The algebra is called a pre--algebra if it satisfies all conditions to be a -algebra except that its norm need not be complete.
Pre-C^*-Algebra
See also
C-*-AlgebraThis entry contributed by Mohammad Sal Moslehian
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References
Murphy, G. J. C-*-Algebras and Operator Theory. New York: Academic Press, 1990.Cite this as:
Moslehian, Mohammad Sal. "Pre-C^*-Algebra." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Pre-C-Star-Algebra.html