Calkin images of Fourier convolution operators with slowly oscillating symbols
Cláudio A. Fernandes, Alexei Yu. Karlovich, Yuri I. Karlovich
aa r X i v : . [ m a t h . F A ] A ug Calkin images of Fourier convolutionoperators with slowly oscillating symbols
C. A. Fernandes, A. Yu. Karlovich and Yu. I. Karlovich
Abstract.
Let Φ be a C ∗ -subalgebra of L ∞ ( R ) and SO ⋄ X ( R ) be theBanach algebra of slowly oscillating Fourier multipliers on a Banachfunction space X ( R ). We show that the intersection of the Calkin imageof the algebra generated by the operators of multiplication aI by func-tions a ∈ Φ and the Calkin image of the algebra generated by the Fourierconvolution operators W ( b ) with symbols in SO ⋄ X ( R ) coincides with theCalkin image of the algebra generated by the operators of multiplicationby constants. Mathematics Subject Classification (2010).
Primary 47G10, Secondary42A45, 46E30.
Keywords.
Fourier convolution operator, Fourier multiplier, multiplica-tion operator, slowly oscillating function, Calkin algebra, Calkin image.
1. Introduction
Let F : L ( R ) → L ( R ) denote the Fourier transform( F f )( x ) := b f ( x ) := Z R f ( t ) e itx dt, x ∈ R , and let F − : L ( R ) → L ( R ) be the inverse of F ,( F − g )( t ) = 12 π Z R g ( x ) e − itx dx, t ∈ R . It is well known that the Fourier convolution operator W ( a ) := F − a F (1.1)is bounded on the space L ( R ) for every a ∈ L ∞ ( R ). This work was partially supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia (Portu-guese Foundation for Science and Technology) through the project UID/MAT/00297/2019(Centro de Matem´atica e Aplica¸c˜oes). The third author was also supported by the SEP-CONACYT Project A1-S-8793 (M´exico).
C. A. Fernandes, A. Yu. Karlovich and Yu. I. KarlovichLet X ( R ) be a Banach function space and X ′ ( R ) be its associate space.Their technical definitions are postponed to Section 2.1. The class of Banachfunction spaces is very large. It includes Lebesgue, Orlicz, Lorentz spaces,variable Lebesgue spaces and their weighted analogues (see, e.g., [4, 6]). Let B ( X ( R )) denote the Banach algebra of all bounded linear operators actingon X ( R ), let K ( X ( R )) be the closed two-sided ideal of all compact operatorsin B ( X ( R )), and let B π ( X ( R )) = B ( X ( R )) / K ( X ( R )) be the Calkin algebraof the cosets A π := A + K ( X ( R )), where A ∈ B ( X ( R )).If X ( R ) is separable, then L ( R ) ∩ X ( R ) is dense in X ( R ) (see Lemma 2.1below). A function a ∈ L ∞ ( R ) is called a Fourier multiplier on X ( R ) if theconvolution operator W ( a ) defined by (1.1) maps L ( R ) ∩ X ( R ) into X ( R )and extends to a bounded linear operator on X ( R ). The function a is calledthe symbol of the Fourier convolution operator W ( a ). The set M X ( R ) ofall Fourier multipliers on X ( R ) is a unital normed algebra under pointwiseoperations and the norm k a k M X ( R ) := (cid:13)(cid:13) W ( a ) (cid:13)(cid:13) B ( X ( R )) . For a unital C ∗ -subalgebra Φ of the algebra L ∞ ( R ), we consider thequotient algebra MO π (Φ) consisting of the cosets[ aI ] π := aI + K ( X ( R ))of multiplication operators by functions in Φ: MO π (Φ) := { [ aI ] π : a ∈ Φ } = { aI + K ( X ( R )) : a ∈ Φ } . For a unital Banach subalgebra Ψ of the algebra M X ( R ) , we also considerthe quotient algebra CO π (Ψ) consisting of the cosets[ W ( b )] π := W ( b ) + K ( X ( R ))of convolution operators with symbols in the algebra Ψ: CO π (Ψ) := { [ W ( b )] π : b ∈ Ψ } = { W ( b ) + K ( X ( R )) : b ∈ Ψ } . It is easy to see that MO π (Φ) and CO π (Ψ) are commutative unitalBanach subalgebras of the Calkin algebra B π ( X ( R )). It is natural to refer tothe algebras MO π (Φ) and CO π (Ψ) as the Calkin images of the algebras MO (Φ) = { aI : a ∈ Φ } ⊂ B ( X ( R )) , CO (Ψ) = { W ( b ) : b ∈ Ψ } ⊂ B ( X ( R )) , respectively. The algebras MO (Φ) and CO (Ψ) are building blocks of thealgebra of convolution type operators A (Φ , Ψ; X ( R )) = alg B ( X ( R )) (cid:8) aI, W ( b ) : a ∈ Φ , b ∈ Ψ (cid:9) , the smallest closed subalgebra of B ( X ( R )) that contains the algebras MO (Φ)and CO (Ψ).Let SO ⋄ be the C ∗ -algebra of slowly oscillating functions and SO ⋄ X ( R ) be the Banach algebra of all slowly oscillating Fourier multipliers on the space X ( R ), which are defined below in Sections 2.5–2.7. The third author provedalkin images of Fourier convolution operators 3in [23, Lemma 4.3] in the case of Lebesgue spaces L p ( R , w ), 1 < p < ∞ , withMuckenhoupt weights w ∈ A p ( R ) that MO π ( SO ⋄ ) ∩ CO π ( SO ⋄ L p ( R ,w ) ) = MO π ( C ) , (1.2)where MO π ( C ) := { [ cI ] π : c ∈ C } . (1.3)This result allowed him to describe the maximal ideal space of the commu-tative Banach algebra A π ( SO ⋄ , SO ⋄ L p ( R ,w ) ; L p ( R , w )) = A ( SO ⋄ , SO ⋄ L p ( R ,w ) ; L p ( R , w )) / K ( L p ( R , w ))(see [23, Theorem 3.1]). In turn, this description plays a crucial role in thestudy of the Fredholmness of operators in more general algebras of con-volution type operators with piecewise slowly oscillating data on weightedLebesgue space L p ( R , w ) (see [23, 24, 25]).Recall that the (non-centered) Hardy-Littlewood maximal function M f of a function f ∈ L ( R ) is defined by( M f )( x ) := sup I ∋ x | I | Z I | f ( y ) | dy, where the supremum is taken over all intervals I ⊂ R of finite length con-taining x . The Hardy-Littlewood maximal operator M defined by the rule f
7→ M f is a sublinear operator.The aim of this paper is to extend (1.2) to the case of separable Ba-nach function spaces such that the Hardy-Littlewood maximal operator M is bounded on X ( R ) and on its associate space X ′ ( R ) and to the case ofarbitrary algebras of functions Φ ⊂ L ∞ ( R ) in place of SO ⋄ .The following statement extends [23, Lemma 4.3]. Theorem 1.1 (Main result).
Let X ( R ) be a separable Banach function spacesuch that the Hardy-Littlewood maximal operator M is bounded on the space X ( R ) and on its associate space X ′ ( R ) . If Φ is a unital C ∗ -subalgebra of L ∞ ( R ) , then MO π (Φ) ∩ CO π ( SO ⋄ X ( R ) ) = MO π ( C ) , (1.4) where MO π ( C ) is defined by (1.3) . This result is one more step towards the study of Fredholm propertiesof convolution type operators with discontinuous data on Banach functionspaces more general than weighted Lebesgue spaces initiated in the authorsworks [8, 9, 10].One can expect, by analogy with the case of weighted Lebesgue spaces,that, for instance, K ( X ( R )) ⊂ A ( SO ⋄ , SO ⋄ X ( R ) ; X ( R )) and that the quotientalgebra A π ( SO ⋄ , SO ⋄ X ( R ) ; X ( R )) = A ( SO ⋄ , SO ⋄ X ( R ) ; X ( R )) / K ( X ( R ))is commutative. It seems, however, that the proofs of both hypotheses will re-quire tools, which are not available in the setting of general Banach functionspaces. We plan to return to these questions in a forthcoming work, restricting C. A. Fernandes, A. Yu. Karlovich and Yu. I. Karlovichourselves to particular Banach function spaces, like rearrangement-invariantspaces with Muckenhoupt weights or variable Lebesgue spaces, where inter-polation theorems are available.The paper is organized as follows. In Section 2, we collect necessaryfacts on Banach function spaces and Fourier multipliers on them. Further,we recall the definition of the C ∗ -algebra SO ⋄ of slowly oscillating functionsand introduce the Banach algebra of slowly oscillating Fourier multipliers SO ⋄ X ( R ) on a Banach function spaces X ( R ). In Section 3, we discuss thestructure of the maximal ideal spaces M ( SO ⋄ ) and M ( SO ⋄ X ( R ) ) of the C ∗ -algebra SO ⋄ of slowly oscillating functions and the Banach algebra SO ⋄ X ( R ) of slowly oscillating Fourier multipliers on a Banach function space X ( R ).In particular, we show that the fibers M t ( SO ⋄ ) of M ( SO ⋄ ) over the points t ∈ ˙ R := R ∪ {∞} can be identified with the fibers M t ( SO t ), where SO t isthe C ∗ -algebra of all bounded continuous functions on ˙ R \ { t } that slowlyoscillate at the point t . An analogous result is also obtained for the fibers ofthe maximal ideal spaces of algebras of slowly oscillating Fourier multiplierson a Banach function space X ( R ). In Section 4, we show that the maximalideal spaces of the algebras MO π (Φ) and CO π (Ψ) are homeomorphic to themaximal ideal spaces of the algebras Φ and Ψ, respectively, where Φ is a unital C ∗ -subalgebra of L ∞ ( R ) and Ψ is a unital Banach subalgebra of M X ( R ) . InSection 5, we recall the definition of a limit operator (see [26] for a generaltheory of limit operators), as well as, a known fact about limit operators ofcompact operators acting on Banach function spaces. Further, we calculatethe limit operators of the Fourier convolution operator W ( b ) with a slowlyoscillating symbol b ∈ SO ⋄ X ( R ) . Finally, gathering the above mentioned resultson limit operators, we prove Theorem 1.1.
2. Preliminaries
The set of all Lebesgue measurable complex-valued functions on R is denotedby M ( R ). Let M + ( R ) be the subset of functions in M ( R ) whose values liein [0 , ∞ ]. The Lebesgue measure of a measurable set E ⊂ R is denoted by | E | and its characteristic function is denoted by χ E . Following [4, Chap. 1,Definition 1.1], a mapping ρ : M + ( R ) → [0 , ∞ ] is called a Banach functionnorm if, for all functions f, g, f n ( n ∈ N ) in M + ( R ), for all constants a ≥ E of R , the following properties hold:(A1) ρ ( f ) = 0 ⇔ f = 0 a.e. , ρ ( af ) = aρ ( f ) , ρ ( f + g ) ≤ ρ ( f ) + ρ ( g ) , (A2) 0 ≤ g ≤ f a.e. ⇒ ρ ( g ) ≤ ρ ( f ) (the lattice property) , (A3) 0 ≤ f n ↑ f a.e. ⇒ ρ ( f n ) ↑ ρ ( f ) (the Fatou property) , (A4) | E | < ∞ ⇒ ρ ( χ E ) < ∞ , (A5) | E | < ∞ ⇒ Z E f ( x ) dx ≤ C E ρ ( f )alkin images of Fourier convolution operators 5with C E ∈ (0 , ∞ ) which may depend on E and ρ but is independent of f .When functions differing only on a set of measure zero are identified, the set X ( R ) of all functions f ∈ M ( R ) for which ρ ( | f | ) < ∞ is called a Banachfunction space. For each f ∈ X ( R ), the norm of f is defined by k f k X ( R ) := ρ ( | f | ) . Under the natural linear space operations and under this norm, the set X ( R )becomes a Banach space (see [4, Chap. 1, Theorems 1.4 and 1.6]). If ρ is aBanach function norm, its associate norm ρ ′ is defined on M + ( R ) by ρ ′ ( g ) := sup (cid:26)Z R f ( x ) g ( x ) dx : f ∈ M + ( R ) , ρ ( f ) ≤ (cid:27) , g ∈ M + ( R ) . It is a Banach function norm itself [4, Chap. 1, Theorem 2.2]. The Banachfunction space X ′ ( R ) determined by the Banach function norm ρ ′ is called theassociate space (K¨othe dual) of X ( R ). The associate space X ′ ( R ) is naturallyidentified with a subspace of the (Banach) dual space [ X ( R )] ∗ . As usual, let C ∞ ( R ) denote the set of all infinitely differentiable functionswith compact support. Lemma 2.1 ( [8, Lemma 2.1] and [22, Lemma 2.12(a)] ). If X ( R ) is a separableBanach function space, then the sets C ∞ ( R ) and L ( R ) ∩ X ( R ) are dense inthe space X ( R ) . Let S ( R ) be the Schwartz space of rapidly decreasing smooth functionsand let S ( R ) denote the set of functions f ∈ S ( R ) such that their Fouriertransforms F f have compact support. Theorem 2.2 ( [10, Theorem 4] ). Let X ( R ) be a separable Banach functionspace such that the Hardy-Littlewood maximal operator M is bounded on X ( R ) . Then the set S ( R ) is dense in the space X ( R ) . M X ( R ) of Fourier multipliers The following result plays an important role in this paper.
Theorem 2.3 ( [21, Corollary 4.2] and [8, Theorem 2.4] ). Let X ( R ) be a sepa-rable Banach function space such that the Hardy-Littlewood maximal operator M is bounded on X ( R ) and on its associate space X ′ ( R ) . If a ∈ M X ( R ) , then k a k L ∞ ( R ) ≤ k a k M X ( R ) . (2.1) The constant on the right-hand side of (2.1) is best possible. Inequality (2.1) was established earlier in [18, Theorem 1] with someconstant on the right-hand side that depends on the space X ( R ).Since (2.1) is available, an easy adaptation of the proof of [12, Proposi-tion 2.5.13] leads to the following (we refer to the proof of [18, Corollary 1]for details). C. A. Fernandes, A. Yu. Karlovich and Yu. I. Karlovich Corollary 2.4.
Let X ( R ) be a separable Banach function space such that theHardy-Littlewood maximal operator M is bounded on X ( R ) and on its asso-ciate space X ′ ( R ) . Then the set of Fourier multipliers M X ( R ) is a Banachalgebra under pointwise operations and the norm k · k M X ( R ) . Let V ( R ) be the Banach algebra of all functions a : R → C with finite totalvariation V ( a ) := sup n X i =1 | a ( t i ) − a ( t i − ) | , where the supremum is taken over all finite partitions −∞ < t < t < · · · < t n < + ∞ of the real line R and the norm in V ( R ) is given by k a k V = k a k L ∞ ( R ) + V ( a ) . Theorem 2.5.
Let X ( R ) be a separable Banach function space such that theHardy-Littlewood maximal operator M is bounded on X ( R ) and on its as-sociate space X ′ ( R ) . If a ∈ V ( R ) , then the convolution operator W ( a ) isbounded on the space X ( R ) and k W ( a ) k B ( X ( R )) ≤ c X k a k V (2.2) where c X is a positive constant depending only on X ( R ) . This result follows from [17, Theorem 4.3].For Lebesgue spaces L p ( R ), 1 < p < ∞ , inequality (2.2) is usually calledStechkin’s inequality, and the constant c L p is calculated explicitly: c L p = k S k B ( L p ( R )) = tan (cid:16) π p (cid:17) if 1 < p ≤ , cot (cid:16) π p (cid:17) if 2 ≤ p < ∞ , (2.3)where S is the Cauchy singular integral operator given by( Sf )( x ) := 1 πi lim ε → Z R \ ( x − ε,x + ε ) f ( t ) t − x dt. We refer to [7, Theorem 2.11] for the proof of (2.2) in the case of Lebesguespaces L p ( R ) with c L p = k S k B ( L p ( R )) and to [13, Chap. 13, Theorem 1.3]for the calculation of the norm of S given in the second equality in (2.3).For Lebesgue spaces with Muckenhoupt weights L p ( R , w ), the proof of The-orem 2.5 with c L p ( w ) = k S k B ( L p ( R ,w )) is contained in [5, Theorem 17.1].Further, for variable Lebesgue spaces L p ( · ) ( R ), Theorem 2.5 with c L p ( · ) = k S k B ( L p ( · ) ( R )) was obtained in [20, Theorem 2].alkin images of Fourier convolution operators 7 Let ˙ R = R ∪ {∞} . For a set E ⊂ ˙ R and a function f : ˙ R → C in L ∞ ( R ), letthe oscillation of f over E be defined byosc( f, E ) := ess sup s,t ∈ E | f ( s ) − f ( t ) | . Following [3, Section 4] and [24, Section 2.1], [25, Section 2.1], we say thata function f ∈ L ∞ ( R ) is slowly oscillating at a point λ ∈ ˙ R if for every r ∈ (0 ,
1) or, equivalently, for some r ∈ (0 , x → osc (cid:0) f, λ + ([ − x, − rx ] ∪ [ rx, x ]) (cid:1) = 0 if λ ∈ R , lim x → + ∞ osc (cid:0) f, [ − x, − rx ] ∪ [ rx, x ] (cid:1) = 0 if λ = ∞ . For every λ ∈ ˙ R , let SO λ denote the C ∗ -subalgebra of L ∞ ( R ) defined by SO λ := n f ∈ C b ( ˙ R \ { λ } ) : f slowly oscillates at λ o , where C b ( ˙ R \ { λ } ) := C ( ˙ R \ { λ } ) ∩ L ∞ ( R ).Let SO ⋄ be the smallest C ∗ -subalgebra of L ∞ ( R ) that contains all the C ∗ -algebras SO λ with λ ∈ ˙ R . The functions in SO ⋄ are called slowly oscil-lating functions. SO λ of three times continuously differentiable slowly oscillatingfunctions For a point λ ∈ ˙ R , let C ( R \ { λ } ) be the set of all three times continuouslydifferentiable functions a : R \ { λ } → C . Following [24, Section 2.4] and [25,Section 2.3], consider the commutative Banach algebras SO λ := (cid:26) a ∈ SO λ ∩ C ( R \ { λ } ) : lim x → λ ( D kλ a )( x ) = 0 , k = 1 , , (cid:27) equipped with the norm k a k SO λ := X j =0 j ! (cid:13)(cid:13) D kλ a (cid:13)(cid:13) L ∞ ( R ) , where ( D λ a )( x ) = ( x − λ ) a ′ ( x ) for λ ∈ R and ( D λ a )( x ) = xa ′ ( x ) for λ = ∞ . Lemma 2.6.
For every λ ∈ ˙ R , the set SO λ is dense in the C ∗ -algebra SO λ .Proof. In view of [2, Lemma 2.3], the set SO ∞∞ := n f ∈ SO ∞ ∩ C ∞ b ( R ) : lim x →∞ ( D k ∞ f )( x ) = 0 , k ∈ N o (2.4)is dense in the Banach algebra SO ∞ . Here C ∞ b ( R ) denotes the set of allinfinitely differentiable functions f : R → C , which are bounded with alltheir derivatives. Note that SO ∞∞ can be equivalently defined by replacing C ∞ b in (2.4) by C ∞ , because f ∈ SO ∞ is bounded and its derivatives f ( k ) arebounded for all k ∈ N in view of lim x →∞ ( D k ∞ f )( x ) = 0. Since SO ∞∞ ⊂ SO ∞ ,this completes the proof in the case λ = ∞ . C. A. Fernandes, A. Yu. Karlovich and Yu. I. KarlovichIf λ ∈ R , then by [25, Corollary 2.2], the mapping T a = a ◦ β λ , where β λ : ˙ R → ˙ R is defined by β λ ( x ) = λx − x + λ , (2.5)is an isometric isomorphism of the algebra SO λ onto the algebra SO ∞ . Henceeach function a ∈ SO λ can be approximated in the norm of SO λ by functions c n = b n ◦ β − λ , where b n ∈ SO ∞∞ for n ∈ N and β − λ ( y ) = λy + 1 λ − y = x, x, y ∈ ˙ R . (2.6)It remains to show that c n ∈ SO λ . Taking into account (2.5)–(2.6), we obtainfor y = β λ ( x ) ∈ R \ { λ } and x = β − λ ( y ) ∈ R :( D λ c n )( y ) = b ′ n (cid:0) β − λ ( y ) (cid:1) λ + 1 y − λ = − b ′ n ( x )( x + λ ) , (2.7)( D λ c n )( y ) = b ′′ n (cid:0) β − λ ( y ) (cid:1) ( λ + 1) y − λ − b ′ n (cid:0) β − λ ( y ) (cid:1) λ + 1 y − λ = − b ′′ n ( x )( x + λ )( λ + 1) + b ′ n ( x )( x + λ ) , (2.8)( D λ c n )( y ) = b ′′′ n (cid:0) β − λ ( y ) (cid:1) ( λ + 1) y − λ − b ′′ n (cid:0) β − λ ( y ) (cid:1) ( λ + 1) y − λ + b ′ n (cid:0) β − λ ( y ) (cid:1) λ + 1 y − λ = − b ′′′ n ( x )( x + λ )( λ + 1) + 2 b ′′ n ( x )( x + λ )( λ + 1) − b ′ n ( x )( x + λ ) . (2.9)Since lim x →∞ ( D k ∞ b n )( x ) = 0 for k ∈ { , , } , we see that lim x →∞ x k b ( k ) n ( x ) = 0 for k ∈ { , , } . (2.10)It follows from (2.7)–(2.10) thatlim y → λ ( D kλ c n )( y ) = 0 for k ∈ { , , } . Hence c n ∈ SO λ for all n ∈ N , which completes the proof. (cid:3) The following result leads us to the definition of slowly oscillating Fouriermultipliers.
Theorem 2.7 ( [19, Theorem 2.5] ). Let X ( R ) be a separable Banach functionspace such that the Hardy-Littlewood maximal operator M is bounded on X ( R ) and on its associate space X ′ ( R ) . If λ ∈ ˙ R and a ∈ SO λ , then theconvolution operator W ( a ) is bounded on the space X ( R ) and k W ( a ) k B ( X ( R )) ≤ c X k a k SO λ , (2.11) where c X is a positive constant depending only on X ( R ) . alkin images of Fourier convolution operators 9Let SO λ,X ( R ) denote the closure of SO λ in the norm of M X ( R ) . Further,let SO ⋄ X ( R ) be the smallest Banach subalgebra of M X ( R ) that contains all theBanach algebras SO λ,X ( R ) for λ ∈ ˙ R . The functions in SO ⋄ X ( R ) will be calledslowly oscillating Fourier multipliers. Lemma 2.8.
Let X ( R ) be a separable Banach function space such that theHardy-Littlewood maximal operator M is bounded on X ( R ) and on its asso-ciate space X ′ ( R ) . Then SO ⋄ X ( R ) ⊂ SO ⋄ L ( R ) = SO ⋄ . Proof.
The continuous embedding SO ⋄ X ( R ) ⊂ SO ⋄ L ( R ) (with the embeddingconstant one) follows immediately from Theorem 2.3 and the definitions ofthe Banach algebras SO ⋄ X ( R ) and SO ⋄ L ( R ) . It is clear that SO ⋄ L ( R ) ⊂ SO ⋄ .The embedding SO ⋄ ⊂ SO ⋄ L ( R ) follows from Lemma 2.6. (cid:3)
3. Maximal ideal spaces of the algebras SO ⋄ and SO ⋄ X ( R ) C ∗ -algebras For a C ∗ -algebra (or, more generally, a Banach algebra) A with unit e andan element a ∈ A , let sp A ( a ) denote the spectrum of a in A . Recall that anelement a of a C ∗ -algebra A is said to be positive if it is self-adjoint andsp A ( a ) ⊂ [0 , ∞ ). A linear functional φ on A is said to be a state if φ ( a ) ≥ a ∈ A and φ ( e ) = 1. The set of all states of A isdenoted by S ( A ). The extreme points of S ( A ) are called pure states of A (see, e.g., [15, Section 4.3]).Following [1, p. 304], for a state φ , let G φ ( A ) := { a ∈ A : | φ ( a ) | = k a k A = 1 } and let G + φ ( A ) denote the set of all positive elements of G φ ( A ). Let A and B be C ∗ -algebras such that e ∈ B ⊂ A . Let φ be a state of B . Following [1,p. 310], we say that A is B -compressible modulo φ if for each x ∈ A and each ε > b ∈ G + φ ( B ) and y ∈ B such that k bxb − y k A < ε .Since a nonzero linear functional on a commutative C ∗ -algebra is a purestate if and only if it is multiplicative (see, e.g., [15, Proposition 4.4.1]), weimmediately get the following lemma from [1, Theorem 3.2]. Lemma 3.1.
Let B be a C ∗ -subalgebra of a commutative C ∗ -algebra A . Anonzero multiplicative linear functional φ on B admits a unique extension toa multiplicative linear functional φ ′ on A if and only if A is B -compressiblemodulo φ . For t ∈ ˙ R and ω >
0, let ψ t,ω be a real-valued function in C ( ˙ R ) such that0 ≤ ψ t,ω ( x ) ≤ x ∈ R . Assume that for t ∈ R , ψ t,ω ( s ) = 1 if s ∈ ( t − ω, t + ω ) , ψ t,ω ( s ) = 0 if s ∈ R \ ( t − ω, t + 2 ω ) , t = ∞ , ψ ∞ ,ω ( s ) = 1 if s ∈ R \ ( − ω, ω ) , ψ ∞ ,ω ( s ) = 0 if s ∈ ( − ω, ω ) . Let M ( A ) denote the maximal ideal space of a commutative Banachalgebra A . Lemma 3.2.
For t ∈ ˙ R and ω > , the function ψ t,ω is a positive element ofthe C ∗ -algebras C ( ˙ R ) , SO t , and SO ⋄ .Proof. Since M ( C ( ˙ R )) = ˙ R , it follows from the Gelfand theorem (see, e.g.,[28, Theorem 2.1.3]) that sp C ( ˙ R ) ( ψ t,ω ) = [0 ,
1] for all t ∈ ˙ R and all ω > C ( ˙ R ) ⊂ SO t ⊂ SO ⋄ , we conclude that the functions ψ t,ω for t ∈ ˙ R and ω > C ∗ -algebras C ( ˙ R ), SO t , and SO ⋄ because their spectra in each of these algebras coincide with [0 ,
1] in view of[15, Proposition 4.1.5]. (cid:3) C ∗ -algebra SO ⋄ If B is a Banach subalgebra of A and λ ∈ M ( B ), then the set M λ ( A ) := { ξ ∈ M ( A ) : ξ | B = λ } is called the fiber of M ( A ) over λ ∈ M ( B ). Hence for every Banach algebraΦ ⊂ L ∞ ( R ) with M ( C ( ˙ R ) ∩ Φ) = ˙ R and every t ∈ ˙ R , the fiber M t (Φ) is theset of all multiplicative linear functionals (characters) on Φ that annihilatethe set { f ∈ C ( ˙ R ) ∩ Φ : f ( t ) = 0 } . As usual, for all a ∈ Φ and all ξ ∈ M (Φ),we put a ( ξ ) := ξ ( a ). We will frequently identify the points t ∈ ˙ R with theevaluation functionals δ t defined by δ t ( f ) = f ( t ) for f ∈ C ( ˙ R ) , t ∈ ˙ R . Lemma 3.3.
For every point t ∈ ˙ R , the fibers M t ( SO t ) and M t ( SO ⋄ ) can beidentified as sets: M t ( SO t ) = M t ( SO ⋄ ) . (3.1) Proof.
Since C ( ˙ R ) ⊂ SO t ⊂ SO ⋄ , by the restriction of a multiplicative linearfunctional defined on a bigger algebra to a smaller algebra, we have M ( SO ⋄ ) ⊂ M ( SO t ) ⊂ M ( C ( ˙ R )) , t ∈ ˙ R . (3.2)Since M (Φ) = [ t ∈ ˙ R M t (Φ) for Φ ∈ { SO ⋄ , SO λ : λ ∈ ˙ R } , where M t (Φ) = { ζ ∈ M (Φ) : ζ | C ( ˙ R ) = δ t } , t ∈ ˙ R , (3.3)it follows from (3.2) and (3.3) that M t ( SO ⋄ ) ⊂ M t ( SO t ) , t ∈ R . (3.4)Now fix t ∈ ˙ R and a multiplicative linear functional η ∈ M t ( SO t ). Letus show that the C ∗ -algebra SO ⋄ is SO t -compressible modulo η . Take ε > SO ⋄ , for a function x ∈ SO ⋄ , there are a finite set F ∈ ˙ R and a finite set { x λ ∈ SO λ : λ ∈ F } such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) x − X λ ∈ F x λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( R ) < ε. If t = ∞ , take ω such that0 < ω <
12 min λ ∈ F \{ t } | λ − t | and b := ψ t,ω . Then y := b X λ ∈ F x λ ! b (3.5)is equal to zero outside the interval ( t − ω, t + 2 ω ). Therefore, y ∈ SO t .If t = ∞ , take ω such that ω > max λ ∈ F \{∞} | λ | and b := ψ ∞ ,ω . Then the function y defined by (3.5) is equal to zero on( − ω, ω ) and y ∈ SO ∞ .For t ∈ ˙ R , we have k bxb − y k L ∞ ( R ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) b x − X λ ∈ F x λ ! b (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( R ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) x − X λ ∈ F x λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( R ) < ε. Since b is a positive element of SO t in view of Lemma 3.2, we have b ∈G + η ( SO t ), which completes the proof of the fact that SO ⋄ is SO t -compressiblemodulo the multiplicative linear functional η ∈ M t ( SO t ).In view of Lemma 3.1, there exists a unique extension η ′ of the multi-plicative linear functional η to the whole algebra SO ⋄ . By the definition ofthe fiber M t ( SO ⋄ ), we have η ′ ∈ M t ( SO ⋄ ). Thus, we can identify M t ( SO t )with a subset of M t ( SO ⋄ ): M t ( SO t ) ⊂ M t ( SO ⋄ ) . (3.6)Combining (3.4) and (3.6), we arrive at (3.1). (cid:3) Corollary 3.4.
The maximal ideal space of the commutative C ∗ -algebra SO ⋄ can be identified with the set [ t ∈ ˙ R M t ( SO t ) . The following theorem in a slightly different form is contained in [29, The-orem 2.1.1] and [30, Theorem 3.10]. For the convenience of readers, we giveits proof here.2 C. A. Fernandes, A. Yu. Karlovich and Yu. I. Karlovich
Theorem 3.5.
Let A , B , C be commutative unital Banach algebras with com-mon unit and homomorphic imbeddings A ⊂ B ⊂ C , where A is dense in B . Iffor each functional ϕ ∈ M ( A ) there exists a unique extension ϕ ′ ∈ M ( C ) , thenfor every functional ψ ∈ M ( B ) there exists a unique extension ψ ′ ∈ M ( C ) .Proof. Let ψ ∈ M ( B ). Then ψ := ψ | A ∈ M ( A ). By the hypotheses, thereexists a unique extension ψ := ( ψ ) ′ ∈ M ( C ). Then ψ ( a ) = ψ ( a ) = ψ ( a )for all a ∈ A . Let ψ := ψ | B ∈ M ( B ). Since A ⊂ B , it follows that ψ ( a ) = ψ ( a ) for all a ∈ A . (3.7)On the other hand, functionals ψ, ψ ∈ M ( B ) are continuous on B (see, e.g.,[16, Lemma 2.1.5]). Since A is dense in B , for every b ∈ B there exists asequence { a n } n ∈ N ⊂ A such that k a n − b k B → n → ∞ . It follows fromthis observation and (3.7) that for every b ∈ B , ψ ( b ) = lim n →∞ ψ ( a n ) = lim n →∞ ψ ( a n ) = ψ ( b ) = ψ ( b ) . Thus ψ ∈ M ( C ) is an extension of ψ . This extension is unique by construc-tion. (cid:3) SO t,X ( R ) We start with the following refinement of [25, Lemma 3.4].
Lemma 3.6.
Let t ∈ ˙ R . Then for each functional ϕ ∈ M ( SO t ) there exists aunique extension ϕ ′ ∈ M ( SO t ) . The density of SO t in the Banach algebra SO t essentially used in theproof of [25, Lemma 3.4] is justified in Lemma 2.6. Note that the uniqueness ofan extension was not explicitly mentioned in [25, Lemma 3.4]. However, since M ( SO t ) and M ( SO t ) are Hausdorff spaces (see, e.g., [16, Theorem 2.2.3]),the uniqueness of an extension constructed in the proof of [25, Lemma 3.4]is a consequence of a standard fact from general topology (see, e.g,, [27,Theorem IV.2(b)]).The following lemma is analogous to [25, Lemma 3.5]. Lemma 3.7.
Let X ( R ) be a separable Banach function space such that theHardy-Littlewood maximal operator M is bounded on the space X ( R ) andon its associate space X ′ ( R ) . If t ∈ ˙ R , then the maximal ideal spaces of the C ∗ -algebra SO t and the Banach algebra SO t,X ( R ) can be identified as sets: M ( SO t ) = M ( SO t,X ( R ) ) . (3.8) Proof.
It follows from Theorem 2.3 that SO t ⊂ SO t,X ( R ) ⊂ SO t , where theimbeddings are homomorphic. By the definition of the algebra SO t,X ( R ) , thealgebra SO t is dense in SO t,X ( R ) with respect to the norm of M X ( R ) . Takinginto account these observations and Lemma 3.6, we see that the commutativeBanach algebras A = SO t , B = SO t,X ( R ) , C = SO t alkin images of Fourier convolution operators 13satisfy all the conditions of Theorem 3.5. By this theorem, every multiplica-tive linear functional on SO t,X ( R ) admits a unique extension to a multiplica-tive linear functional on SO t . Hence we can identify M ( SO t,X ( R ) ) with asubset of M ( SO t ): M ( SO t,X ( R ) ) ⊂ M ( SO t ) . (3.9)On the other hand, since SO t,X ( R ) ⊂ SO t , by the restriction of a multiplica-tive linear functional defined on a bigger algebra to a smaller algebra, wehave M ( SO t ) ⊂ M ( SO t,X ( R ) ) . (3.10)Combining inclusions (3.9) and (3.10), we immediately arrive at (3.8). (cid:3) The next lemma is analogous to Lemma 3.3.
Lemma 3.8.
Let X ( R ) be a separable Banach function space such that theHardy-Littlewood maximal operator M is bounded on the space X ( R ) and onits associate space X ′ ( R ) . Then, for every point t ∈ ˙ R , the fibers M t ( SO t,X ( R ) ) and M t ( SO ⋄ X ( R ) ) can be identified as sets: M t ( SO t,X ( R ) ) = M t ( SO ⋄ X ( R ) ) . (3.11) Proof.
Since SO t,X ( R ) ⊂ SO ⋄ X ( R ) for every t ∈ ˙ R , we conclude by the restric-tion of a multiplicative linear functional defined on the bigger algebra to thesmaller algebra that M ( SO ⋄ X ( R ) ) ⊂ M ( SO t,X ( R ) ). Hence M t ( SO ⋄ X ( R ) ) ⊂ M t ( SO t,X ( R ) ) . (3.12)On the other hand, in view of Lemma 3.7, any multiplicative linear functional ξ ∈ M t ( SO t,X ( R ) ) admits a unique extension ξ ′ ∈ M ( SO t ). Moreover, ξ ′ belongs to M t ( SO t ) as well. By Lemma 3.3, the functional ξ ′ ∈ M t ( SO t )admits a unique extension ξ ′′ ∈ M t ( SO ⋄ ). It is clear that the restriction of ξ ′′ to SO ⋄ X ( R ) belongs to M t ( SO ⋄ X ( R ) ). Thus M t ( SO t,X ( R ) ) can be identifiedwith a subset of M t ( SO ⋄ X ( R ) ): M t ( SO t,X ( R ) ) ⊂ M t ( SO ⋄ X ( R ) ) . (3.13)Combining (3.12) and (3.13), we arrive at (3.11). (cid:3) SO ⋄ X ( R ) Now we are in a position to prove that the maximal ideal spaces of the com-mutative Banach algebra SO ⋄ X ( R ) and the C ∗ -algebra SO ⋄ can be identifiedas sets. Theorem 3.9.
Let X ( R ) be a separable Banach function space such that theHardy-Littlewood maximal operator M is bounded on the space X ( R ) and onits associate space X ′ ( R ) . Then the maximal ideal space of the Banach algebra SO ⋄ X ( R ) can be identified with the maximal ideal space of the C ∗ -algebra SO ⋄ : M ( SO ⋄ X ( R ) ) = M ( SO ⋄ ) . Proof.
It follows from Lemmas 3.8, 3.7 and 3.3 that for every t ∈ ˙ R , M t ( SO ⋄ X ( R ) ) = M t ( SO t,X ( R ) ) = M t ( SO t ) = M t ( SO ⋄ ) . Hence M ( SO ⋄ X ( R ) ) = [ t ∈ ˙ R M t ( SO ⋄ X ( R ) ) = [ t ∈ ˙ R M t ( SO ⋄ ) = M ( SO ⋄ ) , which completes the proof. (cid:3)
4. Maximal ideal spaces of the Calkin images of the Banachalgebras MO (Φ) and CO (Ψ) MO π (Φ) We start with the following known result [14, Theorem 2.4] (see also [9,Theorem 3.1]).
Theorem 4.1.
Let X ( R ) be a separable Banach function space and a ∈ L ∞ ( R ) .Then the multiplication operator aI is compact on the space X ( R ) if and onlyif a = 0 almost everywhere on R . The next theorem says that one can identify the maximal ideal spaces ofthe algebras MO π (Φ) and Φ for an arbitrary unital C ∗ -subalgebra of L ∞ ( R ). Theorem 4.2.
Let X ( R ) be a separable Banach function space. If Φ is a unital C ∗ -subalagebra of L ∞ ( R ) , then the maximal ideal spaces of the commutativeBanach algebra MO π (Φ) and the commutative C ∗ -algebra Φ are homeomor-phic: M ( MO π (Φ)) = M (Φ) . Proof.
Consider the mapping F : Φ → MO π (Φ) defined by F ( a ) = [ aI ] π for every a ∈ Φ. It is clear that this mapping is surjective. If [ aI ] π = [ bI ] π for some a, b ∈ Φ, then ( a − b ) I ∈ K ( X ( R )). It follows from Theorem 4.1that a = b a.e. on R . This implies that the mapping F is injective. Thus, F : Φ → MO π (Φ) is an algebraic isomorphism of commutative Banachalgebras. It follows from [16, Lemma 2.2.12] that the maximal ideal spaces M ( MO π (Φ)) and M (Φ) are homeomorphic. (cid:3) CO π (Ψ) The following analogue of Theorem 4.1 for Fourier convolution operators wasobtained recently by the authors [8, Theorem 1.1].
Theorem 4.3.
Let X ( R ) be a separable Banach function space such that theHardy-Littlewood maximal operator M is bounded on X ( R ) and on its asso-ciate space X ′ ( R ) . Suppose that b ∈ M X ( R ) . Then the Fourier convolutionoperator W ( a ) is compact on the space X ( R ) if and only if b = 0 almosteverywhere on R . The next theorem is an analogue of Theorem 4.2 for Fourier multipliers.alkin images of Fourier convolution operators 15
Theorem 4.4.
Let X ( R ) be a separable Banach function space such that theHardy-Littlewood maximal operator M is bounded on X ( R ) and on its asso-ciate space X ′ ( R ) . If Ψ is a unital Banach subalagebra of M X ( R ) , then themaximal ideal spaces of the commutative Banach algebras CO π (Ψ) and Ψ arehomeomorphic: M ( CO π (Ψ)) = M (Ψ) . Proof.
The proof is analogous to the proof of Theorem 4.2. Consider themapping F : Ψ → CO π (Ψ) defined by F ( a ) = [ W ( a )] π for every a ∈ Ψ. Itis obvious that this mapping is surjective. If [ W ( a )] π = [ W ( b )] π for some a, b ∈ Ψ, then W ( a − b ) = W ( a ) − W ( b ) ∈ K ( X ( R )). By Theorem 4.3,we conclude that a = b a.e. on R . Therefore, the mapping F is injective.Thus, F : Ψ → CO π (Ψ) is an algebraic isomorphism of commutative Banachalgebras. In this case it follows from [16, Lemma 2.2.12] that the maximalideal spaces M ( CO π (Ψ)) and M (Ψ) are homeomorphic. (cid:3)
5. Applications of the method of limit operators
Let X ( R ) be a Banach function space. For a sequence of operators { A n } n ∈ N ⊂B ( X ( R )), let s-lim n →∞ A n denote the strong limit of this sequence, if it exists. For λ, x ∈ R , considerthe function e λ ( x ) := e iλx . Let T ∈ B ( X ( R )) and let h = { h n } n ∈ N be a sequence of numbers h n > h n → + ∞ as n → ∞ . The strong limit T h := s- lim n →∞ e h n T e − h n I is called the limit operator of T related to the sequence h = { h n } n ∈ N , if itexists.In our previous paper [9] we calculated the limit operators for all com-pact operators. Lemma 5.1 ( [9, Lemma 3.2] ). Let X ( R ) be a separable Banach function spaceand K be a compact operator on X ( R ) . Then for every sequence { h n } n ∈ N ofpositive numbers satisfying h n → + ∞ as n → ∞ , one has s-lim n →∞ e h n Ke − h n I = 0 on the space X ( R ) . SO ⋄ X ( R ) Now we will calculate the limit operators for the Fourier convolution operatorwith a slowly oscillating symbol.
Theorem 5.2.
Let X ( R ) be a separable Banach function space such that theHardy-Littlewood maximal operator M is bounded on the space X ( R ) and onits associate space X ′ ( R ) . If b ∈ SO ⋄ X ( R ) , then for every ξ ∈ M ∞ ( SO ⋄ ) thereexists a sequence { h n } n ∈ N of positive numbers such that h n → + ∞ as n → ∞ and s-lim n →∞ e h n W ( b ) e − h n I = b ( ξ ) I (5.1) on the space X ( R ) .Proof. This statement is proved by analogy with [25, Lemma 5.1]. In viewof Lemma 2.8, SO ⋄ X ( R ) ⊂ SO ⋄ . Therefore every ξ ∈ M ∞ ( SO ⋄ ) is a multi-plicative linear functional on SO ⋄ X ( R ) , that is, b ( ξ ) is well defined. By thedefinition of SO ⋄ X ( R ) , if b ∈ SO ⋄ X ( R ) , then there is a sequence b m = X λ ∈ F m b m,λ , m ∈ N , where F m ⊂ ˙ R are finite sets and b m,λ ∈ SO λ for λ ∈ F m and all m ∈ N ,such that lim m →∞ k b m − b k M X ( R ) = 0 . (5.2)By Lemma 3.3, M ∞ ( SO ⋄ ) = M ∞ ( SO ∞ ). Fix ξ ∈ M ∞ ( SO ⋄ ) = M ∞ ( SO ∞ ).Assume first that the set B ∞ := { b m, ∞ ∈ SO ∞ : m ∈ N } is not empty. Since the set B ∞ is at most countable, it follows from [2,Corollary 3.3] or [25, Proposition 3.1] that there exists a sequence { h n } n ∈ N such that h n → + ∞ as n → ∞ and ξ ( b m, ∞ ) = lim n →∞ b m, ∞ ( h n ) for all b m, ∞ ∈ B ∞ . (5.3)As the functions b m,λ are continuous at ∞ if λ = ∞ , we see that ξ ( b m,λ ) = b m,λ ( ∞ ) = lim n →∞ b m,λ ( h n ) for all λ ∈ [ m ∈ N F m \ {∞} . (5.4)Combining (5.3) and (5.4), for every m ∈ N , we get ξ ( b m ) = X λ ∈ F m ξ ( b m,λ ) = X λ ∈ F m lim n →∞ b m,λ ( h n )= lim n →∞ X λ ∈ F m b m,λ ( h n ) = lim n →∞ b m ( h n ) . (5.5)If the set B ∞ is empty, we can take an arbitrary sequence { h n } n ∈ N such that h n → + ∞ as n → ∞ .alkin images of Fourier convolution operators 17Let f ∈ S ( R ). Then, by a smooth version of Urysohn’s lemma (see, e.g.,[11, Proposition 6.5]), there is a function ψ ∈ C ∞ ( R ) such that 0 ≤ ψ ≤ F f ⊂ supp ψ and ψ | supp F f = 1. Therefore, for all n ∈ N , e h n W ( b ) e − h n f − b ( ξ ) f = W [ b ( · + h n )] f − ξ ( b ) f = F − [ b ( · + h n ) − ξ ( b )] ψ F f and (cid:13)(cid:13)(cid:0) e h n W ( b ) e − h n − b ( ξ ) (cid:1) f (cid:13)(cid:13) X ( R ) ≤ (cid:13)(cid:13)(cid:2) b ( · + h n ) − ξ ( b ) (cid:3) ψ k M X ( R ) k f k X ( R ) . (5.6)Since M X ( R ) is translation-invariant and ξ ∈ M ∞ ( SO ⋄ ) is a multiplicativelinear functional on SO ⋄ X ( R ) , we infer for all m, n ∈ N that (cid:13)(cid:13)(cid:2) b ( · + h n ) − ξ ( b ) (cid:3) ψ k M X ( R ) ≤ (cid:13)(cid:13)(cid:2) b ( · + h n ) − b m ( · + h n ) (cid:3) ψ k M X ( R ) + (cid:13)(cid:13)(cid:2) b m ( · + h n ) − ξ ( b m ) (cid:3) ψ k M X ( R ) + (cid:13)(cid:13) ξ ( b m ) − ξ ( b ) (cid:3) ψ k M X ( R ) ≤ k b − b m k M X ( R ) k ψ k M X ( R ) + (cid:13)(cid:13)(cid:2) b m ( · + h n ) − ξ ( b m ) (cid:3) ψ k M X ( R ) . (5.7)Fix ε >
0. By Theorem 2.5, k ψ k M X ( R ) < ∞ . It follows from (5.2) thatthere exists a sufficiently large number m ∈ N (which we fix until the end ofthe proof) such that 2 k b − b m k M X ( R ) k ψ k M X ( R ) < ε/ . (5.8)Let Λ := max λ ∈ F m \{∞} | λ | if F m \ {∞} 6 = ∅ , F m \ {∞} = ∅ , let K := supp ψ and k := max {− inf K, sup K } ∈ [0 , ∞ ) . For x ∈ K and n ∈ N , let I n ( x ) be the segment with the endpoints h n and x + h n . Then I n ( x ) ⊂ [ h n − k, h n + k ]. Since h n → + ∞ as n → ∞ , thereexists N ∈ N such that for all n > N , one has I n ( x ) ⊂ [ h n − k, h n + k ] ⊂ (Λ , ∞ ) . For all n > N , we have (cid:13)(cid:13)(cid:2) b m ( · + h n ) − ξ ( b m ) (cid:3) ψ (cid:13)(cid:13) M X ( R ) ≤ (cid:13)(cid:13)(cid:2) b m ( · + h n ) − b m ( h n ) (cid:3) ψ (cid:13)(cid:13) M X ( R ) + (cid:12)(cid:12) b m ( h n ) − ξ ( b m ) (cid:12)(cid:12) k ψ k M X ( R ) , (5.9)where the functions (cid:2) b m ( · + h n ) − b m ( h n ) (cid:3) ψ for n > N belong to SO ∞ because they are three times continuously differentiable functions of compactsupport.By (5.5), there exists N ∈ N such that N ≥ N and for all n > N , (cid:12)(cid:12) b m ( h n ) − ξ ( b m ) (cid:12)(cid:12) k ψ k M X ( R ) < ε/ . (5.10)8 C. A. Fernandes, A. Yu. Karlovich and Yu. I. KarlovichOn the other hand, since (cid:2) b m ( · + h n ) − b m ( h n ) (cid:3) ψ ∈ SO ∞ for all n > N , itfollows from Theorem 2.7 that there exists a constant c X > X ( R ) such that for all n > N , (cid:13)(cid:13)(cid:2) b m ( · + h n ) − b m ( h n ) (cid:3) ψ (cid:13)(cid:13) M X ( R ) ≤ c X (cid:13)(cid:13)(cid:2) b m ( · + h n ) − b m ( h n ) (cid:3) ψ (cid:13)(cid:13) SO ∞ = c X X j =0 j ! (cid:13)(cid:13) D j ∞ (cid:0)(cid:2) b m ( · + h n ) − b m ( h n ) (cid:3) ψ (cid:1)(cid:13)(cid:13) L ∞ ( R ) . (5.11)For all j ∈ { , , , } , we have D j ∞ (cid:0)(cid:2) b m ( · + h n ) − b m ( h n ) (cid:3) ψ (cid:1) = j X ν =0 (cid:18) jν (cid:19) (cid:0) D ν ∞ (cid:2) b m ( · + h n ) − b m ( h n ) (cid:3)(cid:1) (cid:0) D j − ν ∞ ψ (cid:1) . (5.12)It follows from the mean value theorem that (cid:13)(cid:13)(cid:2) b m ( · + h n ) − b m ( h n ) (cid:3) χ K (cid:13)(cid:13) L ∞ ( R ) = sup x ∈ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z x + h n h n b ′ m ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup x ∈ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z x + h n h n tb ′ m ( t ) dtt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup x ∈ K Z I n ( x ) (cid:12)(cid:12) ( D ∞ b m )( t ) (cid:12)(cid:12) dtt ≤ sup t ∈ [ h n − k,h n + k ] (cid:12)(cid:12) ( D ∞ b m )( t ) (cid:12)(cid:12) Z h n + kh n − k dtt ≤ ln h n + kh n − k k D ∞ b m k L ∞ ( R ) . (5.13)It is easy to see that for x ∈ K , (cid:0) D ∞ (cid:2) b m ( · + h n ) − b m ( h n ) (cid:3)(cid:1) ( x ) = xx + h n ( D ∞ b m )( x + h n ) , (5.14) (cid:0) D ∞ (cid:2) b m ( · + h n ) − b m ( h n ) (cid:3)(cid:1) ( x )= x ( x + h n ) ( D ∞ b m )( x + h n ) + xh n ( x + h n ) ( D ∞ b m )( x + h n ) , (5.15)and (cid:0) D ∞ (cid:2) b m ( · + h n ) − b m ( h n ) (cid:3)(cid:1) ( x )= x ( x + h n ) ( D ∞ b m )( x + h n ) + 3 x h n ( x + h n ) ( D ∞ b m )( x + h n )+ xh n − x h n ( x + h n ) ( D ∞ b m )( x + h n ) . (5.16)alkin images of Fourier convolution operators 19It follows from (5.14)–(5.16) that for all n > N , (cid:13)(cid:13)(cid:0) D ∞ (cid:2) b m ( · + h n ) − b m ( h n ) (cid:3)(cid:1) χ K (cid:13)(cid:13) L ∞ ( R ) ≤ kh n − k k D ∞ b m k L ∞ ( R ) , (5.17) (cid:13)(cid:13)(cid:0) D ∞ (cid:2) b m ( · + h n ) − b m ( h n ) (cid:3)(cid:1) χ K (cid:13)(cid:13) L ∞ ( R ) ≤ k ( h n − k ) (cid:13)(cid:13) D ∞ b m (cid:13)(cid:13) L ∞ ( R ) + kh n ( h n − k ) k D ∞ b m k L ∞ ( R ) , (5.18)and (cid:13)(cid:13)(cid:0) D ∞ (cid:2) b m ( · + h n ) − b m ( h n ) (cid:3)(cid:1) χ K (cid:13)(cid:13) L ∞ ( R ) ≤ k ( h n − k ) (cid:13)(cid:13) D ∞ b m (cid:13)(cid:13) L ∞ ( R ) + 3 k h n ( h n − k ) (cid:13)(cid:13) D ∞ b m (cid:13)(cid:13) L ∞ ( R ) + kh n + k h n ( h n − k ) k D ∞ b m k L ∞ ( R ) . (5.19)Since max j ∈{ , , , } (cid:13)(cid:13) D j ∞ ψ (cid:13)(cid:13) L ∞ ( R ) < ∞ , it follows from (5.12)–(5.13) and (5.17)–(5.19) that for all j ∈ { , , , } ,lim n →∞ (cid:13)(cid:13) D j ∞ (cid:0)(cid:2) b m ( · + h n ) − b m ( h n ) (cid:3) ψ (cid:1)(cid:13)(cid:13) L ∞ ( R ) = 0 . (5.20)We deduce from (5.11) and (5.20) that there exists N ∈ N such that N ≥ N and for all n > N , (cid:13)(cid:13)(cid:2) b m ( · + h n ) − b m ( h n ) (cid:3) ψ (cid:13)(cid:13) M X ( R ) < ε/ . (5.21)Combining (5.7)–(5.10) and (5.21), we see that for every f ∈ S ( R ) and every ε > N ∈ N such that for all n > N , (cid:13)(cid:13)(cid:0) e h n W ( b ) e − h n − b ( ξ ) (cid:1) f (cid:13)(cid:13) X ( R ) < ε k f k X ( R ) , whence for all f ∈ S ( R ),lim n →∞ (cid:13)(cid:13)(cid:0) e h n W ( b ) e − h n I − b ( ξ ) I (cid:1) f (cid:13)(cid:13) X ( R ) = 0 . Since S ( R ) is dense in X ( R ) (see Theorem 2.2), this equality immediatelyimplies (5.1) in view of [28, Lemma 1.4.1(ii)], which completes the proof. (cid:3) Since the function e ≡ SO ⋄ X ( R ) , we see that the set ofall constant functions is contained in Φ and in SO ⋄ X ( R ) . Therefore MO π ( C ) ⊂ MO π (Φ) ∩ CO π ( SO ⋄ X ( R ) ) . (5.22)Let A π ∈ MO π (Φ) ∩ CO π ( SO ⋄ X ( R ) ). Then A π = [ aI ] π = [ W ( b )] π ,where a ∈ Φ and b ∈ SO ⋄ X ( R ) . Therefore, there is an operator K ∈ K ( X ( R ))such that aI = W ( b ) + K. (5.23)0 C. A. Fernandes, A. Yu. Karlovich and Yu. I. KarlovichBy Theorem 5.2, for every ξ ∈ M ∞ ( SO ⋄ ) there exists a sequence { h n } n ∈ N ofpositive numbers such that h n → + ∞ as n → ∞ ands-lim n →∞ e h n W ( b ) e − h n I = b ( ξ ) I. (5.24)Equalities (5.23)–(5.24) and Lemma 5.1 imply that aI = s-lim n →∞ e h n ( aI ) e − h n I = s-lim n →∞ e h n ( W ( b ) + K ) e − h n I = b ( ξ ) I. Hence [ aI ] π = [ b ( ξ ) I ] π ∈ MO π ( C ) and MO π (Φ) ∩ CO π ( SO ⋄ X ( R ) ) ⊂ MO π ( C ) . (5.25)Combining (5.22) and (5.25), we arrive at (1.4). (cid:3) Acknowledgment
We would like to thank the anonymous referee for pointing out a gap inthe original version of the paper. To fill in this gap, we strengthened thehypotheses in the main result.
References [1] J. Anderson,
Extensions, restrictions, and representations of states on C ∗ -algebras , Trans. Amer. Math. Soc. (1979), 303–329.[2] M. A. Bastos, A. Bravo, and Yu. I. Karlovich, Convolution type operatorswith symbols generated by slowly oscillating and piecewise continuous matrixfunctions , Oper. Theory Adv. Appl. (2004), 151–174.[3] M. A. Bastos, C. A. Fernandes, and Yu. I. Karlovich, C ∗ -algebras of integraloperators with piecewise slowly oscillating coefficients and shifts acting freely ,Integr. Equ. Oper. Theory (2006), 19–67.[4] C. Bennett and R. Sharpley, Interpolation of Operators , Academic Press,Boston, 1988.[5] A. B¨ottcher, Yu. I. Karlovich, and I. M. Spitkovsky,
Convolution Operatorsand Factorization of Almost Periodic Matrix Functions , Birkh¨auser, Basel,2002.[6] D. Cruz-Uribe and A. Fiorenza,
Variable Lebesgue Spaces , Birkh¨auser/Sprin-ger, New York, 2013.[7] R. V. Duduchava,
Integral Equations with Fixed Singularities , Teubner,Leipzig, 1979.[8] C. A. Fernandes, A. Yu. Karlovich, and Yu. I. Karlovich,
Noncompactness ofFourier convolution operators on Banach function spaces , Ann. Funct. Anal.AFA (2019), 553–561.[9] C. A. Fernandes, A. Yu. Karlovich, and Yu. I. Karlovich, Algebra of convolu-tion type operators with continuous data on Banach function spaces , BanachCenter Publications (2019), 157–171.[10] C. A. Fernandes, A. Yu. Karlovich, and Yu. I. Karlovich,
Fourier convolu-tion operators with symbols equivalent to zero at infinity on Banach func-tion spaces , Proceedings of ISAAC 2019, to appear. Preprint is available atarXiv:1909.13538 [math.FA]. alkin images of Fourier convolution operators 21 [11] G. B. Folland,
A Guide to Advanced Real Analysis , The Mathematical Asso-ciation of America, Washington, DC, 2009.[12] L. Grafakos,
Classical Fourier Analysis , 3rd ed., Springer, New York, 2014.[13] I. Gohberg and N. Krupnik,
One-Dimensional Linear Singular Integral Equa-tions . Vol. II, Birkh¨auser, Basel, 1992.[14] H. Hudzik, R. Kumar, and R. Kumar,
Matrix multiplication operators onBanach function spaces , Proc. Indian Acad. Sci., Math. Sci. (2006), 71–81.[15] R. V. Kadison and J. R Ringrose,
Fundamentals of the Theory of OperatorAlgebras. Vol. I: Elementary Theory , 2nd ed. American Mathematical Society,Providence, RI, 1997.[16] E. Kaniuth,
A Course in Commutative Banach Algebras , Springer, New York,2009.[17] A. Yu. Karlovich,
Maximally modulated singular integral operators and theirapplications to pseudodifferential operators on Banach function spaces , Con-temp. Math. (2015), 165–178.[18] A. Yu. Karlovich,
Banach algebra of the Fourier multipliers on weighted Ba-nach function spaces , Concr. Oper. (2015), 27–36.[19] A. Yu. Karlovich, Commutators of convolution type operators on some Banachfunction spaces , Ann. Funct. Anal. AFA (2015), 191–205.[20] A. Yu. Karlovich, The Stechkin inequality for Fourier multipliers on variableLebesgue spaces , Math. Inequal. Appl. (2015), 1473–1481.[21] A. Karlovich and E. Shargorodsky, When does the norm of a Fourier multi-plier dominate its L ∞ norm? Proc. London Math. Soc. (2019), 901–941.[22] A. Yu. Karlovich and I. M. Spitkovsky,
The Cauchy singular integral operatoron weighted variable Lebesgue spaces , Oper. Theory Adv. Appl. (2014),275–291.[23] Yu. I. Karlovich,
Algebras of convolution-type operators with piecewise slowlyoscillating data on weighted Lebesgue spaces , Mediterr. J. Math. (2017),paper no. 182, 20 p.[24] Yu. I. Karlovich and I. Loreto Hern´andez, Algebras of convolution type op-erators with piecewise slowly oscillating data. I: Local and structural study ,Integr. Equ. Oper. Theory (2012), 377–415.[25] Yu. I. Karlovich and I. Loreto Hern´andez, On convolution type operators withpiecewise slowly oscillating data , Oper. Theory Adv. Appl. (2013), 185–207.[26] V. Rabinovich, S. Roch, and B. Silbermann,
Limit Operators and Their Ap-plications in Operator Theory , Birkh¨auser, Basel, 2004.[27] M. Reed and B. Simon,
Methods of Modern Mathematical Physics. I: Func-tional Analysis , Academic Press, New York, 1980.[28] S. Roch, P. A. Santos, and B. Silbermann,
Non-Commutative Gelfand The-ories. A Tool-kit for Operator Theorists and Numerical Analysts , Springer,Berlin, 2011.[29] I. B. Simonenko,
Local Method in the Theory of Shift Invariant Operators andTheir Envelopes , Rostov Univ. Press, Rostov on Don, 2007 (in Russian). [30] I. B. Simonenko and Chin Ngok Min,
Local Method in the Theory of One-Dimensional Singular Integral Equations with Piecewise Continuous Coeffi-cients. Noetherity , Rostov Univ. Press, Rostov on Don, 1986 (in Russian).C. A. FernandesCentro de Matem´atica e Aplica¸c˜oes,Departamento de Matem´atica,Faculdade de Ciˆencias e Tecnologia,Universidade Nova de Lisboa,Quinta da Torre,2829–516 Caparica, Portugale-mail: [email protected]
A. Yu. KarlovichCentro de Matem´atica e Aplica¸c˜oes,Departamento de Matem´atica,Faculdade de Ciˆencias e Tecnologia,Universidade Nova de Lisboa,Quinta da Torre,2829–516 Caparica, Portugale-mail: [email protected]