aa r X i v : . [ m a t h . F A ] S e p LOOPS IN SU (2) AND FACTORIZATION, II
DOUG PICKRELL
Abstract.
In the prequel to this paper, we proved that for a SU (2 , C ) valuedloop having the critical degree of smoothness (one half of a derivative in the L Sobolev sense), the following statements are equivalent: (1) the Toeplitzand shifted Toeplitz operators associated to the loop are invertible, (2) theloop has a unique triangular factorization, and (3) the loop has a unique rootsubgroup factorization. This hinges on some Plancherel-esque formulas for de-terminants of Toeplitz operators. The main point of this report is is to outlinea generalization of this result to loops of vanishing mean oscillation, and to dis-cuss some consequences. This generalization hinges on an operator-theoreticfactorization of the Toeplitz operators (not simply their determinants). Introduction
This paper concerns the Polish topological groups of maps W / ( S , SU (2)), V M O ( S , SU (2)), and (in a supporting role) M eas ( S , SU (2)) (equivalence classesof SU (2 , C ) valued loops which have one half of a derivative in the L Sobolevsense, are of vanishing mean oscillation, and are measurable, respectively; the basicbackground - such as the Polish topologies of these groups - is recalled in Section 1).In an attempt to motivate the subject matter, we first consider a larger topologicalperspective.Suppose that K is a compact Lie group. The equatorial inclusions S ⊂ S ⊂ S ⊂ S ⊂ ... induce (down arrow) inclusions and (left to right arrow) trace homomorphisms ofgroups ... → C ∞ ( S , K ) → C ∞ ( S , K ) → C ∞ ( S , K ) → C ∞ ( S , K ) ↓ ↓ ↓ ↓ ... → W / ( S , K ) → W ( S , K ) → W / ( S , K ) ↓ ↓ ↓ ↓ ... → V M O ( S , K ) → V M O ( S , K ) → V M O ( S , K ) ↓ ↓ ↓ ↓ ... M eas ( S , K ) M eas ( S , K ) M eas ( S , K )The groups of smooth maps are Frechet Lie groups (see Section 3.2 of [13]), hencewe know what they look like locally, and their global topology can be analyzedusing conventional methods of algebraic topology.For the groups W d/ ( S d , K ) ⊂ V M O ( S d , K ) ⊂ M eas ( S d , K ), generic group ele-ments are not continuous mappings (Recall that s = d/ L exponent:the Sobolev embedding W s,L ( S d ) → C ( S d ) holds for s > d/ s = d/ ∈ K and considering theset of maps with image in this neighborhood, fails in this context because generic group elements are locally unbounded and hence this set is not an open neighbor-hood of 1 ∈ W d/ ( S d , K ) (or V M O , or
M eas ). For similar reasons conventionalmethods of algebraic topology do not apply to understand the global topology.Nonetheless it is important to understand the local and global topology of these(Polish) mapping groups; see [2], [3], [4], and references, for foundational work inthis direction and further motivation. The happiest possibility (this is pure specu-lation) is that W d/ ( S d , K ) and V M O ( S d , K ) are topological manifolds (they aredefinitely not smooth Lie groups as Polish topological groups), and the inclusions C ∞ ( S d , K ) → W d/ ( S d , K ) → V M O ( S d , K )are homotopy equivalences. A homotopy inverse to the first map can plausibly beconstructed using a (short time) negative gradient flow for a conformally invariantenergy function on W d/ ( S d , K ) - the usual energy when d = 2 - but to my knowl-edge this has not been done. A guiding idea is that VMO is the maximal categoryof maps which behave like continuous maps. This is exemplified by the existence oftrace maps for VMO (see [3]) and the nonexistence of trace maps for measurablemaps in the above diagram. More directly relevant to this paper, in the specialcase d = 1, the global topology for the smooth loop space is intimately related tothe map(0.1) C ∞ ( S , K ) → F red ( H + ) : g → A ( g )where A ( g ) is the Toeplitz operator with symbol g (see chapter 6 of [13]); the pointis that V M O ( S , K ) is the natural domain. Remark.
M eas ( S d , K ) is an outlier in this topological digression. Since its def-inition depends only upon the Lebesgue measure class of S d , it is isomorphic to M eas ([0 , , K ), and it is a contractible space.In this paper d = 1 (unless noted otherwise), and we mainly consider K = SU (2), rather than a general simply connected compact Lie group, because themain issues are more of an analytical than a Lie theoretic character (we will outlinethe extension of the theory at the end of the paper). In the prequel to this paper, weshowed that for g ∈ W / ( S , SU (2)), the following statements are equivalent: (1)the Toeplitz and shifted Toeplitz operators associated to g are invertible, (2) g has aunique triangular factorization, and (3) g has a unique root subgroup factorization(we will review this in Section 2). The key to this, and in truth the more interestingpoint, is that there exist various explicit expressions for det ( A ( g ) A ( g − )) (akin tothe Plancherel formula in linear Fourier analysis, and Szego’s theorems in the theoryof Verblunsky coefficients). A corollary of this is that W / ( S , SU (2)) is a (Polish)topological manifold (which is not smooth).The main point of this paper is investigate extensions of this theory to VMO,and some qualified extensions to the measurable (or L ) context. In the VMOcontext, we have a Fredholm Toeplitz operator A ( g ), and a section det ( A ( g )) of adeterminant bundle, but the scalar expression det ( A ( g ) A ( g − )) is identically zero inthe complement of W / ( S , SU (2)). The theory extends because, as we essentiallyobserved in [1] (we will need a slight refinement), there is actually a factorizationof A ( g ), as an operator, in root subgroup coordinates. Remark.
In the notation of Theorem 1.4 of [1], we will show that A ( k ∗ e χ k ) = A ( k ∗ e χ − ) A ( e χ + χ + k ) OOPS IN SU (2) AND FACTORIZATION, II 3 and we will see that it is relatively easy to analyze the two Toeplitz operators onthe right hand side (when the symbols are bounded). If we had observed this in[1], then we could have eliminated Lemmas 4 and 5 in Section 5.1, and this wouldhave greatly simplified the exposition. The caveat in the context of this paper isthat even though exp ( χ ) is bounded, when χ ∈ V M O ( S , i R ), it is not generallythe case that exp ( χ ± ) are bounded.This paper is largely expository. There are a number of loose ends which wehave not tied off. It also goes without saying that the approach of this paper, usingfactorization, is special to d = 1.0.1. Plan of the Paper.
In Section 1 we establish basic notation and recall somebackground results (concerning for example the operator theoretic origin of thetopologies for the various spaces of loops).In Section 2 we succinctly outline the main results from [9] for loops into SU (2) := SU (2 , C ) with critical degree of smoothness in the L Sobolev sense (the W / the-ory). These results hinge on various Plancherel-esque identities (which are relatedto the existence of a (Kac-Moody) central extension for the group of loops whichhave half a derivative). Our original intention was to consider non-generic loops,and note how this shows that C ∞ ( S , SU (2)) → W / ( S , SU (2)) is a homotopyequivalence (following the outline of [13]), but we have not completed this.In Section 3 we consider measurable maps, which we refer to as the L theory.In this context we are not pursuing topological results. The point is that thisis the edge of deterministic results. Our main interest is actually to understandprobabilistic statements which are just beyond this edge ([11]).In Section 4 we consider maps of vanishing mean oscillation. In this case wewant to argue that V M O ( S , SU (2)) is a topological manifold, but there are severalanalytic issues which I cannot resolve.1. Notation and Background If f ( z ) = P f n z n , then we will write f = f − + f + f + where f − ( z ) = P n< f n z n and f + ( z ) = P n> f n z n , ( f ) = f + f + , and f ∗ ( z ) = P f ∗− n z n , where w ∗ = ¯ w is the complex conjugate of the complex number w . If theFourier series is convergent at a point z ∈ S , then f ∗ ( z ) is the conjugate of thecomplex number f ( z ). If f ∈ H (∆), then f ∗ ∈ H (∆ ∗ ), where ∆ is the open unitdisk, ∆ ∗ is the open unit disk at ∞ , and H ( U ) denotes the space of holomorphicfunctions for a domain U ⊂ C . W / ( S , C ) denotes the Hilbert space of (equivalence classes of) functions f ( z )which have half a derivative in the L Sobolev sense, i.e. in terms of its Fourierseries, P ∞ n = −∞ | n || f n | < ∞ , and V M O ( S ) denotes the Banach space of (equiv-alence classes of Lebesgue) measurable functions which are of vanishing mean os-cillation. The precise norms for these spaces are not important for our purposes. M eas ( S , C ) denotes equivalence classes of Lebesgue measurable functions with thetopology corresponding to convergence in (Lebesgue) measure (which is separableand induced by a complete metric, see below). w / denotes the Hilbert space of complex sequences ζ such that P ∞ k =1 k | ζ k | < ∞ . DOUG PICKRELL L fin SU (2) ( L fin SL (2 , C )) denotes the group consisting of functions S → SU (2)( SL (2 , C ), respectively) having finite Fourier series, with pointwise multiplication.For example, for ζ ∈ C and n ∈ Z , the function S → SU (2) : z → a ( ζ ) (cid:18) ζz − n − ¯ ζz n (cid:19) , where a ( ζ ) = (1 + | ζ | ) − / , is in L fin SU (2).As in the introduction, consider the groups W d/ ( S d , SU (2)) ⊂ V M O ( S d , SU (2)) ⊂ M eas ( S d , SU (2))As a digression, in the topologies induced by the Banach algebras L ∞ ∩ W / , QC := L ∞ ∩ V M O , and L ∞ , respectively, these are Banach Lie groups. Howeverin all cases smooth loops are not dense, and there are uncountably many connectedcomponents (Convergence in each of these Banach algebras implies uniform con-vergence, hence the identity component in each case consists of classes which havecontinuous representatives; this is not what we are interested in). In this paperwe will always view W d/ ( S d , SU (2)), V M O ( S d , SU (2)), and M eas ( S d , SU (2))as topological groups with the complete separable (Polish) topologies induced by W d/ , V M O , and convergence in probability, respectively. For measurable maps,the bijection
M eas ( S d , U (2)) → { unitary multiplication operators on L ( S d , C ) } is a homeomorphism with respect to the convergence in probability topology andthe strong (or weak) topology for multiplication operators (see Section 2 of [7]).Now suppose that d = 1. In this setup the inclusions L fin SU (2) ⊂ C ∞ ( S , SU (2)) ⊂ W / ( S , SU (2)) ⊂ V M O ( S , SU (2)) ⊂ M eas ( S , SU (2))are dense. The first two inclusions are homotopy equivalences. One of the goals ofthis paper is to show the third is a homotopy equivalence. The fourth is a map intoa contractible space.Suppose that g ∈ L ( S , SL (2 , C )). A triangular factorization of g is a factor-ization of the form(1.1) g = l ( g ) m ( g ) a ( g ) u ( g ) , where l = (cid:18) l l l l (cid:19) ∈ H (∆ ∗ , SL (2 , C )) , l ( ∞ ) = (cid:18) l ( ∞ ) 1 (cid:19) ,l has a L radial limit, m = (cid:18) m m − (cid:19) , m ∈ S , a ( g ) = (cid:18) a a − (cid:19) , a > u = (cid:18) u u u u (cid:19) ∈ H (∆ , SL (2 , C )) , u (0) = (cid:18) u (0)0 1 (cid:19) , and u has a L radial limit. Note that (1.1) is an equality of measurable functionson S . A Birkhoff (or Wiener-Hopf, or Riemann-Hilbert) factorization is a factor-ization of the form g = g − g g + , where g − ∈ H (∆ ∗ , ∞ ; SL (2 , C ) , g ∈ SL (2 , C ), g + ∈ H (∆ , SL (2 , C ) , g ± have L radial limits on S . Clearly g has atriangular factorization if and only if g has a Birkhoff factorization and g has atriangular factorization, in the usual sense of matrices. OOPS IN SU (2) AND FACTORIZATION, II 5 As in [13], consider the polarized Hilbert space(1.2) H := L ( S , C ) = H + ⊕ H − , where H + = P + H consists of L -boundary values of functions holomorphic in ∆.If g ∈ L ∞ ( S , SL (2 , C )), we write the bounded multiplication operator defined by g on H as(1.3) M g = (cid:18) A ( g ) B ( g ) C ( g ) D ( g ) (cid:19) where A ( g ) = P + M g P + is the (block) Toeplitz operator associated to g and so on.If g has the Fourier expansion g = P g n z n , g n = (cid:18) a n b n c n d n (cid:19) , then relative to thebasis for H :(1.4) ..ǫ z, ǫ z, ǫ , ǫ , ǫ z − , ǫ z − , .. where { ǫ , ǫ } is the standard basis for C , the matrix of M g is block periodic ofthe form(1.5) . . . . . . ... a b a b | a b .... c d c d | c d .... a − b − a b | a b .... c − d − c d | c d .. − − − − − − − − − .. a − b − a − b − | a b .... c − d − c − d − | c d ... . . . . . . From this matrix form, it is clear that, up to equivalence, M g has just two types of“principal minors”, the matrix representing A ( g ), and the matrix representing theshifted Toeplitz operator A ( g ), the compression of M g to the subspace spannedby { ǫ i z j : i = 1 , , j > } ∪ { ǫ } .Given the polarization H = H + ⊕ H − and a symmetrically normed ideal I ⊂L ( H ), there is an associated Banach ∗ -algebra, L ( I ) , which consists of boundedoperators on H , represented as two by two matrices as in (1.3), such that the norm(1.6) | (cid:18) A D (cid:19) | L + | (cid:18) BC (cid:19) | I is finite. The ∗ -operation is the usual adjoint operation. The corresponding unitarygroup is U ( I ) = U ( H ) ∩ L ( I ) ;it is referred to as a restricted unitary group in [13] ( Geometrically this group isthe group of automorphisms of a Grassmannian (Finsler) symmetric space modeledon I ). There are two standard topologies on U ( I ) . The first is the induced Banachtopology, and in this topology U ( I ) has the additional structure of a Banach Liegroup. The second (more important for our purposes) is the Polish topology τ KM for which convergence means that for g n , g ∈ U ( I ) , g n → g if and only if g n → g strongly and (cid:18) B n C n (cid:19) → (cid:18) BC (cid:19) in I DOUG PICKRELL
Remark.
For the unitary group of a countably infinite dimensional Hilbert space,the group of unitary operators with either the strong or with the operator normtopology is contractible. Consequently the algebraic topology of U ( I ) is the samefor the first and second topologies. But we are always interested in the second(Polish) topology.For the subalgebra of multiplication operators M ap ( S , L ( C )) ⊂ L ( L p ) Peller has shown that the norm (1.6) is equivalent to | F | L ∞ + | F | B /p where B /p is the Besov p-norm (if p = ∞ , then the Besov space is replaced by V M O , and the result in this case is due to Hartmann). But as we pointed out above,it is not desirable to impose uniform convergence on rough loops. In the followingstatement we are using the second Polish topology of the previous paragraph.
Proposition 1. (a) W / ( S , SU (2)) → U ( L ) ( H + ⊕ H − ) is a homeomorphismonto its image.(b) V M O ( S , SU (2)) → U ( L ∞ ) ( H + ⊕ H − ) is a homeomorphism onto its image.(c) U ( L ∞ ) ( H + ⊕ H − ) → F red ( H + ) is a homotopy equivalence. This is described in chapter 6 of [13]; part (c) is essentially Proposition (6.2.4)of [13].Given a countably infinite dimensional Hilbert space such as H + , Quillen con-structed a holomorphic determinant line bundle Det → F red ( H + ) and a canonicalholomorphic section det which vanishes on the complement of invertible operators.This induces a determinant bundle A ∗ Det → V M O ( S , SU (2))(There is a discussion of this, and references, at the end of Section 7.7 of [13]). Corollary 1.
For
V M O ( S , SU (2)) the set of loops with invertible Toeplitz oper-ators is defined by the equation det ( A ( g )) = 0 , hence is open (The same applies forthe shifted Toeplitz operator).Remark. The point of this paper is to explore the meaning of the condition that A ( g ) is invertible for loops. The claim is that det ( A ( g )) = 0 is the right kind ofdomain to consider for our coordinate charts. What is the analogue (or analogues)of this for M ap ( S d , K )? When d is odd one can consider analogues of Toeplitzoperators. But we do not see how to parameterize this kind of set.2. Review of the W / Theory
Theorem 2.1.
Suppose that k : S → SU (2) is Lebesgue measurable. The follow-ing are equivalent:(I.1) k ∈ W / ( S , SU (2)) and is of the form k ( z ) = (cid:18) a ( z ) b ( z ) − b ∗ ( z ) a ∗ ( z ) (cid:19) , z ∈ S , where a, b ∈ H (∆) , a (0) > , and a and b do not simultaneously vanish at a pointin ∆ . OOPS IN SU (2) AND FACTORIZATION, II 7 (I.2) k has a (root subgroup) factorization, in the sense that k ( z ) = lim n →∞ a ( η n ) (cid:18) − ¯ η n z n η n z − n (cid:19) .. a ( η ) (cid:18) − ¯ η η (cid:19) for a.e. z ∈ S , where ( η i ) ∈ w / and the limit is understood in the W / sense.(I.3) k has triangular factorization of the form (cid:18) P nj =0 ¯ y j z − j (cid:19) (cid:18) a a − (cid:19) (cid:18) α ( z ) β ( z ) γ ( z ) δ ( z ) (cid:19) , where a > , and α , β ∈ W / .Suppose that k : S → SU (2) is Lebesgue measurable. The following are equiv-alent:(II.1) k ∈ W / ( S , SU (2)) and is of the form k ( z ) = (cid:18) d ∗ ( z ) − c ∗ ( z ) c ( z ) d ( z ) (cid:19) , z ∈ S , where c, d ∈ H (∆) , c (0) = 0 , d (0) > , and c and d do not simultaneously vanishat a point in ∆ .(II.2) k has a (root subgroup) factorization of the form k ( z ) = lim n →∞ a ( ζ n ) (cid:18) ζ n z − n − ¯ ζ n z n (cid:19) .. a ( ζ ) (cid:18) ζ z − − ¯ ζ z (cid:19) for a.e. z ∈ S , where ( η i ) ∈ w / and the limit is understood in the W / sense.(II.3) k has triangular factorization of the form (cid:18) P ∞ j =1 x ∗ j z − j (cid:19) (cid:18) a a − (cid:19) (cid:18) α ( z ) β ( z ) γ ( z ) δ ( z ) (cid:19) where a > , and γ , δ ∈ W / .Remark. There is a
P SU (1 , H (∆) / C ∂ → H (∆) : f + → Θ := ∂f + . This representation is essentially unitary, where the norm of f + is the square root of R ∂f + ∧ ∗ ∂f + . To say that f + ( z ) ∈ W / ( S ) and is holomorphic in ∆ is equivalentto saying that ∂f + ∈ H (∆) and square integrable (in the natural sense which wehave just defined). This comment applies to the conditions we are imposing on a , b , c , d , x, y in the statement of the theorem. Idea of the Proof.
For k ∈ L fin SU (2) , these correspondences are algebraic. Tobe more precise, given a sequence ζ as in II.2 with a finite number of nonzero terms,there are explicit polynomial expressions for x , α , β , γ and δ , and (2.2) a = Y k> (1 + | ζ k | ) Conversely, given k as in II.1 or II.3, the sequence ζ can be recovered recursivelyfrom the Taylor expansion c /d = γ /δ = ( − ζ ) z + ( − ζ )(1 + | ζ | ) z + (cid:16) ( − ζ )(1 + | ζ | )(1 + | ζ | ) + ( − ζ ζ )(1 + | ζ | ) (cid:17) z +(( − ζ )(1 + | ζ | )(1 + | ζ | )(1 + | ζ | ) + (1 + | ζ | )( ζ ζ (1 + | ζ | ) DOUG PICKRELL +2 ζ ζ ζ (1 + | ζ | ) + ζ ζ )) z + ... (This method of recovering the root subgroup coordinates is in [10] , and it is aspecial case of Theorem 8 of [1] . I do not know how to write down a closed rationalexpression for ζ in terms of the coefficients in the triangular decomposition for k ).There are similar formulas involving k .The fact that these algebraic correspondences continuously extend to analyticcorrespondences depends on the following Plancherel-esque formulas (which explainthe interest in root subgroup coordinates). For k i as in Theorem 2.1, (2.3) det ( A ( k ) ∗ A ( k )) = det (1 − C ( k ) ∗ C ( k )) = det (1+ ˙ B ( y ) ∗ ˙ B ( y )) − = Y i ≥ (1+ | η i | ) − i and (2.4) det ( A ( k ) ∗ A ( k )) = det (1 − C ( k ) ∗ C ( k )) = det (1+ ˙ B ( x ) ∗ ˙ B ( x )) − = Y k ≥ (1+ | ζ k | ) − k where in the third expressions, x and y are viewed as multiplication operators on H = L ( S ) , with Hardy space polarization. In (2.3), the first two terms arenonzero iff k ∈ W / , the third is nonzero iff y ∈ W / , and the third is nonzeroiff η ∈ w / .Why are the limits in I.2 and II.2 actually W / limits (as opposed to simplypointwise a.e.)? The answer is that det ( A ( g ) A ( g − )) is a continuous positive def-inite function on W / . The positive definite function associated to the vacuumvector for the so called basic representation of the Kac-Moody central extensionof W / ( S , SU (2)) is the section det ( A ) , viewed as a function on the central ex-tension. For the basic representation tensored with its dual, the positive definitefunction is the scalar function det ( A ( g ) A ( g − )) . The continuity of this function isequivalent to the strong operator continuity of the corresponding unitary represen-tation (see chapter 13 of [6] for background; note that in this reference groups areoften assumed to be locally compact, but for this particular result, local compactnessis not used). Theorem 2.2.
Suppose g ∈ W / ( S , SU (2)) . The following are equivalent:(i) The (block) Toeplitz operator A ( g ) and shifted Toeplitz operator A ( g ) areinvertible.(ii) g has a triangular factorization g = lmau .(iii) g has a (root subgroup) factorization of the form g ( z ) = k ∗ ( z ) (cid:18) e χ ( z ) e − χ ( z ) (cid:19) k ( z ) where k and k are as in Theorem 2.1 and χ ∈ W / ( S , i R ) . Idea of the Proof.
The equivalence of (i) and (ii) is standard (see also (2.7)below).Suppose that g ∈ L fin SU (2) . If g has a root subgroup factorization as in (iii),one can directly find the triangular factorization (see Proposition 2 below), and fromthis explicit expression, one can see how to recover the factors η, χ, ζ (incidentally, η and ζ have finitely many nonzero terms, but this is not so for χ ). OOPS IN SU (2) AND FACTORIZATION, II 9 As was the case for Theorem 2.1, the fact that these correspondences extend toanalytic correspondences depends on a number of Plancherel-esque identities. For g ∈ W / ( S , SU (2)) satisfying the conditions in Theorem 2.2, (2.5) det ( A ( g ) ∗ A ( g )) = ∞ Y i =0 | η i | ) i ! × ∞ Y j =1 e − j | χ j | × ∞ Y k =1 | ζ k | ) k ! (2.6) det ( A ( g ) ∗ A ( g )) = ∞ Y i =0 | η i | ) i +1 ! × ∞ Y j =1 e − j | χ j | × ∞ Y k =1 | ζ k | ) k − ! (where A is the shifted Toeplitz operator) (2.7) a ( g ) = det ( A ( g ) ∗ A ( g )) det ( A ( g ) ∗ A ( g )) = ∞ Y i =0 | η i | ) ! × ∞ Y k =1 (1 + | ζ k | ) ! Remark.
Note that because g is unitary, i.e. g − = g ∗ on S , parts (i) and (ii)are obviously inversion invariant, and this does not depend on the hypothesis that g ∈ W / (as we will see below, it is true more generally for g ∈ V M O ). Onthe other hand part (iii) is not obviously inversion invariant, reflecting the factthat inversion invariance apparently depends on the hypothesis that g ∈ W / (sothat we can use the identities (2.5) and (2.6) to prove that the existence of a rootsubgroup factorization implies invertibility of the Toeplitz determinants). A centralquestion related to the generalizations to follow is whether the hypothesis g ∈ W / is crucial for inversion invariance of root subgroup factorization.A corollary of Theorem 2.2 is that W / ( S , SU (2)) is a topological Hilbertmanifold. Thus we know what this space looks like locally. To prove that the inclu-sion C ∞ ( S , SU (2)) → W / ( S , SU (2)) is a homotopy equivalence, one approachmight be to mimic the method in Proposition 8.6.6 of [13]. However for this weneed a theory of root subgroup factorization which applies to lower strata. Thishas not been carried out in detail.3. The L Theory
The Plancherel-esque formulas (2.3), (2.4), (2.5), and (2.6) are useless for non- W / loops, because the quantities vanish. On the other hand the formulas (2.2)and (2.7) hint at L generalizations. Theorem 3.1.
Suppose that k : S → SU (2) is Lebesgue measurable. The follow-ing are equivalent:(I.1) k is of the form k ( z ) = (cid:18) a ( z ) b ( z ) − b ∗ ( z ) a ∗ ( z ) (cid:19) , z ∈ S , where a, b ∈ H (∆) , a (0) > , a and b do not simultaneously vanish at a point in ∆ , and b a −∗ ∈ L .(I.2) k has a (root subgroup) factorization, in the sense that k ( z ) = lim n →∞ a ( η n ) (cid:18) − ¯ η n z n η n z − n (cid:19) .. a ( η ) (cid:18) − ¯ η η (cid:19) for a.e. z ∈ S , where ( η i ) ∈ l .(I.3) k has triangular factorization of the form (cid:18) P nj =0 ¯ y j z − j (cid:19) (cid:18) a a − (cid:19) (cid:18) α ( z ) β ( z ) γ ( z ) δ ( z ) (cid:19) , where a > .For k satisfying these conditions, a − = ∞ Y i =0 (1 + | η i | ) = | α | + | β | on S .Suppose that k : S → SU (2) is Lebesgue measurable. The following are equiv-alent:(II.1) k is of the form k ( z ) = (cid:18) d ∗ ( z ) − c ∗ ( z ) c ( z ) d ( z ) (cid:19) , z ∈ S , where c, d ∈ H (∆) , c (0) = 0 , d (0) > , c and d do not simultaneously vanish at apoint in ∆ , and c d −∗ ∈ L .(II.2) k has a (root subgroup) factorization of the form k ( z ) = lim n →∞ a ( ζ n ) (cid:18) ζ n z − n − ¯ ζ n z n (cid:19) .. a ( ζ ) (cid:18) ζ z − − ¯ ζ z (cid:19) for a.e. z ∈ S , where ( ζ i ) ∈ l .(II.3) k has triangular factorization of the form (cid:18) P ∞ j =1 x ∗ j z − j (cid:19) (cid:18) a a − (cid:19) (cid:18) α ( z ) β ( z ) γ ( z ) δ ( z ) (cid:19) where a > .For k satisfying these conditions, a = ∞ Y k =1 (1 + | ζ k | ) − = | γ | + | δ | on S .Proof. The two sets of conditions are equivalent; they are intertwined by the outerinvolution σ of LSL (2 , C ) given by(3.1) σ ( (cid:18) a bc d (cid:19) ) = (cid:18) d cz − bz a (cid:19) . We will consider the second set.We first do some preliminary calculations. Assume II.3, i.e. k has triangularfactorization k = (cid:18) P ∞ j =1 x ∗ j z − j (cid:19) (cid:18) a a − (cid:19) (cid:18) α ( z ) β ( z ) γ ( z ) δ ( z ) (cid:19) where a >
0. The special unitarity of k implies(3.2) a α + x ∗ a − γ = a − δ ∗ , a β + x ∗ a − δ = − a − γ ∗ SU (2) AND FACTORIZATION, II 11 and(3.3) a − ( γ ∗ γ + δ ∗ δ ) = 1(3.2) implies α = − a − x ∗ γ + a − δ ∗ and β = − a − x ∗ δ − a − γ ∗ Applying the ( · ) projection to each of these, we obtain α = 1 − ( X ∗ γ ) + and β = − ( X ∗ δ ) . From these identities it is straightforward to compute that on S (3.4) | α | + | β | = a − (1 + | x | )The proof of the equivalence of II.1 and II.3 is the same as in the W / and willnot be repeated.II.3 implies II.2 because of (2.2). The crux of the matter is to understand whyII.2 implies II.1 and II.3. I basically proved this in [9], but I missed one elementarypoint at the very end of the argument. Lemma 1.
Suppose that ζ = ( ζ n ) ∈ l . Let k ( N )2 = (cid:18) d ( N ) ∗ − c ( N ) ∗ c ( N ) d ( N ) (cid:19) := N Y n =1 a ( ζ n ) ! (cid:18) ζ N z − N − ¯ ζ N z N (cid:19) .. (cid:18) ζ z − − ¯ ζ z (cid:19) Then c ( N ) and d ( N ) converge uniformly on compact subsets of ∆ to holomorphicfunctions c = c ( ζ ) and d = d ( ζ ) , respectively, as N → ∞ . The functions c and d have radial limits at a.e. point of S , c and d are uniquely determined by theseradial limits, k ( z ) = k ( ζ )( z ) := (cid:18) d ( ζ ) ∗ ( z ) − c ( ζ ) ∗ ( z ) c ( ζ )( z ) d ( ζ )( z ) (cid:19) ∈ M eas ( S , SU (2 , C ))The fact that c ( N ) and d ( N ) converge uniformly on compact subsets of ∆ toholomorphic functions c = c ( ζ ) and d = d ( ζ ), respectively, as N → ∞ is thecontent of Lemma 1 of [9]. We also observed that for the radial limits on S , dd ∗ + cc ∗ ≤
1. The fact that k actually has values in SU (2) is a consequence ofthe following elementary lemma. Lemma 2.
Suppose that f n ∈ L ∞ H (∆) and f n converges uniformly on compactsubsets to f ∈ L ∞ H (∆) . Then there exists a subsequence f n j which convergespointwise a.e. on S to f . This completes the proof of Lemma 1.Lemma 1 completes the proof that II.2 implies II.1, and hence the proof ofTheorem 3.1. (cid:3)
Theorem 3.2.
Suppose that g : S → SU (2) is measurable, and consider thefollowing conditions:(i) A ( g ) and A ( g ) are invertible.(ii) g has a triangular factorization, g = lmau , and the operator R : C [ z ] ⊗ C → C [ z ] ⊗ C : ψ + → M l − ◦ P ◦ M u − ( ψ + ) (mapping a polynomial to a polynomial) extends to a bounded operator on H + .(ii’) g has a triangular factorization. (iii) g and g − have (root subgroup) factorizations of the form g = k ( η ) ∗ (cid:18) e χ e − χ (cid:19) k ( ζ ) g − = k ( η ′ ) ∗ (cid:18) e χ ′ e − χ ′ (cid:19) k ( ζ ′ ) where k and k are as in Conjecture 3.1 and exp ( − χ + ) , exp ( − χ ′ + ) ∈ L .(i) and (ii) are equivalent, and (ii’) and (iii) are equivalent. Remarks. (a) Suppose that g = (cid:18) e χ e − χ (cid:19) , where without loss of generality weassume χ = 0 . In this case R = A ( e − χ + ) A ( e χ ∗ + ) : ψ + → F ( F ∗ ψ + ) + where F = e χ + ∈ L ( S ) , F ∈ H (∆ , C × ) , and /F ∈ L .(b) It is important to note that conditions (i), (ii), and (ii’) are invariant withrespect interchange of g and g − . It is for this reason that we have imposed acondition on both g and its inverse in part (iii). This was not necessary in the W / case.Proof. It is known that (i) and (ii) are equivalent; see Theorem 5.1 (page 109) of[8].To show that (ii’) and (iii) are equivalent, we recall some more formulas.
Proposition 2. . Suppose that g = k ( η ) ∗ e χ k ( ζ ) . Then g = l ( g ) m ( g ) a ( g ) u ( g ) ,where l ( g ) = (cid:18) l l l l (cid:19) = (cid:18) α ∗ − ( Y ∗ α ) − β ∗ − ( Y ∗ β ) − (cid:19) (cid:18) e − χ ∗ + e χ ∗ + (cid:19) (cid:18) M − (cid:19) m ( g ) = (cid:18) e χ e − χ (cid:19) , a ( g ) = (cid:18) a a − (cid:19) = (cid:18) a a
00 ( a a ) − (cid:19) u ( g ) = (cid:18) u u u u (cid:19) = (cid:18) M (cid:19) (cid:18) e χ + e − χ + (cid:19) (cid:18) − ( X ∗ γ ) + − ( X ∗ δ ) γ δ (cid:19) and M = ( a m ) − e χ ∗ + Y + e χ + X ∗ Proof.
Given the triangular factorizations for k and k , g equals (cid:18) α β γ δ (cid:19) ∗ (cid:18) Y (cid:19) (cid:18) a a e − χ ∗ + + χ + χ +
00 ( a a e − χ ∗ + + χ + χ + ) − (cid:19) (cid:18) X ∗ (cid:19) (cid:18) α β γ δ (cid:19) (3.5) = (cid:18) α ∗ γ ∗ β ∗ δ ∗ (cid:19) (cid:18) e − χ ∗ + e χ ∗ + (cid:19)(cid:18) e χ ∗ + Y (cid:19) (cid:18) a a e χ
00 ( a a e χ ) − (cid:19) (cid:18) e χ + X ∗ (cid:19) (cid:18) e χ + e − χ + (cid:19) (cid:18) α β γ δ (cid:19) OOPS IN SU (2) AND FACTORIZATION, II 13 The product of the middle three factors is upper triangular, and it is easy to findits triangular factorization: (cid:18) e χ ∗ + Y (cid:19) (cid:18) a a e χ
00 ( a a e χ ) − (cid:19) (cid:18) e χ + X ∗ (cid:19) = (cid:18) M − (cid:19) (cid:18) a a e χ
00 ( a a e χ ) − (cid:19) (cid:18) M (cid:19) where M = ( a a ) − e − − χ ∗ + + χ ) Y + e χ + X ∗ It remains to explain the formulas in the proposition for γ ∗ , δ ∗ , α and β in(3.5). The special unitarity of k implies(3.6) a α + x ∗ a − γ = a − δ ∗ , a β + x ∗ a − δ = − a − γ ∗ and(3.7) a − ( γ ∗ γ + δ ∗ δ ) = 1(3.6) implies α = − a − x ∗ γ + a − δ ∗ and β = − a − x ∗ δ − a − γ ∗ Applying the ( · ) projection to each of these, we obtain α = 1 − ( X ∗ γ ) + and β = − ( X ∗ δ ) .The formulas for γ and δ are derived in a similar way. (cid:3) Given g as in (iii), we have a candidate expression for the triangular factorizationas in the proposition. We need to show that the l ( g ) and u ( g ) factors are L . Sinceon S | α | + | β | = a − and | γ | + | δ | = a The first column of l ( g ) and the second row of u ( g ) are L iff exp ( Re ( χ − )) = exp ( − Re ( χ + )) ∈ L The second column of l ( g ) and the first row of u ( g ) appear to be hopeless. Buthere is the key fact: g has a triangular factorization iff g − has a triangular factor-ization. It is not a priori clear (and it is probably not true) that in this generality g has a root subgroup factorization iff g − has a root subgroup factorization. Butnow we are assuming both have root subgroup factorizations. So the argument goesthrough. This is kind of miraculous. (cid:3) Remark.
There are at least two reasons why we cannot draw any conclusions aboutwhat
M eas ( S , SU (2)) looks like locally from what we have done. First we havenot shown that the measurable loops which satisfy the conditions (ii’) and (iii) inthe previous theorem form an open set. Secondly we have to assume that both g and g − have root subgroup factorizations. This means that there are verycomplicated compatibility relations involving the pairs of parameters ζ and ζ ′ ,and η and η ′ . There are also complicated relations involving χ and χ ′ . The up-shot is that root subgroup factorization does not seem to help understand what M eas ( S , SU (2)) looks like locally (which is not surprising, because this group isisomorphic to M eas ([0 , , SU (2)), hence has nothing to do with S ). The
V M O
Theory
We begin by recalling basic facts about the abelian case,
V M O ( S , S ). Thenotion of degree (or winding number) can be extended from C to V M O ( S , S )(see Section 3 of [4] for an amazing variety of formulas, and further references, orpages 98-100 of [8]). Also given λ ∈ V M O ( S , S ), we view λ as a multiplicationoperator on H = L ( S ), with the Hardy polarization. We write ˙ A ( λ ) for theToeplitz operator, and so on (with the dot), to avoid confusion with the matrixcase. Lemma 3.
There is an exact sequence of topological groups → πi Z → V M O ( S , i R ) exp → V M O ( S , S ) degree → Z → . Moreover degree ( λ ) = − index ( ˙ A ( λ )) . This is implicit on pages 100-101 of [8]. The important point is that a
V M O function cannot have jump discontinuities. This implies that the kernel of exp is2 πi Z . Thus the sequence in the statement of the Lemma is continuous and exact. Remark.
This should be contrasted with the measurable case. The short exactsequence 0 → Z → R → T → → M eas ([0 , , Z ) → M eas ([0 , , R ) → M eas ([0 , , T ) → M eas ([0 , , Z ) is notdiscrete, and (just as the unitary group of an infinite dimensional Hilbert spaceis contractible - in either the strong operator or norm topology) M eas ([0 , , T ) iscontractible. Conjecture.
Suppose that k : S → SU (2) is Lebesgue measurable. The followingare equivalent:(I.1) k ∈ V M O is of the form k ( z ) = (cid:18) a ( z ) b ( z ) − b ∗ ( z ) a ∗ ( z ) (cid:19) , z ∈ S , where a, b ∈ H (∆) , a (0) > , a and b do not simultaneously vanish at a point in ∆ (I.2) k has a (root subgroup) factorization, in the sense that k ( z ) = lim n →∞ a ( η n ) (cid:18) − ¯ η n z n η n z − n (cid:19) .. a ( η ) (cid:18) − ¯ η η (cid:19) for a.e. z ∈ S , where P η i z i ∈ V M O .(I.3) k has triangular factorization of the form (cid:18) P nj =0 ¯ y j z − j (cid:19) (cid:18) a a − (cid:19) (cid:18) α ( z ) β ( z ) γ ( z ) δ ( z ) (cid:19) , where a > and α , β ∈ V M O .Suppose that k : S → SU (2) is Lebesgue measurable. The following are equiv-alent:(II.1) k ∈ V M O is of the form k ( z ) = (cid:18) d ∗ ( z ) − c ∗ ( z ) c ( z ) d ( z ) (cid:19) , z ∈ S , OOPS IN SU (2) AND FACTORIZATION, II 15 where c, d ∈ H (∆) , c (0) = 0 , d (0) > , c and d do not simultaneously vanish at apoint in ∆ .(II.2) k has a (root subgroup) factorization of the form k ( z ) = lim n →∞ a ( ζ n ) (cid:18) ζ n z − n − ¯ ζ n z n (cid:19) .. a ( ζ ) (cid:18) ζ z − − ¯ ζ z (cid:19) for a.e. z ∈ S , where P ζ k z k ∈ V M O .(II.3) k has triangular factorization of the form (cid:18) P ∞ j =1 x ∗ j z − j (cid:19) (cid:18) a a − (cid:19) (cid:18) α ( z ) β ( z ) γ ( z ) δ ( z ) (cid:19) where a > and γ , δ ∈ V M O . Idea of the Proof.
We consider the second set of equivalences. The equivalenceof II.1 and II.3 is basically obvious, because of the VMO condition of γ and δ inII.3.Theorem 3.1 implies that if ζ ∈ l , then the limit k ( z ) = lim n →∞ a ( ζ n ) (cid:18) ζ n z − n − ¯ ζ n z n (cid:19) .. a ( ζ ) (cid:18) ζ z − − ¯ ζ z (cid:19) exists for a.e. z ∈ S and defines a SU (2) loop. We need to identify the conditionon the sequence ( ζ k ) which is equivalent to saying that k ∈ V M O . We conjecturethere is a simple condition: P ζ k z k ) ∈ V M O .Remark.
Assuming the truth of the conjecture, this means that the set of k we areconsidering is parameterized by V M OA , holomorphic functions in the disk withVMO boundary values. This is a linear space.
Conjecture.
Suppose that g ∈ V M O ( S , SU (2)) . The following are equivalent:(a) A ( g ) and A ( g ) are invertible.(b) g has a triangular factorization.(c) g has a (root subgroup) factorization of the form g = k ( η ) ∗ (cid:18) e χ e − χ (cid:19) k ( ζ ) where k and k as in Theorem 3.1, χ ∈ V M O ( S ; i R ) and exp ( − χ + ) ∈ L ( S ) . Idea of the Proof.
The equivalence of parts (a) and (b) is true more generallyfor g ∈ QC ( S , SL (2 , C )) (see (b) of Remark of [9] ).Now suppose that (a) and (b) hold. Note that these conditions are inversioninvariant. Theorem 4 implies that there are l sequences η, χ, ζ and a factorization g = k ( η ) ∗ exp ( χ ) k ( ζ ) , where exp( − χ + ) ∈ L ( S ) . Note that V M O is not analgebra, hence not a decomposing algebra. It is not true that g ∈ V M O implies thatthe triangular factors l, u are VMO. So our claim that the k and k factors areVMO is not evident.Now suppose that g has a root subgroup factorization as in part (c). Our strategywill be to show that A ( g ) and A ( g ) are invertible. In the following two lemmas weinitially assume χ ± are bounded. We need to use the hypothesis that exp ( − χ + ) is L to somehow get around this limitation when we consider a general g . Lemma 4.
Assume χ ± are bounded. Then A ( k ∗ e χ k ) = A ( k ∗ e χ − ) A ( e χ k ) and same is true for A .Proof. The first statement is equivalent to showing that B ( k ∗ e χ − ) C ( e χ k ) van-ishes. Applied to (cid:18) f f (cid:19) ∈ H + , this equals B ( (cid:18) e χ − a ∗ − e − χ − b e χ − b ∗ e − χ − a (cid:19) ) C ( (cid:18) e χ + d ∗ − e − χ + c ∗ e − χ + c e − χ + d (cid:19) ) (cid:18) f f (cid:19) = (cid:20)(cid:18) e χ − a ∗ − e − χ − b e χ − b ∗ e − χ − a (cid:19) (cid:18) ( e χ + d ∗ f − e − χ + c ∗ f ) − (cid:19)(cid:21) + = (cid:20)(cid:18) e χ − a ∗ ( e χ + d ∗ f − e − χ + c ∗ f ) − e χ − b ∗ ( e χ + d ∗ f − e − χ + c ∗ f ) − (cid:19)(cid:21) + = 0This proves the first statement.For the second statement involving A , we are considering a polarization for H where H + now has orthonormal basis { ǫ i z j : i = 1 , , j > } ∪ { ǫ } (see (1.4)).We let B , C denote the Hankel operators relative to this shifted polarization. Wemust show B ( k ∗ e χ − ) C ( e χ k ) vanishes. The calculation is basically the same,but it depends on some normalizations in a subtle way. Applied to (cid:18) f f (cid:19) ∈ H + ,this equals B ( (cid:18) e χ − a ∗ − e − χ − b e χ − b ∗ e − χ − a (cid:19) ) C ( (cid:18) e χ + d ∗ − e − χ + c ∗ e − χ + c e − χ + d (cid:19) ) (cid:18) f f (cid:19) = B ( (cid:18) e χ − a ∗ − e − χ − b e χ − b ∗ e − χ − a (cid:19) ) (cid:18) ( e χ + d ∗ f − e − χ + c ∗ f ) − (cid:19) where the vanishing of the second entry uses the fact that c (0) = 0. This nowequals (cid:18) [ e χ − a ∗ ( e χ + d ∗ f − e − χ + c ∗ f ) − ] [ e χ − b ∗ ( e χ + d ∗ f − e − χ + c ∗ f ) − ] + (cid:19) = 0This proves the second statement. (cid:3) Lemma 5.
Assume χ ± are bounded. Then A ( k ∗ e χ − ) and A ( e χ k ) are injective,and similarly for A .Proof. The four statements are all proved in the same way. We consider the secondassertion concerning A . Suppose that A ( (cid:18) e χ + d ∗ − e − χ + c ∗ e − χ + c ∗ e − χ + d (cid:19) ) (cid:18) f f (cid:19) = 0This implies (cid:18) [ e χ + ( d ∗ f − c ∗ f )] + e − χ + ( c f + d f ) (cid:19) ) = 0The second component implies c f + d f = 0, and this implies (cid:18) f f (cid:19) = g (cid:18) d − c (cid:19) where g is holomorphic in the disk. Plug this into the first component to obtain[ e χ + g ( d d ∗ + c c ∗ )] + = [ e χ + g ] + = 0which implies g = 0. Thus f = 0. (cid:3) OOPS IN SU (2) AND FACTORIZATION, II 17 Remark.
The set U of χ ∈ V M O ( S ; i R ) such that exp ( − χ + ) ∈ L ( S ) is an openneighborhood of 0 ∈ V M O ( S ; i R ). Together with Remark 4, this means that thetruth of Conjectures 4 and 4 would imply that the open neighborhood of the identityin V M O ( S , SU (2)) in Conjecture 4 is parameterized by V M OA × U × V M OA .Assuming this bijection is a homeomorphism, this suggests that
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