Derivations from the disc algebra into natural modules Yemon Choi University of Saskatchewan Banach Algebras and Applications 2013 Chalmers University, 4th August 2013 0 / 16
S ETTING THE SCENE D = the open unit disc in C ; T = ∂ D the unit circle. O ( D ) = the algebra of holomorphic functions D → C . The disc algebra A ( D ) = { f ∈ C ( D ) and f | D ∈ O ( D ) } ∼ = { f ∈ C ( T ) : � f ( n ) = 0 for all n < 0 } Fundamental example in Banach algebra theory. We know many things about A ( D ) , but not so much about its continuous Hochschild cohomology. This talk explores a small corner. 1 / 16
W HY CARE ABOUT A ( D ) ? Completion in a natural norm of the polynomial ring C [ z ] , which has a basic role in commutative algebra/algebraic geometry. Has connections with operator theory (e.g. von Neumann’s inequality). Serves as a prototype/motivation for more exotic Banach algebras of functions with analytic structure. Indirect motivation: the paper M. C. W HITE , Injective modules over uniform algebras . Proc. LMS 73 (1996) 155–184. 2 / 16
D ERIVATIONS Let A be an algebra and M an A -bimodule. A derivation is a linear map D : A → M satisfying D ( ab ) = a · D ( b ) + D ( a ) · b for all a , b ∈ A . (In this talk, all derivations etc are tacitly norm-continuous .) Der ( A , M ) = space of derivations A → M . In this talk, only consider symmetric bimodules ( a · m = m · a ). 3 / 16
If D ∈ Der ( A ( D ) , M ) an easy induction gives D ( Z n ) = nZ n − 1 · D ( Z ) ( n ≥ 1 ) so by linearity D ( f ) = f ′ · D ( Z ) for every f ∈ C [ z ] . 4 / 16
If D ∈ Der ( A ( D ) , M ) an easy induction gives D ( Z n ) = nZ n − 1 · D ( Z ) ( n ≥ 1 ) so by linearity D ( f ) = f ′ · D ( Z ) for every f ∈ C [ z ] . Easy result Let M be a C ( T ) -bimodule and D : A ( D ) → M a derivation. Then D = 0. Hence: Der ( A ( D ) , C ( T )) = 0; and Der ( A ( D ) , A ( D )) = 0. 4 / 16
D ERIVATIONS INTO C ( T ) /A ( D ) Notation Define P + : L 1 ( T ) → O ( D ) and P − : L 1 ( T ) → Conj O ( D ) by ( P + f )( z ) = ∑ � ( P − f )( z ) = ∑ � f ( n ) z n f ( − n ) z n , n ≥ 1 n ≥ 1 For k ∈ L 1 ( T ) let ∂ k be the formal/distributional derivative of k : ∂ k ( n ) = in � � k ( n ) ( n ∈ Z ) H p denotes Hardy space on the disc/circle (1 ≤ p ≤ ∞ ). 5 / 16
E XPLICIT DESCRIPTION OF Der ( A ( D ) , C ( T ) /A ( D )) The following theorem paraphrases results of A LEKSANDROV –P ELLER , IMRN 1996. Theorem Let k ∈ C ( T ) . Then the following are equivalent: 1 � f ′ k + A ( D ) � C ( T ) /A ( D ) � � f � ∞ for all f ∈ C [ z ] ; ∂ k ∈ ( H 1 ) ∗ , i.e. 2 �� � � � � � � � sup T ∂ k ( z ) h ( z ) | dz | � : h ∈ C [ z ] , � h � 1 ≤ 1 < ∞ � With a bit more work: Der ( A ( D ) , C ( T ) /A ( D )) ∼ = P − C ( T ) via D k ↔ ∂ P − k 6 / 16
M ORE REMARKS ON THIS RESULT Can present the proof without using real-variable techniques. Instead, use an isomorphism between P − L ∞ ( T ) and a space of multipliers on some Hilbert space of analytic functions. Most proofs of the theorem use, implicitly or explicitly, factorization theorems for derivatives of functions in H p . D AVIDSON –P AULSEN , JRAM 1997 had an operator-theoretic viewpoint, and gave a proof that gives more. 7 / 16
M Y ORIGINAL MOTIVATION Interested in the second Hochschild cohomology group H 2 ( A ( D ) , A ( D )) = Z 2 ( A ( D ) , A ( D )) B 2 ( A ( D ) , A ( D )) Long known (e.g. J OHNSON , MAMS 1972) that H 2 ( A ( D ) , A ( D )) � = 0, in contrast with what one gets for cohomology of C [ z ] . Seems that there is nothing more in the literature on H 2 ( A ( D ) , A ( D )) . . . 8 / 16
T HEOREMS IN THE REAR VIEW MIRROR APPEAR CLOSER THAN THEY ARE Theorem (C., unpublished, circa 2002) H 2 ( A ( D ) , A ( D )) is a Banach space, and there is a bounded linear map ψ : H 2 ( A ( D ) , A ( D )) → Der ( A ( D ) , C ( T ) /A ( D )) which is injective with dense range. 9 / 16
T HEOREMS IN THE REAR VIEW MIRROR APPEAR CLOSER THAN THEY ARE Theorem (C., unpublished, circa 2002) H 2 ( A ( D ) , A ( D )) is a Banach space, and there is a bounded linear map ψ : H 2 ( A ( D ) , A ( D )) → Der ( A ( D ) , C ( T ) /A ( D )) which is injective with dense range. Question Is the map ψ surjective? Answers will be gratefully received! 9 / 16
O UTLINE OF THE PROOF By general theory (W HITE , PLMS 1996) or direct averaging arguments, H n ( A ( D ) , L ∞ ( T )) = 0 for all n ≥ 1. In the case n = 2 we can actually show H 2 ( A ( D ) , C ( T )) = 0 (borrowing ideas from J OHNSON , MAMS 1972) and prove that B 2 ( A ( D ) , A ( D )) is closed in Z 2 ( A ( D ) , A ( D )) . Now use Z 2 ( A ( D ) , A ( D )) ֒ → Z 2 ( A ( D ) , C ( T )) = B 2 ( A ( D ) , C ( T )) and Der ( A ( D ) , C ( T )) = 0 to define a continuous linear map H 2 ( A ( D ) , A ( D )) → Der ( A ( D ) , C ( T ) /A ( D )) . [“Excision argument”] Finally, check this map is injective with dense range. � 10 / 16
A N A ( D ) - MODULE OF MULTIPLIERS Let Λ be the measure on D given by d Λ ( z ) : = 4 log 1 | z | dx dy ; and let for all k ∈ H 2 } . C = { h ∈ O ( D ) : � hk � L 2 ( Λ ) � � k � L 2 ( T ) Note that if f ∈ H ∞ and h ∈ C then fh ∈ C . 11 / 16
A N A ( D ) - MODULE OF MULTIPLIERS Let Λ be the measure on D given by d Λ ( z ) : = 4 log 1 | z | dx dy ; and let for all k ∈ H 2 } . C = { h ∈ O ( D ) : � hk � L 2 ( Λ ) � � k � L 2 ( T ) Note that if f ∈ H ∞ and h ∈ C then fh ∈ C . Equip C with the natural norm, and let Ω be the closure of C [ z ] inside C . This is a Banach A ( D ) -module. Theorem (Paraphrase of known result) Equip P + L ∞ ( T ) with the quotient norm induced from L ∞ ( T ) . Then f �→ f ′ is a continuous bijection from P + L ∞ ( T ) onto C . 11 / 16
Remark This result can be proved by concatenating two hard results: P + L ∞ ( T ) ∼ = BMOA ∼ = C where the second isomorphism is given by differentiation of functions. But there are proofs which just use complex analysis and Green’s function identities. Corollary � f ′ � C � � f � ∞ for all f ∈ C [ z ] . Thus f �→ f ′ gives a non-zero, bounded derivation d Ω : A ( D ) → Ω . 12 / 16
N ON - COMPACT DERIVATIONS ? H EATH , PhD thesis 2008 studied when various derivations on commutative Banach algebras are compact. To my knowledge, the only recorded example of a non-compact derivation from a uniform algebra to a symmetric module is a road-runner-ish construction of J. F EINSTEIN . 13 / 16
N ON - COMPACT DERIVATIONS ? H EATH , PhD thesis 2008 studied when various derivations on commutative Banach algebras are compact. To my knowledge, the only recorded example of a non-compact derivation from a uniform algebra to a symmetric module is a road-runner-ish construction of J. F EINSTEIN . Observation (C.) The derivation d Ω : A ( D ) → Ω is not even weakly compact. Idea of the proof. Fix f ∈ H ∞ ( D ) , then look at ( f n ) ⊂ A ( D ) given by f n ( z ) = f ( r n z ) , where r n ր 1. If d Ω were weakly compact we’d end up with f ′ ∈ Ω . Now choose f with hindsight so this can’t happen. 13 / 16
A UNIVERSAL PROPERTY Proposition ( M ORRIS , PhD thesis 1993) If D : A ( D ) → M is a derivation and f ∈ A ( D ) then � D ( f ) � ≤ 2 e � f � P + L ∞ Corollary (Universal derivation) Given D : A ( D ) → M, there is a bounded linear map Ω → M sending h to h · D ( Z ) , and a factorization of D through d Ω : A ( D ) → Ω . Remark Any unital CBA will have a “universal symmetric module for derivations”: see R UNDE , GMJ 1992. The point is that for A ( D ) we have an explicit description of a universal module. 14 / 16
E XPLICIT DESCRIPTION OF Der ( A ( D ) , A ( D ) ∗ ) Theorem ( C.–H EATH , PAMS 2011) Let h ∈ O ( D ) , h ( 0 ) = 0 , and suppose h ′ ∈ H 1 . Then � T f ′ ( z ) g ( z ) h ( z ) | dz | D h ( f )( g ) = defines a bounded derivation D h : A ( D ) → A ( D ) ∗ , with � D h � � � h ′ � 1 . Moreover, every D ∈ Der ( A ( D ) , A ( D ) ∗ ) has the form D h for a unique h as above. Consequences Every D h is compact. Every D h is 2-summing, and we can write down an explicit Pietsch control measure in terms of h . 15 / 16
R EMAINING PROBLEMS Determining H 2 ( A ( D ) , A ( D )) explicitly Does every derivation D : A ( D ) → C ( T ) /A ( D ) admit a bounded linear lifting A ( D ) → C ( T ) ? Can we use dilation techniques applied to (Foguel–)Hankel operators? Finding the universal derivation for other R ( X ) Some partial results for X a circular domain (C. + H EATH , unpublished) but nothing yet for infinitely connected domains. Function algebras on polydiscs or Euclidean balls Which algebras do we choose as “suitable” generalizations of A ( D ) ? Noncommutative versions What are the right questions for the NC disc algebra? 16 / 16
Recommend
More recommend
