On the Existence of an Absolutely Minimal Norm in a Banach Algebra
نویسنده
چکیده
1. The norm || • || in a Banach algebra A is said to be minimal [l ] if, given any other norm || •||1 in A (with respect to which A need not be complete), the condition ||a||iá||a|| for each oG^4 implies that ||a[|i = ||a||. We shall say that || •|| is absolutely minimal if, given any other norm ||-||i whatever in A, then ||a||iè||a|| for each aEA. An absolutely minimal norm is of course minimal. Among the several examples of Banach algebras possessing a minimal norm, we mention (i) any complex 2?*-algebra [l, Theorem 10], (ii) any left c.c. 5#-algebra [3, Theorem 3.2], (iii) any Banach algebra of operators on a Banach space X which contains all operators in X of finite rank with the uniform topology [l Theorem 8]. The following example of a commutative Banach algebra with the stronger property of an absolutely minimal norm is due to Kaplansky [2, Theorem 6j:
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