Arens algebras, associated with commutative von Neumann algebras (Q1128329)

The classification of the Arens algebras associated with the commutative von Neumann algebras is given. Let \(M\) be a von Neumann algebra with a faithful normal finite trace \(\mu\). The \(L^p(X,\mu)\) space \((1\leq p<\infty)\) is defined as the space of all \(\mu\)-measurable operators affiliated with \(M\) for which \(\mu((x^\ast x)^p)<\infty\). The Arens algebra \(L^\omega(M, \mu)\) is then defined as an intersection \[ L^\omega(M,\mu)=\bigcap_{1\leq p<\infty}L^p(M,\mu). \] It is shown that \(L^\omega(M,\nu)\) and \(L^\omega(M,\mu)\) are \(\ast\)-isomorphic whenever \(M\) and \(N\) are non-atomic commutative von Neumann algebras acting on a separable Hilbert space with faithful normal finite traces \(\mu\) and \(\nu\), respectively. The main result of the paper fully describes the situation when \(L^\omega(M,\mu)\) and \(L^\omega(M,\nu)\) are \(\ast\)-isomorphic for a non-atomic commutative \(\sigma\)-finite von Neumann algebra \(M\) in terms of the decomposition of its projection Boolean algebra into a direct sum of homogeneous parts. Interesting consequences of this result are given.