Partial *-algebras generated by a family of unbounded operators on Hibert spaces

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We have studied a family of unbounded operators equipped with a partial multiplication (partial Op^*-algebra), and obtained the following results :(1) We studied the structure of partial Op^*-algebras generated by finite commutative symmetric operators.(2) We studied cyclic vectors for partial Op^*-algebras.(3) We investigated the self-adjointness and the standardness of (paxtial) Op^*-algebras and unbounded *-representations of *-algebras.(4) We investigated the self-adjoint extension and standard extension of unbounded *-representations of *-algebras.(5) We studied invariant positive sesquilinear forms on partial *-algebras and considered the non-commutative integration theory for partial Op^*-algebras.(6) We extended the Tomita-Takesaki theory for von Neumann algebras to partial Op^*-algebras, and applied it to the Wightman quantum field theory.