Completely bounded module maps and the Haagerup tensor product (Q1180633)

For two subalgebras \({\mathcal A}\) and \({\mathcal B}\) of the operator algebra \(B(H)\) and a left \({\mathcal A}\)-submodule, right \({\mathcal B}\)-submodule \({\mathcal E}\) of \(B(H)\) a mapping \(\Phi:{\mathcal E}\to B(H)\) is called a module map if \(\Phi(aeb)=a\Phi(e)b\) for \(a\in{\mathcal A}\), \(b\in{\mathcal B}\), \(e\in{\mathcal E}\). The author finds most general conditions on \({\mathcal A}\) and \({\mathcal B}\) ensuring the complete boundedness of a module map. The Haagerup representation theorem is obtained also with minimal restrictions on \({\mathcal A}\) and \({\mathcal B}\). Obtained results are applied to some problems of the cohomology theory. The paper is concluded with the investigation of the Haagerup's tensor product \(B(H)\otimes_ hB(H)\). The main result claims that this algebra satisfies a strong version of Tomita's commutant theorem for von Neumann algebras and the slice map property.