Norming and related algebras

Author:
B. E. Johnson

Journal:
Trans. Amer. Math. Soc. **220** (1976), 37-58

MSC:
Primary 46J10; Secondary 02H25

DOI:
https://doi.org/10.1090/S0002-9947-1976-0415326-6

MathSciNet review:
0415326

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Abstract: The first result of the paper is that the question of defining a submultiplicative seminorm on the commutative unital algebra is equivalent to that of putting a nontrivial submultiplicative seminorm on the algebra of infinitesimals in some nonstandard model of *C*. The extent to which the existence of such a norm on one implies the existence for others is investigated. Using the continuum hypothesis it is shown that the algebras of infinitesimals are isomorphic and that if such an algebra has a submultiplicative norm (or, equivalently, seminorm) then, for any totally ordered field containing R, the R-algebra of infinitesimals in has a norm. A result of Allan is extended to show that in the particular case when is a certain field of Laurent series in several (possibly infinitely many) unknowns then the infinitesimals have a submultiplicative seminorm.

**[1]**G. R. Allan,*Embedding the algebra of formal power series in a Banach algebra*, Proc. London Math. Soc. (3)**25**(1972), 329-340. MR**46**#4201. MR**0305071 (46:4201)****[2]**W. G. Bade and P. C. Curtis, Jr.,*Homomorphisms of commutative Banach algebras*, Amer. J. Math.**82**(1960), 589-608. MR**22**#8354. MR**0117577 (22:8354)****[3]**W. F. Donoghue, Jr.,*The lattice of invariant subspaces of a completely continuous quasi-nilpotent transformation*, Pacific J. Math.**7**(1957), 1031-1035. MR**19**, 1066. MR**0092124 (19:1066f)****[4]**L. Gillman and M. Jerison,*Rings of continuous functions*, University Ser. in Higher Math., Van Nostrand, Princeton, N.J., 1960. MR**22**#6994. MR**0116199 (22:6994)****[5]**L. Gillman,*General topology and its relations to modern analysis and algebra*. II, Proc. Second Prague Topological Sympos., 1966.**[6]**L. Hörmander,*An introduction to complex analysis in several variables*, Van Nostrand, Princeton, N.J., 1966. MR**34**#2933. MR**0203075 (34:2933)****[7]**N. Jacobson,*Lectures in abstract algebra*. Vol. III:*Theory of fields and Galois theory*, Van Nostrand, Princeton, N.J., 1964. MR**30**#3087. MR**0172871 (30:3087)****[8]**B. E. Johnson and A. M. Sinclair,*Continuity of linear operators commuting with continuous linear operators*. II, Trans. Amer. Math. Soc.**146**(1969), 533-540. MR**40**#4791. MR**0251564 (40:4791)****[9]**A. M. Sinclair,*Homomorphisms from*-*algebras*, Proc. London Math. Soc. (3)**29**(1974), 435-452. MR**0358368 (50:10834)**

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DOI:
https://doi.org/10.1090/S0002-9947-1976-0415326-6

Article copyright:
© Copyright 1976
American Mathematical Society