Factorization in S L n (R) with elementary matrices when R is the disk algebra and the Wiener algebra
aa r X i v : . [ m a t h . A C ] M a y FACTORIZATION IN SL n ( R ) WITH ELEMENTARYMATRICES WHEN R IS THE DISK ALGEBRA AND THEWIENER ALGEBRA
AMOL SASANE
Abstract.
Let R be the polydisc algebra or the Wiener algebra. Itis shown that the group SL n ( R ) is generated by the subgroup of ele-mentary matrices with all diagonal entries 1 and at most one nonzerooff-diagonal entry. The result an easy consequence of the deep resultdue to Ivarsson and Kutzschebauch [4]. Introduction
Let R be a commutative unital ring. Let I n denote the n × n identitymatrix, that is the square matrix with all diagonal entries equal to 1 ∈ R and off-diagonal entries equal to 0 ∈ R . Recall that an elementary matrix E ij ( α ) over R is a matrix of the form I n + α e ij , where i = j , α ∈ R , and e ij is the n × n matrix whose entry in the i th row and j th column is 1 andall other entries are zeros. Let SL n ( R ) be the group of all n × n matriceswhose entries are elements of R and whose determinant is 1. Let E n ( R ) bethe subgroup of SL n ( R ) generated by the elementary matrices.A classical question in commutative algebra is the following: Question 1.1. Is SL n ( R ) equal to E n ( R )?The answer to this question depends on the ring R , and here is a list ofa few known results.(1) If R = C , then the answer is “Yes”, and this is standard exercise inlinear algebra; see for example [1, Exercise 18.(c), page 71].(2) Let R be the polynomial ring C [ z , · · · , z n ] in the indeterminates z , · · · , z n with complex coefficients.If n = 1, then the answer is “Yes”, and this follows from the Eu-clidean Division Algorithm in C [ z ].If n = 2, then the answer is “No”, and [2] gave the following coun-terexample: (cid:20) z z z − z − z z (cid:21) ∈ SL ( C [ z , z ]) \ E ( C [ z , z ]) . Mathematics Subject Classification.
Primary 46J10; Secondary 15A23, 15A54.
Key words and phrases.
Gromov’s Vasertein Problem, Banach algebras, polydisc alge-bra, Wiener algebra.
For n ≥
3, the answer is “Yes”, and this is the K -analogue of Serre’sConjecture, which is the Suslin Stability Theorem [5].(3) The case of R being a ring of continuous functions was considered in[6]. Let C ( X ; C ) be the ring of continuous complex-valued functionson the finite-dimensional normal topological space X with pointwiseoperations. C b ( X ; C ) denotes the subring of C ( X ; C ) consisting of bounded functions. It was shown in [6] that for R = C ( X ; C ) or C b ( X ; C ), the answer is “Yes” if there is no homotopy obstruction.Indeed, if E is an elementary matrix, then ( X ∋ ) x E ( x ) ∈ SL n ( C )is null-homotopic (to the constant map x I n : X → SL n ( C )).So it follows that if π ( F ) denotes the homotopy class of the map x F ( x ) : X → SL n ( C ) corresponding to F ∈ SL n ( R ), then anecessary condition for F ∈ E n ( C ( X ; C )) is that π ( F ) = 0. It turnsout that this condition is also sufficient, and this is the content of[6, Theorem 4].(4) Based on the above result, it is natural to consider the question alsofor the ring O ( X ) of holomorphic functions on Stein spaces in C n .This was posed as an explicit open problem by Gromov in [3], andwas recently solved by Ivarsson and Kutzschebauch [4]. The mainresult in [4] is the following: Theorem 1.2 ([4]) . If X is a finite-dimensional reduced Stein spaceand F : X → SL n ( C ) is a holomorphic mapping that is null-homotopic,then there exists a natural number K and holomorphic mappings G , · · · , G K : X → C m ( m − / such that F can be written as a prod-uct of upper and lower diagonal unipotent matrices F ( x ) = M ( G ( x )) · · · M K ( G K ( x )) , x ∈ X, where the matrices M j ( G j ( x )) are defined by M j ( G j ( x )) := . . . G j ( x ) 1 if j is odd,while M j ( G j ( x )) := G j ( x ) . . . if j is even.In particular, the assumption of null-homotopy is always satisfied if X is contractible. We wish to consider Question 1.1 for commutative, semisimple, unitalcomplex Banach algebras R . A special case is when R = C b ( X ; R ), where X is a compact Hausdorff topological space, and item (3) above describes theanswer in this special case. Motivated by this, we formulate the followingquestion/conjecture, but first we introduce some convenient notation. ACTORIZATION IN SL n ( R ) 3 Let R be a commutative, semisimple, unital complex Banach algebra withmaximal ideal space denoted by X R , equipped with the weak- ∗ topologyinduced from the dual space R ∗ := L ( R ; C ) of R .Let b · : R → C ( X R ; C ) denote the Gelfand transform. For F ∈ SL n ( R ),let b F be the matrix with elements in C ( X R ; C ) obtained by taking theGelfand transform of the entries of F , and π ( b F ) denotes the homotopy classof ϕ b F ( ϕ ) : X R → SL n ( C ) . Conjecture 1.3.
Let R be a commutative, semisimple, unital complex Ba-nach algebra. F ∈ SL n ( R ) belongs to E n ( R ) if and only if π ( b F ) = 0.We consider Question 1.1 for two important Banach algebras of holomor-phic functions: the polydisc algebra A ( D n ) and the Wiener algebra W + ( D n ).Let D := { z ∈ C : | z | < } and D := { z ∈ C : | z | ≤ } . Let d ∈ N . The Wiener algebra W + ( D n ) is the Banach algebra defined by W + ( D d ) = ∞ X k =0 · · · ∞ X k d =0 a ( k ,...,k d ) z k · · · z k d d : ∞ X k =0 · · · ∞ X k d =0 | a ( k ,...,k d ) | < ∞ , with pointwise addition and multiplication, and the k · k -norm given by k f k = ∞ X k =0 · · · ∞ X k d =0 | a ( k ,...,k d ) | , f = ∞ X k =0 · · · ∞ X k d =0 a ( k ,...,k d ) z k · · · z k d n . The polydisc algebra A ( D d ) is the Banach algebra of all continuous functions f : D d → C which are holomorphic in D d , with pointwise addition andmultiplication, and the supremum norm k · k ∞ given by k f k ∞ := sup ( z ,...,z d ) ∈ D d | f ( z , . . . , z d ) | , f ∈ A ( D d ) . The ball algebra A ( B d ) is defined similarly, with the polydisc D d replacedby the ball B d := { ( z · · · , z d ) ∈ C d : | z | + · · · + | z d | ≤ } . For a n × n matrix F with entries in A ( D d ), A ( B d ) or W + ( D d ), we define k F k := n X i,j =1 k F ij k ∞ , where F ij denotes the entry in the i th row and j th column of F . Then k F G k ≤ k F kk G k , for n × n matrices F, G with entries from any of theBanach algebras A ( D d ), A ( B d ) or W + ( D d ).Our main result is the following. Theorem 1.4. If R = A ( D d ) , A ( B d ) or W + ( D d ) , then SL n ( R ) = E n ( R ) . AMOL SASANE If R = A ( D d ) or W + ( D d ), then in both cases, the maximal ideal space X R can be identified with D d as a topological space. Similarly, X A ( B d ) = B d . IfConjecture 1.3 is true, then Theorem 1.4 follows from the observation that D d , B d are contractible (since then π ( b F ) is always trivial).We will derive our main result as a consequence of the result from [4]quoted above, and [6, Lemma 9] reproduced below. Lemma 1.5 ([6]) . Let R be a commutative topological unital ring such thatthe set of invertible elements of R is open in R . If F ∈ SL n ( R ) is sufficientlyclose to I n , then F belongs to E n ( R ) . Proof of Theorem 1.4
Proof.
We will simply prove the result in the case of the disc algebra A ( D d );the proofs in the cases of the ball algebra A ( B d ) and the Wiener algebrabeing analogous.Let F ∈ SL n ( A ( D d ). Let r ∈ (0 ,
1) (to be determined later). Define F r ( z , · · · , z d ) := F ( rz , · · · , rz d ) , ( z , · · · , z d ) ∈ D d . As F r ∈ O ( r D d ), and det F r ≡
1, it follows from Theorem 1.2 (since r D d isa contractible Stein domain) that there are elementary matrices G , · · · , G K belonging to E n ( O ( r D d )) such that F r = E · · · E K ∈ E n ( O ( 1 r D d )) ⊂ E n ( A ( D d )) . Thus F ( I n + F − ( F r − F )) = F r ∈ E n ( A ( D d )). As det F = det F r = 1, itfollows that also det( I n + F − ( F r − F )) = 1. We will be done if we manageto show that I n + F − ( F r − F ) ∈ E n ( A ( D d )) too. But this is clear byLemma 1.5, since (cid:13)(cid:13)(cid:13)(cid:16) I n + F − ( F r − F ) (cid:17) − I n (cid:13)(cid:13)(cid:13) = k F − ( F r − F ) k ≤ k F − kk F r − F k , and we can make k F r − F k as small as we like by choosing r close enoughto 1. (cid:3) Remark 2.1.
The above proof also works for some other Banach algebras ofsmooth functions contained in the polydisc algebra, for example, if N ∈ N ,the Banach algebra ∂ − N A ( D d ) of all functions f ∈ A ( D d ) whose complexpartial derivatives of all orders up to N belong to A ( D d ), with the norm k f k ∂ − N A ( D d ) := X α + ··· + α d ≤ N α ! . . . α d ! sup ( z , ··· ,z d ) ∈ D d (cid:12)(cid:12)(cid:12)(cid:12) ∂ α + ··· + α d f∂z α . . . ∂z α d d ( z , · · · , z d ) (cid:12)(cid:12)(cid:12)(cid:12) . In light of Theorem 1.4, it is natural to ask the analogous question alsofor the Hardy algebra. Recall that if U is an open set in C d , then the Hardyalgebra H ∞ ( U ) is the Banach algebra of all complex-valued functions on U that are bounded and holomorphic in U . ACTORIZATION IN SL n ( R ) 5 Conjecture 2.2. SL n ( H ∞ ( U )) = E n ( H ∞ ( U )) if U is the polydisc D d oropen unit ball U = B d := { ( z · · · , z d ) ∈ C d : | z | + · · · + | z d | < } . References [1] Michael Artin.
Algebra . Prentice Hall, Englewood Cliffs, NJ, 1991.[2] Paul M. Cohn. On the structure of the GL of a ring. Institut des Hautes ´EtudesScientifiques. Publications Math´ematiques , No. 30, 5-53, 1966.[3] Michael Gromov. Oka’s principle for holomorphic sections of elliptic bundles.
Journalof the American Mathematical Society , 2:851-897, no. 4, 1989.[4] Bj¨orn Ivarsson and Frank Kutzschebauch. Holomorphic factorization of mappingsinto SL n ( C ). Annals of Mathematics (2), 175:45-69, no. 1, 2012.[5] Andrei A. Suslin. The structure of the special linear group over rings of polynomials.
Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya , 41:235-252, no. 2, 1977.[6] Leonid N. Vaserstein. Reduction of a matrix depending on parameters to a diago-nal form by addition operations.
Proceedings of the American Mathematical Society ,103:741-746, no. 3, 1988.
Department of Mathematics, Faculty of Science, Lund University, Sweden.
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