Abstract
Unbounded *-representations of *-algebras are studied. Representations called self-adjoint representations are defined in analogy to the definition of a self-adjoint operator. It is shown that for self-adjoint representations certain pathologies associated with commutant and reducing subspaces are avoided. A class of well behaved self-adjoint representations, called standard representations, are defined for commutative *-algebras. It is shown that a strongly cyclic self-adjoint representation of a commutative *-algebra is standard if and only if the representation is strongly positive, i.e., the representations preserves a certain order relation. Similar results are obtained for *-representations of the canonical commutation relations for a finite number of degrees of freedom.
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References
Bourbaki, N.: Éléments de mathématiques Livre V, Espaces vectoriels topologiques. Paris: Herman 1955.
Dixmier, J.: LesC*-algébras et leurs représentations. Paris: Gauthier-Villars 1964.
Dunford, N., Schwartz, J. T.: Linear operators part II. New York: Interscience Pub. 1963.
Gelfand, M., Vilenkin, N. Ya.: Generalized functions, Vol. IV. New York: Academic Press 1964.
Jaffe, A. M.: Dynamics of a cut-offλ φ4 field theory. Thesis Princeton University (1965).
Naimark, M. A.: Normed Rings. Groningen, The Netherlands: P. Noordholf N. V. 1964.
Nelson, E.: Analytic vectors. Ann. Math.70, 572–615 (1959).
Reed, M.: A garding domain for quantum fields. Commun. math. Phys.14, 336–346 (1969).
Segal I. E.: Irreducible representations of operator algebras. Bull. Am. Math. Soc.53, 73–88 (1947).
-- Decomposition of operator algebras, II. Memoirs of the American Mathematical Society, No. 9, New York (1951).
Sherman, T.: Positive linear functionals on *-algebras of unbounded operators. J. Math. Anal. Appl.22, 285–318 (1968).
Shohat, J. A., Tamarkin, J. D.: The problem of moments, surveys, No. 1. Am. Math. Soc. New York (1943).
Simon, B.: Distributions and their Hermite expansions. Preprint.
Stone, M. H.: Linear transformation in Hilbert space and their applications to analysis. Am. Math. Soc. Colloquium Pub., Vol. 15, New York (1932).
Streater, R. F., Wightman, A. S.: P.C.T., spin and statistics and all that. New York: Benjamin 1964.
Riesz, F., Nagy, B. Sz.: Functional analysis. New York: Fredrich Ungar Pub. 1955.
Neumann, J. von: Die Eindeutigkeit der Schrödingerschen Operatoren. Math. Ann.104, 570–578 (1931).
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Work supported in part by U.S. Atomic Energy Commission under Contract AT(30-1)-2171 and by the National Science Foundation.
Alfred P. Sloan Foundation Fellow.
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Powers, R.T. Self-adjoint algebras of unbounded operators. Commun.Math. Phys. 21, 85–124 (1971). https://doi.org/10.1007/BF01646746
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DOI: https://doi.org/10.1007/BF01646746