Norming 𝐶(𝑈) and related algebras

Johnson, B. E.

doi:10.1090/S0002-9947-1976-0415326-6

Norming $C(U)$ and related algebras
HTML articles powered by AMS MathViewer

by B. E. Johnson PDF

Trans. Amer. Math. Soc. 220 (1976), 37-58 Request permission

Abstract:

The first result of the paper is that the question of defining a submultiplicative seminorm on the commutative unital ${C^\ast }$ algebra $C(\Omega )$ 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 $C(\Omega )$ 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 $\mathfrak {k}$ containing R, the R-algebra of infinitesimals in $\mathfrak {k}$ has a norm. A result of Allan is extended to show that in the particular case when $\mathfrak {k}$ is a certain field of Laurent series in several (possibly infinitely many) unknowns then the infinitesimals have a submultiplicative seminorm.

References

G. R. Allan, Embedding the algebra of formal power series in a Banach algebra, Proc. London Math. Soc. (3) 25 (1972), 329–340. MR 305071, DOI 10.1112/plms/s3-25.2.329
W. G. Bade and P. C. Curtis Jr., Homomorphisms of commutative Banach algebras, Amer. J. Math. 82 (1960), 589–608. MR 117577, DOI 10.2307/2372972
W. F. Donoghue Jr., The lattice of invariant subspaces of a completely continuous quasi-nilpotent transformation, Pacific J. Math. 7 (1957), 1031–1035. MR 92124
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199

General topology and its relations to modern analysis and algebra

Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
Nathan Jacobson, Lectures in abstract algebra. Vol III: Theory of fields and Galois theory, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London-New York, 1964. MR 0172871
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 251564, DOI 10.1090/S0002-9947-1969-0251564-X
Allan M. Sinclair, Homomorphisms from $C^{\ast }$-algebras, Proc. London Math. Soc. (3) 29 (1974), 435–452. MR 358368, DOI 10.1112/plms/s3-29.3.435