Unbounded *-representations of partial *-algebras

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*-algebras (O^*-algebras) of closable operators in Hilbert space have been studied from the pure mathematical situation and the physical applications. A survey of the theory of O^*-algebras may be found in the monograph of K. Schmudgen "Unbounded Operator Algebras and Representation Theory, Operator Theory: Advances and Applications vol. 37, Birkhauser(1990)" and the lecture note of A. Inoue "Tomita-Takasaki Theory in Algebras of Unbounded Operator Algebras, Lecture Notes in Mathematics, 1699, Springer(1995)".Furthermore, partial *-algebras (partial O^*-algebras) of closable operators have been studied from the mathematical situations and the physical applications, but it seems that the systematic studies are insufficient. In order to proceed the studies of *-representations of partial *-algebras, we have investigated the following.(1) What is a well-behaved *-representation in unbounded *-representations of (partial) *-algebras?(2) Systemic studies of weights of partial O^*-algebras.(3) Applications of (1) and (2) to quantum physics.Thus we (A. Inoue, J. P. Antoine and C. Trapani) have published a book "Partial *-algebras and Their Operator Realizations, Mathematics and Its Applications vol. 553" from Kluwer Academic Publishers.