The extension class and KMS states for Cuntz-Pimsner algebras of some bi-Hilbertian bimodules. (English) Zbl 1370.19004
A \(C^*\)-correspondence \(E\) over a \(C^*\)-algebra \(A\) produces a Toeplitz algebra \(\mathcal{T}_E\) and a Cuntz-Pimsner algebra \(\mathcal{O}_E\), which is a quotient of the Toeplitz algebra. If the left \(A\)-action on \(E\) is faithful, then the kernel of the quotient map from \(\mathcal{T}_E\) to \(\mathcal{O}_E\) is the \(C^*\)-algebra \(\mathcal{K}_E\) of compact operators on the Fock module, which is a full Hilbert \(A\)-module, hence Morita equivalent to \(A\). The resulting extension is used to compute the K-theory and KK-theory for the Cuntz-Pimsner algebra \(\mathcal{O}_E\). This computation becomes very explicit if the class of the Toeplitz extension in \(KK_1(\mathcal{O}_E,A)\) is known. This article describes explicit odd KK-cycles for this KK-class in certain cases.
First, if \(E\) is a self-Morita equivalence, then there is an explicit unbounded odd KK-cycle representing the class of the Toeplitz extension in \(KK_1(\mathcal{O}_E,A)\).
The main goal of the article is to describe a bounded odd KK-cycle that represents this class if \(E\) is bi-Hilbertian, that is, it also carries a left inner product for which the right action of \(A\) is by adjointable operators. This setting was previously considered by T. Kajiwara et al. [J. Funct. Anal. 215, No. 1, 1–49 (2004; Zbl 1067.46053)] to define an analogue of the Jones index for a correspondence. If \(E\) is bi-Hilbertian and satisfies a further technical assumption, then a certain conditional expectation \(\mathcal{O}_E\to A\) is defined. This is the main ingredient to describe a bounded odd KK-cycle that represents the class of the Toeplitz extension in \(KK_1(\mathcal{O}_E,A)\). The conditional expectation is related to a one-parameter group of automorphisms of \(\mathcal{T}_E\), which descends to \(\mathcal{O}_E\), and which differs from the usual gauge action.
First, if \(E\) is a self-Morita equivalence, then there is an explicit unbounded odd KK-cycle representing the class of the Toeplitz extension in \(KK_1(\mathcal{O}_E,A)\).
The main goal of the article is to describe a bounded odd KK-cycle that represents this class if \(E\) is bi-Hilbertian, that is, it also carries a left inner product for which the right action of \(A\) is by adjointable operators. This setting was previously considered by T. Kajiwara et al. [J. Funct. Anal. 215, No. 1, 1–49 (2004; Zbl 1067.46053)] to define an analogue of the Jones index for a correspondence. If \(E\) is bi-Hilbertian and satisfies a further technical assumption, then a certain conditional expectation \(\mathcal{O}_E\to A\) is defined. This is the main ingredient to describe a bounded odd KK-cycle that represents the class of the Toeplitz extension in \(KK_1(\mathcal{O}_E,A)\). The conditional expectation is related to a one-parameter group of automorphisms of \(\mathcal{T}_E\), which descends to \(\mathcal{O}_E\), and which differs from the usual gauge action.
Reviewer: Ralf Meyer (Göttingen)
MSC:
| 19K35 | Kasparov theory (\(KK\)-theory) |
| 46L08 | \(C^*\)-modules |
| 46L30 | States of selfadjoint operator algebras |
Keywords:
Kasparov module; extension; Cuntz-Pimsner algebra; Toeplitz extension; KMS state; conditional expectationCitations:
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