Spectral Synthesis and Ideal Theory Lecture 2 Eberhard Kaniuth - PowerPoint PPT Presentation

Spectral Synthesis and Ideal Theory Lecture 2 Eberhard Kaniuth University of Paderborn, Germany Fields Institute, Toronto, March 28, 2014 Fields Institute, Toronto, March 28, 2014 Eberhard Kaniuth (University of Paderborn, Germany) Spectral Synthesis and Ideal Theory / 21

Synthesis Notions A a regular and semisimple commutative Banach algebra. For a closed subset E of ∆( A ), let j ( E ) = { a ∈ A : � a has compact support disjoint from E } . Then, if I is any ideal in A with h ( I ) = E , j ( E ) ⊆ I ⊆ k ( E ) . Definition E is called a set of synthesis or spectral set if j ( E ) = k ( E ) (equivalently, I = k ( E ) for any closed ideal I with h ( I ) = E ). We say that spectral synthesis holds for A if every closed subset of ∆( A ) is a set of synthesis. Fields Institute, Toronto, March 28, 2014 Eberhard Kaniuth (University of Paderborn, Germany) Spectral Synthesis and Ideal Theory / 21

Definition E ⊆ ∆( A ) closed is called Ditkin set if a ∈ aj ( E ) for every a ∈ k ( E ). Thus • Every Ditkin set is a set of synthesis • ∅ is a Ditkin set if and only if given a ∈ A and ǫ > 0, there exists b ∈ A such that � b has compact support and � a − ab � ≤ ǫ (in this case we also say that A satisfies Ditkin’s condition at infinity ) A is called Tauberian if the set of all a ∈ A such that � a has compact support, is dense in A . Thus • A is Tauberian if and only if ∅ is a set of synthesis. Fields Institute, Toronto, March 28, 2014 Eberhard Kaniuth (University of Paderborn, Germany) Spectral Synthesis and Ideal Theory / 21

When does Spectral Synthesis hold for A ? Spectral synthesis holds for C 0 ( X ), X a locally compact Hausdorff space Spectral synthesis does not hold for C n [ a , b ], n ≥ 1: singletons { t } , t ∈ [ a , b ], are not sets of synthesis Remark Suppose that spectral synthesis holds for A . Then a ∈ aA for each a ∈ A . Proof: Let E = { ϕ ∈ ∆( A ) : ϕ ( a ) = 0 } . Then E is closed in ∆( A ) and E = h ( aA ). Thus a ∈ k ( E ) = aA since E is of synthesis. The condition that a ∈ aA for every a ∈ A is satisfied, if A has an approximate identity. Fields Institute, Toronto, March 28, 2014 Eberhard Kaniuth (University of Paderborn, Germany) Spectral Synthesis and Ideal Theory / 21

Lemma Let A be a regular and semisimple commutative Banach algebra and E an open and closed subset of ∆( A ) . 1 If A is Tauberian and a ∈ aA for every a ∈ k ( E ) , then E is a set of synthesis. 2 If A satisfies Ditkin’s condition at infinity, then E is a Ditkin set. Proof of (2) Have to show that a ∈ aj ( E ) for each a ∈ k ( E ): • E open and closed = ⇒ h ( j ( E ) + j (∆( A ) \ E )) = E ∩ (∆( A ) \ E ) = ∅ and hence j ( ∅ ) ⊆ j ( E ) + j (∆( A ) \ E ) • ∅ Ditkin ⇒ for every a ∈ A , there exist sequences ( u n ) n ⊆ j ( E ) and ( v n ) n ⊆ j (∆( A ) \ E ) such that a ( u n + v n ) → a • let a ∈ k ( E ): then � av n = � a � v n vanishes on E and on ∆( A ) \ E , hence av n = 0. So a = lim n →∞ au n ∈ aj ( E ), as required. Fields Institute, Toronto, March 28, 2014 Eberhard Kaniuth (University of Paderborn, Germany) Spectral Synthesis and Ideal Theory / 21

From the first assertion of the lemma and the above remark it follows Corollary Suppose that ∆( A ) is discrete and A is Tauberian. Then spectral synthesis holds for A if and only if a ∈ aA for each a ∈ A. Corollary Let G be a compact abelian group. Then spectral synthesis holds for L 1 ( G ) . Proof. • L 1 ( G ) has an approximate identity • L 1 ( G ) is Tauberian • � G = ∆( L 1 ( G )) is discrete since G is compact. Fields Institute, Toronto, March 28, 2014 Eberhard Kaniuth (University of Paderborn, Germany) Spectral Synthesis and Ideal Theory / 21

The Example of L. Schwartz Theorem For n ≥ 3 , the sphere S n − 1 = { y ∈ R n : � y � = 1 } ⊆ ∆( L 1 ( R n )) fails to be a set of synthesis for L 1 ( R n ) . Remark (1) L. Schwartz [Sur une propri´ et´ e de synth` ese spectrale dans les groupes noncompacts, C.R. Acad. Sci. Paris 227 (1948), 424-426] proved this result for n = 3, but the proof works for all n ≥ 3. (2) S 1 ⊆ R 2 is a set of synthesis for L 1 ( R 2 ) [C. Herz, Spectral synthesis for the circle , Ann. Math. 68 (1958), 709-712] Fields Institute, Toronto, March 28, 2014 Eberhard Kaniuth (University of Paderborn, Germany) Spectral Synthesis and Ideal Theory / 21

Proof of Schwartz’ Theorem R n with R n through y → γ y , where γ y ( x ) = � x , y � for x ∈ R n . Identify � � • � 1 R n f ( x ) e − i � x , y � dx , f ∈ L 1 ( R n ) f ( y ) = (2 π ) n / 2 � g ∈ L 1 ( � 1 R n g ( y ) e i � x , y � dy , • ˇ R n ) g ( x ) = (2 π ) n / 2 f ) ∧ = f in L 2 ( R n ), hence • f ∈ L 1 ( � R n ) ∩ L 2 ( � R n ) and ˇ f ∈ L 1 ( R n ), then (ˇ f ) ∧ ( x ) = f ( x ) for all x ∈ R n if f is continuous (ˇ Lemma Let D ( R 3 ) denote the set of all functions in L 1 ( R 3 ) ∩ C 0 ( R 3 ) with the property that all partial derivatives exist and are in L 1 ( R 3 ) ∩ C 0 ( R 3 ) . Then � f ) ∧ = f for every f ∈ D ( R 3 ) . f ∈ L 1 ( R 3 ) and (ˇ Fields Institute, Toronto, March 28, 2014 Eberhard Kaniuth (University of Paderborn, Germany) Spectral Synthesis and Ideal Theory / 21

Lemma Let S = S 2 and I = k ( S ) ⊆ L 1 ( R n ) , and � � f ∈ D ( R n ) and ∂ � f f ∈ I : � J = = 0 on S . ∂ y 1 Then J is an ideal in L 1 ( R 3 ) and h ( J ) = S. To show that J � = I , it suffices to construct a bounded linear functional F on L 1 ( R 3 ) such that F ( J ) = { 0 } , but F ( I ) � = { 0 } . Such an F can be constructed as follows: There exists a unique probability measure µ on S , which is invariant under orthogonal transformations. Define a function φ on R 3 by � e − i � x , y � d µ ( y ) . φ ( x ) = S Fields Institute, Toronto, March 28, 2014 Eberhard Kaniuth (University of Paderborn, Germany) Spectral Synthesis and Ideal Theory / 21

Then the function x → x 1 φ ( x ) on R 3 is continuous and bounded. More precisely, it can be shown that | x 1 φ ( x ) | ≤ � x � · | φ ( x ) | ≤ 4 π x ∈ R 3 . 3 , The required functional F can now be defined by � f ∈ L 1 ( R 3 ) . F ( f ) = R 3 f ( x ) x 1 φ ( x ) dx , Since � ∂ � f ( y ) = ( − ix 1 f ( x )) ∧ ( y ) = R 3 ( − ix 1 ) f ( x ) e − i � x , y � dx , ∂ y 1 we have �� � � � ∂ � f R 3 x 1 f ( x ) e − i � x , y � dx i ( y ) d µ ( y ) = d µ ( y ) ∂ y 1 S S �� � � � e − i � x , y � d µ ( y ) = R 3 x 1 f ( x ) dx = R 3 f ( x ) x 1 φ ( x ) dx = F ( f ) . S Thus F ( f ) = 0 for every f ∈ J . Fields Institute, Toronto, March 28, 2014 Eberhard Kaniuth (University of Paderborn, Germany) Spectral Synthesis and Ideal Theory / 21

To show that F ( I ) � = { 0 } , consider the function √ 2) 3 e −� x � 2 − e 1 / 4 e −� x � 2 / 2 , x ∈ R 3 . f ( x ) = ( Then f ∈ L 1 ( R 3 ), and f ( y ) = e −� y � 2 / 4 − e 1 / 4 e −� y � 2 / 2 . � Hence � f ( y ) = 0 if � y � = 1, i.e. f ∈ I . We claim that F ( L a f ) � = 0 for some a ∈ R 3 (note that L a f ∈ I since I is a closed ideal). For arbitrary f , we have � � ⇒ ∂ � f ( y ) + ∂ � L a f f L a f ( y ) = e i � a , y � � � i a 1 � ( y ) = e i � a , y � f ( y ) = ( y ) . ∂ y 1 ∂ y 1 If f ∈ I , then � f ( y ) = 0 for y ∈ S , and hence � � ∂ � e i � a , y � ∂ � L a f f F ( L a f ) = i ( y ) d µ ( y ) = i ( y ) d µ ( y ) . ∂ y 1 ∂ y 1 S S Fields Institute, Toronto, March 28, 2014 Eberhard Kaniuth (University of Paderborn, Germany) Spectral Synthesis and Ideal Theory / 21

Now, for our special function f , ∂ � f ( y ) = − 1 2 y 1 e −� y � 2 / 4 + y 1 e 1 / 4 e −� y � 2 / 2 ∂ y 1 and hence, for y ∈ S , ∂ � f ( y ) = 1 2 y 1 e − 1 / 4 y 1 . ∂ y 1 Finally, take a = ( π, 0 , 0); then with c = 1 2 e − 1 / 4 , � e i π y 1 y 1 d µ ( y ) F ( L a f ) = i c S � � = i c y 1 cos( π y 1 ) µ ( y ) − c y 1 sin( π y 1 ) µ ( y ) . S S The first integral is zero since ( y 1 , y 2 , y 3 ) → ( − y 1 , y 2 , y 3 ) is an orthogonal transformation. So � F ( L a f ) = c y 1 sin( π y 1 ) µ ( y ) . S Since y 1 sin( π y 1 ) > 0 whenever y 1 � = 0 , 1 , − 1, it follows that F ( L a f ) � = 0. Fields Institute, Toronto, March 28, 2014 Eberhard Kaniuth (University of Paderborn, Germany) Spectral Synthesis and Ideal Theory / 21

Theorem 2 ⌋ , let I k denote the Let I = k ( S n − 1 ) ⊆ L 1 ( R n ) , and for 1 ≤ k ≤ ⌊ n +1 closed ideal of L 1 ( R n ) generated by all convolution products f 1 ∗ f 2 ∗ . . . ∗ f k , f j ∈ I. Then I = I 1 ⊇ I 2 ⊇ . . . ⊇ I ⌊ n +1 2 ⌋ = j ( S n − 1 ) . • All the inclusions are proper • The ideals I k are the only rotation invariant closed ideals of L 1 ( R n ) with hull equal to S n − 1 . N.Th. Varopoulos, Spectral synthesis on spheres , Math. Proc. Camb. Phil. Soc. 62 (1966), 379-387. Fields Institute, Toronto, March 28, 2014 Eberhard Kaniuth (University of Paderborn, Germany) Spectral Synthesis and Ideal Theory / 21