Algebras of unbounded operators called O~* -algebras have been studying from the pure mathematical situations (operator theory, topological *-algebras, representations of Lie algebras etc.) and the physical applications (the Wightman. quantum field theory, unbounded CCR-algebras, quantum groups etc.). To proceed such studies it is important to study the Tomita-Takesaki theory in O~*-algebras. Weights on O~*-algebras (that is, linear functionals that take positive, but not necessarily finite valued) are naturally appeared in the studies of the unbounded Tomita-Takesaki theory and the quantum physics. Thus it is significant to study weights on O~*-algebras for the structure of O~*-algebras and the physical applications. Further, the weights on O~*-algebras occasion some pathological phenomena which don't occur for weights on C~*- and W~*-algebras. From this viewpoint we should study systematically weights on O~*-algebras.
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