Geodesic
The spectral theorem in Banach algebras
Glasgow mathematical journal, Tome 13 (1972) no. 1, pp. 49-55

Voir la notice de l'article provenant de la source Cambridge University Press

The concept of a hermitian element of a Banach algebra was first introduced by Vidav [21] who proved that, if a Banach algebra A has “enough” hermitian elements, then A can be renormed and given an involution to make it a stellar algebra. (Following Bourbaki [5] we shall use the expression “stellar algebra” in place of the term “C*-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. 
Plafker, Stephen. The spectral theorem in Banach algebras. Glasgow mathematical journal, Tome 13 (1972) no. 1, pp. 49-55. doi: 10.1017/S0017089500035758
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