This paper, the first of two, is devoted to a systematic study of partial *-algebras of closable operators in a Hilbert space (partial Op*-algebras). After setting up the basic definitions, we describe canonical extensions of partial Op*-algebras by closure and introduce a new bounded commutant, called quasi-weak. We initiate a theory of abelian partial *-algebras. As an application, we analyze thoroughly the partial Op*-algebras generated by a single closed symmetric operator.
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Jean-Pierre Antoine, Atsushi Inoue, Camillo Trapani, Partial *-Algebras of Closable Operators. I. The Basic Theory and the Abelian Case. Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, pp. 359–395DOI 10.2977/PRIMS/1195171084