A seminorm with square property on a complex associative algebra is submultiplicative

Sebestyén, Zoltán

doi:10.1090/S0002-9939-01-06278-5

A seminorm with square property on a complex associative algebra is submultiplicative
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Proc. Amer. Math. Soc. 130 (2002), 1993-1996 Request permission

Abstract:

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.

References

Richard Arens, Moshe Goldberg, and W. A. J. Luxemburg, Multiplicativity factors for seminorms. II, J. Math. Anal. Appl. 170 (1992), no. 2, 401–413. MR 1188561, DOI 10.1016/0022-247X(92)90026-A
Bernard Aupetit, Propriétés spectrales des algèbres de Banach, Lecture Notes in Mathematics, vol. 735, Springer, Berlin, 1979 (French). MR 549769
S. J. Bhatt and D. J. Karia, Uniqueness of the uniform norm with an application to topological algebras, Proc. Amer. Math. Soc. 116 (1992), no. 2, 499–503. MR 1097335, DOI 10.1090/S0002-9939-1992-1097335-4
S. J. Bhatt, A seminorm with square property on a Banach algebra is submultiplicative, Proc. Amer. Math. Soc. 117 (1993), no. 2, 435–438. MR 1128724, DOI 10.1090/S0002-9939-1993-1128724-8
Robert S. Doran and Victor A. Belfi, Characterizations of $C^*$-algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 101, Marcel Dekker, Inc., New York, 1986. The Gel′fand\mhy Naĭmark theorems. MR 831427
Zoltán Magyar and Zoltán Sebestyén, On the definition of $C^\ast$-algebras. II, Canad. J. Math. 37 (1985), no. 4, 664–681. MR 801420, DOI 10.4153/CJM-1985-035-7
Theodore W. Palmer, Banach algebras and the general theory of $^*$-algebras. Vol. I, Encyclopedia of Mathematics and its Applications, vol. 49, Cambridge University Press, Cambridge, 1994. Algebras and Banach algebras. MR 1270014, DOI 10.1017/CBO9781107325777
Z. Sebestyén, Every $C^{\ast }$-seminorm is automatically submultiplicative, Period. Math. Hungar. 10 (1979), no. 1, 1–8. MR 506161, DOI 10.1007/BF02018366