Tensor products of unbounded operator algebras

Abstract

The term $GW^*$-algebra means a generalized $W^*$-algebra and corresponds to an unbounded generalization of a standard von Neumann algebra. It was introduced by the second named author in 1978 for developing the Tomita-Takesaki theory in algebras of unbounded operators. In this note we consider tensor products of unbounded operator algebras resulting in a $GW^*$-algebra. Existence and uniqueness of the $GW^*$-tensor product is encountered, while ``properly $W^*$-infinite" $GW^*$-algebras are introduced and their structure is investigated.