is a pair of
C*-algebras \(A_1\) and \(A_2\) that contain a common C*-subalgebra \(B\).
In addition, conditional expectations from \(A_1\) and \(A_2\) onto the
subalgebra \(B\) are fixed. The article introduces a variant of the
reduced amalgamated free product. This construction agrees with
Voiculescu’s reduced amalgamated free product in case the
expectations to \(B\) are GNS-faithful, but not in general. In
particular, if both expectations to \(B\) are homomorphisms, then the
new reduced free product coincides with the pullback \(A_1 \oplus_B
A_2\), whereas Voiculescu’s definition gives just \(B\). The new
reduced amalgamated free product has the advantage that it is always
KK-equivalent to the full amalgamated free product. This is the
first main theorem in this article.
The particular factorisation of the identity on \(A_1 *_B A_2\) through the new reduced amalgamated free product is then used to simplify the proof that \(A_1 *_B A_2\) is KK-equivalent to the homotopy pushout of the diagram \(A_1 \leftarrow B \rightarrow A_2\). So far, this was only known under extra assumptions. The KK-equivalence between the amalgamated free product is equivalent to six-term exact sequences that relate the KK-groups \(\mathrm{KK}_*(A_1 *_B A_2,C)\) or \(\mathrm{KK}_*(C,A_1 *_B A_2)\) for another C*-algebra \(C\) to the corresponding KK-groups for \(A_1 \oplus A_2\) and \(B\) instead of \(A_1 *_B A_2\). The KK-equivalence between reduced and full amalgamated products also implies that amalgamated free products of K-amenable (quantum) groups are again K-amenable.
The particular factorisation of the identity on \(A_1 *_B A_2\) through the new reduced amalgamated free product is then used to simplify the proof that \(A_1 *_B A_2\) is KK-equivalent to the homotopy pushout of the diagram \(A_1 \leftarrow B \rightarrow A_2\). So far, this was only known under extra assumptions. The KK-equivalence between the amalgamated free product is equivalent to six-term exact sequences that relate the KK-groups \(\mathrm{KK}_*(A_1 *_B A_2,C)\) or \(\mathrm{KK}_*(C,A_1 *_B A_2)\) for another C*-algebra \(C\) to the corresponding KK-groups for \(A_1 \oplus A_2\) and \(B\) instead of \(A_1 *_B A_2\). The KK-equivalence between reduced and full amalgamated products also implies that amalgamated free products of K-amenable (quantum) groups are again K-amenable.
Reviewer: Ralf Meyer (Göttingen)
MSC:
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 19K35 | Kasparov theory (\(KK\)-theory) |
Keywords:
amalgamated free product; conditional expectation; reduced free product; KK-theory; six-term exact sequence; K-amenableReferences:
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