# zbMATH — the first resource for mathematics

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)
Full Text: