Iterating the Pimsner construction (Q997838)

The author considers a pair \(E_1, E_2\) of right-Hilbert \(A\)-\(A\) bimodules (otherwise known as \(C^*\)-correspondences over \(A\)) equipped with a fixed isomorphism \(\chi : E_1 \otimes_A E_2 \to E_2 \otimes_A E_1\). He uses the canonical inclusions of \(A\) into the algebras \(\mathcal{T}_{E_1}\) and \(\mathcal{O}_{E_1}\) associated to \(E_1\) in [\textit{M.\,V.\thinspace Pimsner}, Fields Inst.\ Commun.\ 12, 189--212 (1997; Zbl 0871.46028)] to form right-Hilbert \(A\)-\(A\) bimodules \(E_2 \otimes_A \mathcal{T}_{E_1}\) and \(E_2 \otimes_A \mathcal{O}_{E_1}\). By iterating Pimsner's constructions, he therefore obtains \(C^*\)-algebras \[ \mathcal{T}_{E_2 \otimes_A \mathcal{T}_{E_1}}, \quad \mathcal{T}_{E_2 \otimes_A \mathcal{O}_{E_1}}, \quad \mathcal{O}_{E_2 \otimes_A \mathcal{T}_{E_1}}, \quad\text{and}\quad \mathcal{O}_{E_2 \otimes_A \mathcal{O}_{E_1}}. \] Under the assumption that the \(E_i\) are full and finitely generated, the author establishes a number of relationships between these \(C^*\)-algebras and the ones obtained in the same manner after reversing the rôles of \(E_1\) and \(E_2\): \[ \mathcal{T}_{E_2 \otimes_A \mathcal{T}_{E_1}} \cong \mathcal{T}_{E_1 \otimes_A \mathcal{T}_{E_2}}, \quad \mathcal{T}_{E_1 \otimes_A \mathcal{O}_{E_2}} \cong \mathcal{O}_{E_2 \otimes_A \mathcal{T}_{E_1}}, \quad\text{and}\quad \mathcal{O}_{E_2 \otimes_A \mathcal{O}_{E_1}} \cong \mathcal{O}_{E_1 \otimes_A \mathcal{O}_{E_2}}. \] The isomorphism \(\chi\) is precisely what is needed to form a product system \(E\) of Hilbert bimodules over \(\mathbb{Z}^2\) in the sense of [\textit{N.\,J.\thinspace Fowler}, Pac.\ J.\ Math.\ 204, No.\,2, 335--375 (2002; Zbl 1059.46034)]. The author shows that \(\mathcal{T}_{E_2 \otimes_A \mathcal{T}_{E_1}}\) coincides with Fowler's \(\mathcal{T}_E\) and that \(\mathcal{O}_{E_2 \otimes_A \mathcal{O}_{E_1}}\) coincides with Fowler's \(\mathcal{O}_E\). In the final section, the author obtains some exact sequences in \(K\)-theory, building on results of [\textit{R.\,G.\thinspace Douglas} and \textit{R.\,Howe}, Trans.\ Am.\ Math.\ Soc.\ 158, 203--217 (1971; Zbl 0224.47015)] and [\textit{M.\,Pimsner} and \textit{D.\,Voiculescu}, J.~Oper.\ Theory 4, No.\,1, 93--118 (1980; Zbl 0474.46059)]. The author first presents a nine-term commuting diagram with exact rows and columns involving the \(C^*\)-algebras and Hilbert bimodules constructed earlier in the paper. He obtains from this diagram two short exact sequences for which the corresponding six-term sequences in \(K\)-theory can be used to obtain information about \(\mathcal{O}_{E_2 \otimes_A \mathcal{O}_{E_1}}\). The author illustrates this point with a number of examples.