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Green’s theorem for crossed products by Hilbert \(C^*\)-bimodules. (English) Zbl 1385.46042

Crossed products of \(C^*\)-algebras by Hilbert bimodules were defined in [B. Abadie et al., Trans. Am. Math. Soc. 350, No. 8, 3043–3054 (1998; Zbl 0899.46053)] as follows. Given a Hilbert \(A-A\) bimodule \(X\) and a \(C^*\)-algebra \(C\), a covariant pair from \(_AX_A\) to \(C\) is a pair of maps \((\varphi,\psi)\) where \(\varphi: A\to C\) is a \(*\)-morphism and \(\psi: X\to C\) a linear map satisfying \[ \psi(a\cdot x)=\varphi(a)\psi(x),\quad \varphi(\langle x,y\rangle_L)=\psi(x)\psi(y)^*, \]
\[ \psi(x\cdot a)=\psi(x)\varphi(a),\quad \varphi(\langle x,y\rangle_R)=\psi(x)^*\psi(y), \] for all \(a\in A\), \(x,y\in X\). That is, the pair preserves the Hilbert bimodule structure, considering on \(C\) the trivial Hilbert \(C-C\) bimodule structure. A crossed product of a \(C^*\)-algebra \(A\) by a Hilbert \(A-A\) bimodule \(X\) is a \(C^*\)-algebra \(A\rtimes X\) together with a covariant pair \((\iota_A,\iota_X)\) from \(_AX_A\) to \(A\rtimes X\) satisfying the following universal property: for any covariant pair \((\varphi,\psi)\) from \(_AX_A\) to a \(C^*\)-algebra \(C\), there exists a unique \(*\)-morphism \(\varphi\rtimes\psi: A\rtimes X\to C\) such that \(\varphi=(\varphi\rtimes\psi)\circ\iota_A\) and \(\psi=(\varphi\rtimes\psi)\circ\iota_X\). The crossed product exists and is unique up to isomorphism. The maps \(\iota_A\) and \(\iota_X\) are injective, so that we may consider \(A,X\subseteq A\rtimes X\) and the induced \(*\)-morphism \(\varphi\rtimes\psi\) as an extension of the covariant pair \((\varphi,\psi)\).
If \(X_B\) is a right Hilbert \(B\)-module, \(C\) a \(C^*\)-algebra and \(\sigma: C\to B\) a \(*\)-isomorphism, then we denote \(X_{\sigma}\) the right Hilbert module over \(C\) obtained by considering on the vector space \(X\) the operations \[ x\cdot_{\sigma}c=x\cdot\sigma(c) \quad\text{and}\quad \langle x,y\rangle^{ \sigma}=\sigma^{-1}( \langle x,y\rangle )\qquad\text{for }c\in C,\,x,y\in X. \] If, in addition, \(X\) is a Hilbert \(A-B\) bimodule, then \(X_{\sigma}\) is a Hilbert \(A-C\) bimodule with the original left structure. The module \(X\) is right full iff \(X_{\sigma}\) is. Given Hilbert bimodules \(_{A_i}{X_i}_{B_i}\) for \(i=1,\ldots,n\), then \(\bigoplus X_i\) is a Hilbert \(\bigoplus A_i-\bigoplus B_i\) bimodule with point-wise operations. The bimodule \(\bigoplus X_i\) is right full iff \(X_i\) is for all \(i=1,\dots,n\). Let \(_{A_1}{X_1}_{A_2}, {}_{A_2}{X_2}_{A_3}, \dots, {}_{A_n}{X_n}_{A_1}\) be Hilbert bimodules; we can make \(\bigoplus X_i\) into a Hilbert bimodule over \(\bigoplus A_i\) twisting the right action in the previous constriction with the isomorphism \(\sigma: A_1\oplus A_2\oplus\cdots\oplus A_n\to A_2\oplus\cdots\oplus A_n\oplus A_1\) given by \[ \sigma(a_1,a_2,\dots,a_n)=(a_2,\dots,a_n,a_1)\quad\text{for }a_k\in A_k. \]
Let \(_{A_1}{X_1}_{A_2}, {}_{A_2}{X_2}_{A_3},\dots, {}_{A_n}{X_n}_{A_1}\) be right full Hilbert bimodules and consider their twisted sum \((X_1\otimes\cdots\otimes X_n)_{\sigma}\) as above. In this paper, the author proves the following Morita equivalence \[ A_1\rtimes(X_1\otimes\cdots\otimes X_n)\sim(A_1\oplus\cdots\oplus A_n)\rtimes (X_1\oplus\cdots\oplus X_n)_{\sigma}. \] This fact can be considered as a generalization of Green’s theorem [D. P. Williams, Crossed products of \(C^*\)-algebras. Providence, RI: American Mathematical Society (AMS) (2007; Zbl 1119.46002)] for crossed products by Hilbert \(C^*\)-bimodules.

MSC:

46L08 \(C^*\)-modules
46L55 Noncommutative dynamical systems
46L05 General theory of \(C^*\)-algebras
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