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The spectral theorem in Banach algebras

  • Stephen Plafker (a1)
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The concept of a hermitian element of a Banach algebra was first introduced by Vidav [21] who proved that, if a Banach algebra π’œ has β€œenough” hermitian elements, then π’œ can be renormed and given an involution to make it a stellar algebra. (Following Bourbaki [5] we shall use the expression β€œstellar algebra” in place of the term β€œC*-algebra”.) This theorem was improved by Berkson [2], Glickfeld [10] and Palmer [17]. The improvements consist of removing hypotheses from Vidav's original theorem and in showing that Vidav's new norm is in fact the original norm of the algebra. Lumer [13] gave a spatial definition of a hermitian operator on a Banach space E and proved it to be equivalent to Vidav's definition when one considers the Banach algebra 𝓛(E) of continuous linear mappings of E into E.

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Copyright
COPYRIGHT: Β© Glasgow Mathematical Journal Trust 1972
References
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1.Berkson, E., A characterization of scalar type operators on reflexive Banach spaces, Pacific J. Math. 13 (1963), 365–373.
2.Berkson, E., Some characterizations of C*-algebras, Illinois J. Math. 10 (1966), 1–8.
3.Berkson, E., Hermitian projections and orthogonality in Banach spaces, Proc London Math. Soc. (3)24(1972), 101–118.
4.Bohnenblust, H. and Karlin, S., Geometrical properties of the unit sphere of Banach algebras, Ann. of Math. 62 (1955), 217–229.
5.Bourbaki, N., Theories spectrales, Chapitres 1 et 2 (Paris, 1967).
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13.Lumer, G., Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961), 29–43.
14.Lumer, G., Spectral operators, hermitian operators and bounded groups, Ada. Sci. Math. XXV (1964), 75–85.
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16.Maeda, F.-Y., Generalized spectral operators on locally convex spaces, Pacific J. Math. 13 (1963), 177–192.
17.Palmer, T., Characterization of C*-algebras, Bull. Amer. Math. Soc. 74 (1968), 538–540.
18.Palmer, T., Unbounded normal operators on Banach spaces, Trans. Amer. Math. Soc. 133 (1968), 385–414.
19.Panchapagesan, T., Unitary operators in Banach spaces, Pacific J. Math. 22 (1967), 465–475.
20.Plafker, S., Spectral representations for a general class of operators on a locally convex space, Illinois J. Math. 13 (1969), 573–582.
21.Vidav, I., Eine metrische Kennzeichnung der selbstadjungierten Operatoren, Math. Zeit. 66 (1956), 121–128.
22.Wermer, J., Commuting spectral measures on Hilbert space, Pacific J. Math. 4 (1954), 355–361.
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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