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Multipliers of imprimitivity bimodules and Morita equivalence of crossed products. (English) Zbl 0843.46049
Recall that two \(C^*\)-algebras \(A\) and \(B\) are called Morita equivalent, if there exists an \(A-B\) imprimitivity bimodule \(X\), which is a left Hilbert \(A\)-module and a right Hilbert \(B\)-module with commuting module actions and compatible \(A\)- and \(B\)-valued inner products. Morita equivalent \(C^*\)-algebras have many important properties in common, for instance it has been shown by L. G. Brown, P. Green and M. A. Rieffel [Pac. J. Math. 71, 349-363 (1977; Zbl 0362.46043)] that two separable \(C^*\)-algebras are Morita equivalent if and only if they are stably isomorphic. Following the well-known construction of the multiplier algebra \(M(A)\) of a \(C^*\)-algebra \(A\), we construct the multiplier bimodule \(M(X)\) of an \(A- B\) imprimitivity bimodule \(X\) as the set of compatible pairs of maps \(m_A: A\to X\), \(m_B: B\to X\), representing left and right multiplication by the multiplier \(m= (m_A, m_B)\). \(M(X)\) can be characterized as the universal \(A- B\) bimodule \(M\) containing \(X\) as a submodule satisfying \(A\cdot M\subseteq X\) and \(M\cdot B\subseteq X\), and we show that the multiplier bimodule \(M(X)\) enjoys similar properties as the multiplier algebra of a \(C^*\)-algebra.
The motivation for our construction lies in recent work of S. Baaj and G. Skandalis [\(K\)-Theory, 2, No. 6, 683-721 (1989; Zbl 0683.46048)] and H. H. Bui [J. Funct. Anal. 123, No. 1, 59-98 (1994; Zbl 0815.46059)] on Morita equivalence of crossed products by coactions. In our language, a Morita equivalence for two coactions \(\delta_A\) and \(\delta_B\) of a locally compact group \(G\) on the Morita equivalent \(C^*\)-algebras \(A\), \(B\) is a coaction \(\delta_X: X\to M(X\otimes C^*_r(G))\) of \(G\) on the imprimitivity bimodule \(X\) connecting \(\delta_A\) and \(\delta_B\) in a natural way. Both Baaj-Skandalis and Bui proved that Morita equivalent coactions \(\delta_A\) and \(\delta_B\) have Morita equivalent crossed products \(A\times_{\delta_A} G\) and \(B\times_{\delta_B} G\). We use the basic properties of multiplier bimodules to give a relatively easy construction of the imprimitivity bimodule \(X\times_{\delta_X} G\). As a corollary, we also obtain the analoguous result for twisted coactions in the sense of J. Phillips and I. Raeburn [J. Austral. Math. Soc., Ser. A 56, No. 3, 320-344 (1994; Zbl 0805.46067)], first proved by Bui as the main result in his paper cited above.

MSC:
46L55 Noncommutative dynamical systems
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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