Closed ideals in algebras of smooth functions

Hanin Leonid G.

GDML_Books, (1997), p.

Résumé

AbstractA topological algebra admits spectral synthesis of ideals (SSI) if every closed ideal in this algebra is an intersection of closed primary ideals. According to classical results this is the case for algebras of continuous, several times continuously differentiable, and Lipschitz functions. New examples (and counterexamples) of function algebras that admit or fail to have SSI are presented. It is shown that the Sobolev algebra $W_p^{l}(ℝⁿ)$, 1 ≤ p < ∞, has the property of SSI for and only for n = 1 and 2 ≤ n < p. It is also proved that every algebra $C^m Lip φ$ in one variable admits SSI. A unified approach to SSI based on an abstract spectral synthesis theorem for a class of Banach algebras (called D-algebras) defined in terms of point derivations and consisting of functions with “order of smoothness” not greater than 1 is discussed. Within this framework, theorems on SSI for Zygmund algebras $Λ_φ$ in one and two variables not imbedded in C¹ as well as for their separable counterparts $λ_φ$ are obtained. The fact that a Zygmund algebra $Λ_φ$ is a D-algebra is equivalent to a special extension theorem of independent interest which leads to a solution of the spectral approximation problem for the algebras $Λ_φ$ in the cases mentioned above. Closed primary ideals and point derivations in arbitrary Zygmund algebras are described.

CONTENTSIntroduction........................................................................................51. Main definitions and basic examples..............................................72. Closed ideals in Sobolev algebras...............................................10 2.0. Notation...................................................................................10 2.1. Preliminary observations and results.......................................11 2.2. Closed primary ideals..............................................................13 2.3. Spectral synthesis of ideals.....................................................153. Spectral synthesis of ideals in the algebras $C^m Lip φ$............184. D-algebras...................................................................................215. Zygmund algebras.......................................................................26 5.1. Basic properties.......................................................................26 5.2. Extensions, approximations, and traces...................................32 5.3. Closed primary ideals...............................................................40 5.4. Point derivations......................................................................43 5.5. An extension property and spectral synthesis..........................46 5.6. Proof of Theorem 5.1...............................................................48Appendix..........................................................................................52 1. Traces of generalized Lipschitz spaces.......................................53 2. Traces of Zygmund spaces.........................................................58 3. Proof of Proposition 5.2.11..........................................................62References.......................................................................................65

1991 Mathematics Subject Classification: Primary 46E25, 46E35, 46H10, 46J10, 46J15, 46J20; Secondary 26A16, 41A10.

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EUDML-ID : urn:eudml:doc:271128