Ditkin set

A closed subset $E$ of a locally compact space $X$ is called a Ditkin set (with respect to a regular function algebra $\mathcal{A} ( X )$ defined on $X$; cf. Algebra of functions) if each $f \in \mathcal{A} ( X )$ vanishing on $E$ can be approximated, arbitrarily closely, by functions $f g$ with $g \in \mathcal{A} ( X )$ and $g$ vanishing "near" $E$ (i.e. on a neighbourhood of $E$). The notion of a Ditkin set is closely related to, but more restrictive than, that of a set of spectral synthesis (cf. Spectral synthesis): for such a set the requirement is that each $f \in \mathcal{A} ( X )$ vanishing on $E$ can be approximated by functions $g \in \mathcal{A} ( X )$ vanishing near $E$.

The closed ideal of all $f \in \mathcal{A} ( X )$ vanishing on $E$ is usually denoted by $I_E$. Denoting the ideal of all $f \in \mathcal{A} ( X )$ vanishing near $E$ by $J ^ { \circ_E } $ and its closure by $J _ { E }$, one has $J _ { E } \subset I _ { E }$. Now $E$ is a set of spectral synthesis if $J _ { E } = I _ { E }$, whereas $E$ is a Ditkin set if each $f \in I _ { E }$ belongs to the closure of $f J ^ { \circ_E}$ (or, equivalently, to the closure of $f J _ { E }$). It is a famous open problem (as of 2000) whether (in specific cases) each set of spectral synthesis is actually a Ditkin set (this problem may be called the synthesis-Ditkin problem; in [a1] it is called the $C$-set-$S$-set problem).

Ditkin sets were first studied for the Fourier algebra $A ( \widehat { G } ) \cong L ^ { 1 } ( G )$, with the norm defined by $\| \hat { f } \| = \| f \| _ { 1 }$; here, $G$ is any locally compact Abelian group, $\hat { C }$ is its dual group, and $\hat { f }$ is the Fourier transform of $f$ (cf. also Harmonic analysis; Fourier transform). A.P. Calderón (1956) studied this kind of set in an effort to obtain results about sets of spectral synthesis. Therefore, Ditkin sets are sometimes called Calderón sets or $C$-sets; cf. [a4] and [a10], respectively. The name "Ditkin set" , attributed in [a6], p. 183, to C.S. Herz, refers to work of V.A. Ditkin (1910–1987) in his seminal paper [a2]; results from this paper were later studied and generalized in [a11]. In [a9] the term Wiener–Ditkin set is used.

The union of two Ditkin sets is again a Ditkin set; this follows easily from a triangle inequality like $\| f - f g h \| \leq \| f - f g \| + \| f g - f g h \|$. More generally, if a closed set is the union of countably many Ditkin sets, then it is again a Ditkin set. In contrast, for most function algebras it is unknown (as of 2000) whether the union of two sets of spectral synthesis is again of spectral synthesis: this is the famous union problem for this class of sets. Of course, the union problem becomes trivial if the synthesis-Ditkin problem gets a positive answer. Also, if $E _ { 1 }$ and $E _ { 2 }$ are sets of spectral synthesis such that $E _ { 1 } \cap E _ { 2 }$ is a Ditkin set, then $E _ { 1 } \cup E _ { 2 }$ is a set of spectral synthesis.

It is also easy to prove that if the boundary of a closed set $E$ is a Ditkin set, then so is $E$ itself; cf. [a10] (for the case $A ( \hat { G } )$), [a9] or [a8].

Ditkin sets are of particular interest if $\mathcal{A} ( X )$ satisfies Ditkin's condition, i.e. if single points are Ditkin sets for $\mathcal{A} ( X )$. This notion is older than that of a Ditkin set; cf., e.g., [a7], p. 86. If $\mathcal{A} ( X )$ has approximate units (i.e. if each $f \in \mathcal{A} ( X )$ can be approximated by functions $f g$ with $g \in \mathcal{A} ( X )$), then $\mathcal{A} ( X )$ satisfies Ditkin's condition if and only if for each $x \in X$ and each $f \in \mathcal{A} ( X )$ such that $f ( x ) = 0$ (i.e. such that $f$ belongs to the maximal ideal $I _ { x }$), the zero function can be approximated, arbitrarily closely, by functions $f \tau$ with $\tau \in \mathcal{A} ( X )$ and $\tau$ equals $1$ near $x$. It follows that if $\mathcal{A} ( X )$ satisfies Ditkin's condition, then closed scattered sets (cf. Scattered space) are Ditkin sets.

The following results can be found in [a8], Sec. 7.4. First, closed subgroups of $\hat { C }$ are Ditkin sets for the Fourier algebra $A ( \hat { G } )$, and the same result still holds for certain Beurling algebras. Secondly, the following injection theorem for Ditkin sets holds: If $\Gamma$ is a closed subgroup of $\hat { C }$, and $E$ is a closed subset of $\Gamma$, then $E$ is a Ditkin set for $A ( \hat { G } )$ if and only if it is one for $A ( \Gamma ) \cong L ^ { 1 } ( G / H )$, where $H = \Gamma ^ { \perp }$, the subgroup of $G$ orthogonal to $\Gamma$ (cf. also Orthogonality).

In the literature a more restrictive class of sets is also considered, especially in the case of the Fourier algebra $A ( \hat { G } )$. A closed set $E$ is called a strong Ditkin set if there exists a net $( g _ { \alpha } )$ (cf. also Net (directed set)), bounded in the operator norm (i.e. the mapping $\alpha \mapsto \operatorname { sup } \{ \| f g _ { \alpha } \| / \| f \| : f \in I _ { E } \}$ is bounded), such that $\operatorname { im } _ { \alpha } f g _ { \alpha } = f$. If $X$ is metrizable (cf. Metrizable space), then one can require, equivalently, the existence of a sequence $( g _ { n } ) _ { n \geq 1}$ in $J ^ { \circ_E } $ such that $\operatorname { lim } _ { n \rightarrow \infty } f g _ { n } = f$ for all $f \in I _ { E }$, the boundedness in operator norm then being automatically satisfied, by the uniform boundedness theorem (cf. Uniform boundedness)

Strong Ditkin sets were first considered by I. Wik [a12]. Subsequently it was proved that a closed subset $E$ of $\hat { C }$ without interior is a strong Ditkin set for $A ( \hat { G } )$ if and only if E belongs to the coset ring of $\hat { C }$ (cf., e.g., [a4], [a3], [a8] for details).

A closed interval in the circle group $\bf T$ is a strong Ditkin set; cf. [a12]. Therefore, it is essential, for the criterion above, to consider closed sets with empty interior. Also, a line segment in $\mathbf{T} ^ { 2 }$ is not a strong Ditkin set for $A ( \mathbf{T} ^ { 2 } )$, because it has empty interior but does not belong to the coset ring. Consequently, the above-mentioned injection theorem does not hold for strong Ditkin sets.

If $E$ is not a set of spectral synthesis, then only functions $f \in J _ { E }$ have a chance of being approximable in the Ditkin sense. This motivates the following definition, given in [a8]. A closed set $E$ is called a Ditkin set in the wide sense if each $f \in J _ { E }$ can be approximated by functions $f g$ with $g \in J _ { E } ^ { \circ }$. This notion is, in a way, more natural than that of a Ditkin set; but in 1956 it was not yet known that sets not of spectral synthesis abound in the case of the Fourier algebra: Malliavin's result (cf. Spectral synthesis) dates from 1959. It is not known in general (for instance in the case of the Fourier algebra) whether all closed subsets are Ditkin sets in the wide sense. This problem is a natural generalization of the synthesis-Ditkin problem.

References

[a1]	J.J. Benedetto, "Spectral synthesis" , Teubner (1975)
[a2]	V.A. Ditkin, "On the structure of ideals in certain normed rings" Uchen. Zap. Mosk. Gos. Univ. Mat. , 30 (1939) pp. 83–120
[a3]	C.C. Graham, O.C. McGehee, "Essays in commutative harmonic analysis" , Springer (1979)
[a4]	E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 2 , Springer (1970)
[a5]	C.S. Herz, "The sprectral theory of bounded functions" Trans. Amer. Math. Soc. , 94 (1960) pp. 181–232
[a6]	J.-P. Kahane, R. Salem, "Ensembles parfaits et séries trigonométriques" , Hermann (1963)
[a7]	L.H. Loomis, "An introduction to abstract harmonic analysis" , Van Nostrand (1953)
[a8]	H. Reiter, J.D. Stegeman, "Classical harmonic analysis and locally compact groups" , Oxford Univ. Press (2000)
[a9]	H. Reiter, "Classical harmonic analysis and locally compact groups" , Oxford Univ. Press (1968)
[a10]	W. Rudin, "Fourier analysis on groups" , Interscience (1962)
[a11]	G.E. Shilov, "On regular normed rings" Trav. Inst. Math. Steklov , 21 (1947) (In Russian) (English summary)
[a12]	I. Wik, "A strong form of spectral synthesis" Ark. Mat. , 6 (1965) pp. 55–64