Compact endomorphisms of Banach algebras

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

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“…We let φ n denote the n th iterate of φ. If the endomorphism T is compact, then it was shown in [6] that…”

Section: Part I Compact Endomorphismsmentioning

confidence: 99%

Compact Endomorphisms of Banach Algebras of Infinitely Differentiable Functions

Feinstein

Kamowitz

2004

J. London Math. Soc.

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“…On uniform algebras with connected spectrum, compact operators have range contained in a single Gleason part; that is, if φ is the map associated with our composition operator, there exists a positive constant r < 1 such that ρ(φ(x), φ(y)) ≤ r (see [9] or [10]) for all x and y in the spectrum M (A). Thus, we expect noncompactness to be associated with pseudohyperbolic distance tending to 1.…”

Section: Other Essential Norm Estimatesmentioning

confidence: 99%

“…In particular, as shown in [9], if X is a compact connected Hausdorff space, then every nonzero compact endomorphism T of C(X) has the form T f = f (x 0 )1 for some x 0 ∈ X. In the final section of this paper, we will study conditions under which a linear combination of unital endomorphisms is compact.…”

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Norms and essential norms of linear combinations of endomorphisms

Gorkin

Mortini²

2004

Trans. Amer. Math. Soc.

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“…We show that A has a proper closed nonzero two-sided ideal, and we prove a slightly stronger result when A is commutative. There are many Banach algebras which have compact endomorphisms [5]; this can even happen for commutative radical integral domains [3, §5]. On the other hand, some primitive Banach algebras, such as the algebra of compact operators on Hilbert space, have no nonzero two-sided ideals.…”

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Compact endomorphisms and closed ideals in Banach algebras

Grabiner¹

1984

Proc. Amer. Math. Soc.

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Abstract.Every infinite-dimensional Banach algebra with a nonzero compact endomorphism has a proper closed nonzero two-sided ideal. When the algebra is commutative, the ideal is also an ideal in the multiplier algebra.Suppose that A is a Banach algebra with a compact nonzero endomorphism. We show that A has a proper closed nonzero two-sided ideal, and we prove a slightly stronger result when A is commutative. There are many Banach algebras which have compact endomorphisms [5]; this can even happen for commutative radical integral domains [3, §5]. On the other hand, some primitive Banach algebras, such as the algebra of compact operators on Hilbert space, have no nonzero two-sided ideals. Our results show that algebras with no nonzero closed ideals can have no compact nonzero endomorphisms.Our proofs involve applying extensions, which we developed in [4], of Lomonosov's invariant subspace theorems. Apparently the best that one can do with the original versions of Lomonosov's theorem [6,7] is to show that if A has a compact left or right multiplication operator, then it has both left and right nonzero closed ideals. Using very different methods, Esterle [1] has recently made significant contributions to the study of the problem of whether every radical Banach algebra has a nonzero closed ideal. Theorem 1. // the infinite-dimensional Banach algebra A has a compact nonzero endomorphism , then A has a proper closed nonzero ideal.Proof. Let A9 be the Banach algebra formed by adjoining an identity to A if A has no identity, and let A* = A if A has an identity. For each a in A%, let La be the left multiplication operator Lax = ax, and leti? be the algebra of all left multiplication operators La on A for a in A*. Similarly, let ¿% be the algebra of right multiplication operators. To show that A has a proper closed ideal, we show that ££ and 0t have a common proper invariant subspace. We do this by showing that the algebrase and i% satisfy the hypotheses of [4, Theorem (3.2), p. 849].The map a -» La is a continuous linear transformation from A* to the algebra of bounded operators on A. Hence =5? is an operator range algebra in the sense of [4, p. 845] (cf. [2, §3]). Since is a homomorphism, we have La = L^a)^ for every a in

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Compact endomorphisms of Banach algebras

Cited by 29 publications

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Compact Endomorphisms of Banach Algebras of Infinitely Differentiable Functions

Compact Endomorphisms of Banach Algebras of Infinitely Differentiable Functions

Norms and essential norms of linear combinations of endomorphisms

Compact endomorphisms and closed ideals in Banach algebras

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