Compact Endomorphisms of Banach Algebras of Infinitely Differentiable Functions

Abstract

Let (M n ) be a sequence of positive numbers satisfying M 0 = 1 and M n+m M n M m ≥ n + m m for all non-negative integers m, n. We letWith pointwise addition and multiplication, D([0, 1], M ) is a unital commutative semisimple Banach algebra. If lim n→∞ (n!/M n ) 1/n = 0, then the maximal ideal space of the algebra is [0, 1] and every non-zero endomorphism T has the form T f (x) = f (φ(x)) for some selfmap φ of the unit interval. Previously we have shown for a wide class of φ mapping the unit interval to itself that if φ ′ ∞ < 1 then φ induces a compact endomorphism. Here we investigate the extent to which this condition is necessary, and we determine the spectra of all compact endomorphisms of D([0, 1], M ). We also simplify and strengthen some of our earlier results on general endomorphisms of D([0, 1], M ). IntroductionIn two previous papers [5] and [7] we studied endomorphisms of certain algebras of infinitely differentiable functions. In this note, we continue our study, concentrating on algebras of functions on compact intervals in the real line. Let (M n ) be a sequence of positive numbers satisfying M 0 = 1

Quasicompact endomorphisms of commutative semiprime Banach algebras

“…Using a fairly standard technique involving orthogonal idempotents, we will prove the following result, which is a main result of this note. This result extends earlier results of the authors [5,6,7] for commutative semisimple Banach algebras, and results for uniform algebras of Klein [14] and Gamelin, Galindo and Lindström [8,9]. Section 4 contains some results about commutative radical semiprime Banach algebras, while Section 5 presents two examples of commutative semisimple Banach algebras where each quasicompact endomorphism is compact.…”

“…In previous papers the authors [5,6,7] and others [8,9,14] have studied endomorphisms of commutative semisimple Banach algebras and have obtained several general theorems, and also a variety of results pertaining to specific classes of algebras. In this note we extend this discussion to endomorphisms of commutative Banach algebras which are semiprime and not necessarily semisimple.…”

Quasicompact and Riesz endomorphisms of Banach algebras

“…Suppose further that the maximal ideal space of D(X, M) is precisely X. In this case, the proofs of Theorem 2.4 of [6] when X = [0, 1] and Theorem 11 of [7] when X is uniformly regular show that…”

Quasianalyticity in certain Banach function algebras

“…(Note that, in [15], the term analytic was used for those functions on X which extend to be analytic on a neighbourhood of X. This condition is stronger than in [16,17].) Let X be a semi-rectifiable compact plane set, let F be an effective collection of paths in X, and let f ∈ D (∞) F (X).…”

“…This is a generalisation of the term analytic used in [16,17], and is used to find sufficient conditions for maps to induce homomorphisms between the algebras D F (X, M). (Note that, in [15], the term analytic was used for those functions on X which extend to be analytic on a neighbourhood of X.…”