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 L. Gillman, General topology and its relations to modern analysis and algebra. II, Proc. Second Prague Topological Sympos., 1966.
- 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
Additional Information
- © Copyright 1976 American Mathematical Society
- 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