Abstract
Forty years ago, Haag and Kastler [27] introduced the algebraic approach to Quantum Field Theory, and soon their method was extended to statistical mechanics. The rationale was to free the theory from particular realizations, linked to specific models, and to focus on the basic structure, namely, the algebra of observables and states on it. This more abstract language, exploiting the deep mathematical theories of C*-algebras and von Neumann algebras, soon became the standard approach to the mathematically rigorous description of physical systems with infinitely many degrees of freedom. Textbooks flourished and, among many authors, Gérard Emch argued forcefully for the new approach [6].
We review the main steps in the development of partial *-algebras. First we discuss the algebraic structure stemming from the partial multiplication. Then we study in some detail the locally convex partial *-algebras, in particular, the Banach partial *-algebras, and we describe a number of concrete examples. Next we consider the partial *-algebras of closable operators in Hilbert spaces (partial O*-algebras), with a special emphasis on their *-automorphisms. Finally we sketch the representation theory of abstract partial *-algebras and give some instances of possible physical applications.
To Gérard Emch, colleague, friend, editor, on the occasion of his 70th birthday
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Antoine, JP. (2007). Partial *-Algebras, A Tool for the Mathematical Description of Physical Systems. In: Ali, S.T., Sinha, K.B. (eds) Contributions in Mathematical Physics. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-33-0_2
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DOI: https://doi.org/10.1007/978-93-86279-33-0_2
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