A Seminorm with Square Property on a Complex Associative Algebra Is Submultiplicative
نویسنده
چکیده
The result stated in the title is proved as a consequence of an appropriate generalization replacing the square property of a seminorm with a similar weaker property which implies an equivalence to the supnorm of all continuous functions on a compact Hausdorff space also. Theorem. Let p be a seminorm with the square property on a complex (associative) algebra A. Then the following hold for all a, b in A: (1) p(ab− ba) = 0. (2) p(ab) ≤ p(a)p(b). This is a unifom seminorm analogue of [8] or Thm. 6 in [6] that a C∗-seminorm is submultiplicative (and the involution is isometric). We answer a problem posed in [3] and solved in the particular case of Banach algebras [4]. A seminorm on A is a nonnegative function on A satisfying: (i) p(a+ b) ≤ p(a) + p(b) for all a, b in A and (ii) p(λa) = |λ|p(a) for all a, for all scalars λ. The seminorm p is submultiplicative if (iii) p(ab) ≤ p(a)p(b) for all a, b in A. It satisfies the square property [3, 4] if (iv) p(a) = p(a) for all a in A. The above theorem is a consequence of the following. Proposition. Let p be a seminorm on a complex (associative) algebra A satisfying (iv)∗ mp(a) ≤ p(a) ≤Mp(a) for all a in A, where 0 < m ≤M are given constants. Then properties (1) and (2)∗, (2)∗ mp(ab) ≤Mp(a)p(b) for all a, b in A, hold true. Received by the editors September 6, 2000 and, in revised form, January 17, 2001. 2000 Mathematics Subject Classification. Primary 46H05, 46J05.
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