Multinormed semifinite von Neumann algebras, unbounded operators and conditional expectations

The present paper is devoted to classification of multinormed W⁎-algebras in terms of their bounded parts. We obtain a precise description of the bornological predual of a multinormed W⁎-algebra, which is reduced to a multinormed noncommutative L1-space in the semifinite case. A multinormed noncommutative L2-space is obtained as the union space of a commutative domain in a semifinite von Neumann algebra. Multinormed W⁎algebras of type I are described as locally bounded decomposable (unbounded) operators associated with a measurable covering.