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RESEARCH PRODUCT

Biweights on Partial *-Algebras

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This chapter is devoted to the systematic investigation of biweights on partial *-algebras. These are a generalization of invariant positive sesquilinear forms that still allows a Gel’fand—Naĭmark—Segal (GNS) construction of representations. In Section 9.1, we apply this GNS construction for biweights and we obtain *-representations and cyclic vector representations of partial *-algebras, and we give some examples of biweights. Section 9.2 is devoted to the investigation of the Radon—Nikodým theorem and the Lebesgue decomposition theorem for biweights on partial *-algebras. In Section 9.3, we define regular and singular biweights on partial *-algebras and we characterize them with help of the so-called tricom-mutants defined by biweights. Let φ be a biweight with core B(φ) on a partial *-algebra. Suppose that the corresponding GNS representation π φ B is self-adjoint. Then φ decomposes into φ = φ r + φ s , where φ r is a regular biweight and φ s is a singular biweight. In Section 9.4, we define and investigate the notions of admissible and approximately admissible biweights on partial *-algebras. It is shown that a biweight φ is approximately admissible if and only if the GNS representation π φ B is unitarily equivalent to the direct sum of bounded *-representations. Section 9.5 is devoted to the investigation of standard biweights on partial O*-algebras, which allow to develop the Tomita—Takesaki theory in partial O*-algebras. In Section 9.6, finally, we discuss the trace representation of weights on partial O*-algebras.