arXiv:1405.5006v1 [] 20 May 2014

FACTORIZATION IN SLn (R) WITH ELEMENTARY
MATRICES WHEN R IS THE DISK ALGEBRA AND THE
WIENER ALGEBRA
AMOL SASANE
Abstract. Let R be the polydisc algebra or the Wiener algebra. It
is shown that the group SLn (R) is generated by the subgroup of elementary matrices with all diagonal entries 1 and at most one nonzero
off-diagonal entry. The result an easy consequence of the deep result
due to Ivarsson and Kutzschebauch [4].

1. Introduction
Let R be a commutative unital ring. Let In denote the n × n identity
matrix, 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
Eij (α) over R is a matrix of the form In + αeij , where i 6= j, α ∈ R, and
eij is the n × n matrix whose entry in the ith row and jth column is 1 and
all other entries are zeros. Let SLn (R) be the group of all n × n matrices
whose entries are elements of R and whose determinant is 1. Let En (R) be
the subgroup of SLn (R) generated by the elementary matrices.
A classical question in commutative algebra is the following:
Question 1.1. Is SLn (R) equal to En (R)?
The answer to this question depends on the ring R, and here is a list of
a few known results.
(1) If R = C, then the answer is “Yes”, and this is standard exercise in
linear algebra; see for example [1, Exercise 18.(c), page 71].
(2) Let R be the polynomial ring C[z1 , · · · , zn ] in the indeterminates
z1 , · · · , zn with complex coefficients.
If n = 1, then the answer is “Yes”, and this follows from the Euclidean Division Algorithm in C[z].
If n = 2, then the answer is “No”, and [2] gave the following counterexample:


1 + z1 z2
z12
∈ SL2 (C[z1 , z2 ]) \ E2 (C[z1 , z2 ]).
1 − z1 z2
−z22
1991 Mathematics Subject Classification. Primary 46J10; Secondary 15A23, 15A54.
Key words and phrases. Gromov’s Vasertein Problem, Banach algebras, polydisc algebra, Wiener algebra.
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AMOL SASANE

For n ≥ 3, the answer is “Yes”, and this is the K1 -analogue of Serre’s
Conjecture, 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 functions
on the finite-dimensional normal topological space X with pointwise
operations. Cb (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
Cb (X; C), the answer is “Yes” if there is no homotopy obstruction.
Indeed, if E is an elementary matrix, then (X ∋)x 7→ E(x) ∈ SLn (C)
is null-homotopic (to the constant map x 7→ In : X → SLn (C)).
So it follows that if π(F ) denotes the homotopy class of the map
x 7→ F (x) : X → SLn (C) corresponding to F ∈ SLn (R), then a
necessary condition for F ∈ En (C(X; C)) is that π(F ) = 0. It turns
out 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 also
for the ring O(X) of holomorphic functions on Stein spaces in Cn .
This was posed as an explicit open problem by Gromov in [3], and
was recently solved by Ivarsson and Kutzschebauch [4]. The main
result in [4] is the following:
Theorem 1.2 ([4]). If X is a finite-dimensional reduced Stein space
and F : X → SLn (C) is a holomorphic mapping that is null-homotopic,
then there exists a natural number K and holomorphic mappings
G1 , · · · , GK : X → Cm(m−1)/2 such that F can be written as a product of upper and lower diagonal unipotent matrices
F (x) = M1 (G1 (x)) · · · MK (GK (x)),

x ∈ X,

where the matrices Mj (Gj (x)) are defined by


1
0


..
Mj (Gj (x)) := 
 if j is odd,
.
Gj (x)
1

while




Mj (Gj (x)) := 

1

Gj (x)
..

0

.
1




 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, unital
complex Banach algebras R. A special case is when R = Cb (X; R), where X
is a compact Hausdorff topological space, and item (3) above describes the
answer in this special case. Motivated by this, we formulate the following
question/conjecture, but first we introduce some convenient notation.

FACTORIZATION IN SLn (R)

3

Let R be a commutative, semisimple, unital complex Banach algebra with
maximal ideal space denoted by XR , equipped with the weak-∗ topology
induced from the dual space R∗ := L(R; C) of R.
Let b· : R → C(XR ; C) denote the Gelfand transform. For F ∈ SLn (R),
let Fb be the matrix with elements in C(XR ; C) obtained by taking the
Gelfand transform of the entries of F , and π(Fb ) denotes the homotopy class
of ϕ 7→ Fb(ϕ) : XR → SLn (C).

Conjecture 1.3. Let R be a commutative, semisimple, unital complex Banach algebra. F ∈ SLn (R) belongs to En (R) if and only if π(Fb) = 0.

We consider Question 1.1 for two important Banach algebras of holomorn
n
phic functions: the polydisc algebra A(D ) and the Wiener algebra W + (D ).
Let D := {z ∈ C : |z| < 1} and D := {z ∈ C : |z| ≤ 1}. Let d ∈ N. The
n
Wiener algebra W + (D ) is the Banach algebra defined by


∞
∞
∞
∞
X

X
X
X
d
kd
k1
+
···
W (D ) =
a(k1 ,...,kd ) z1 · · · zd :
···
|a(k1 ,...,kd ) | < ∞ ,


k1 =0

kd =0

k1 =0

kd =0

with pointwise addition and multiplication, and the k · k1 -norm given by
kf k1 =

∞
X

···

k1 =0

∞
X

|a(k1 ,...,kd ) |, f =

kd =0

∞
X

···

k1 =0

∞
X

a(k1 ,...,kd ) z1k1 · · · znkd .

kd =0

d

The polydisc algebra A(D ) is the Banach algebra of all continuous functions
d
f : D → C which are holomorphic in Dd , with pointwise addition and
multiplication, and the supremum norm k · k∞ given by
kf k∞ :=

sup

|f (z1 , . . . , zd )|,

d

f ∈ A(D ).

(z1 ,...,zd )∈Dd
d

The ball algebra A(Bd ) is defined similarly, with the polydisc D replaced
by the ball
Bd := {(z1 · · · , zd ) ∈ Cd : |z1 |2 + · · · + |zd |2 ≤ 1}.
d

d

For a n × n matrix F with entries in A(D ), A(Bd ) or W + (D ), we define
kF k :=

n
X

kFij k∞ ,

i,j=1

where Fij denotes the entry in the ith row and jth column of F . Then
kF Gk ≤ kF kkGk, for n × n matrices F, G with entries from any of the
d
d
Banach algebras A(D ), A(Bd ) or W + (D ).
Our main result is the following.
d

d

Theorem 1.4. If R = A(D ), A(Bd ) or W + (D ), then SLn (R) = En (R).

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AMOL SASANE
d

d

If R = A(D ) or W + (D ), then in both cases, the maximal ideal space XR
d
can be identified with D as a topological space. Similarly, XA(Bd ) = Bd . If
Conjecture 1.3 is true, then Theorem 1.4 follows from the observation that
d
D , Bd are contractible (since then π(Fb) 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 that
the set of invertible elements of R is open in R. If F ∈ SLn (R) is sufficiently
close to In , then F belongs to En (R).
2. Proof of Theorem 1.4
d

Proof. We will simply prove the result in the case of the disc algebra A(D );
the proofs in the cases of the ball algebra A(Bd ) and the Wiener algebra
being analogous.
d
Let F ∈ SLn (A(D ). Let r ∈ (0, 1) (to be determined later). Define
(z1 , · · · , zd ) ∈ Dd .

Fr (z1 , · · · , zd ) := F (rz1 , · · · , rzd ),

As Fr ∈ O( 1r Dd ), and det Fr ≡ 1, it follows from Theorem 1.2 (since 1r Dd is
a contractible Stein domain) that there are elementary matrices G1 , · · · , GK
belonging to En (O( 1r Dd )) such that
1
d
Fr = E1 · · · EK ∈ En (O( Dd )) ⊂ En (A(D )).
r
d

Thus F (In + F −1 (Fr − F )) = Fr ∈ En (A(D )). As det F = det Fr = 1, it
follows that also det(In + F −1 (Fr − F )) = 1. We will be done if we manage
d
to show that In + F −1 (Fr − F ) ∈ En (A(D )) too. But this is clear by
Lemma 1.5, since


In + F −1 (Fr − F ) − In = kF −1 (Fr − F )k ≤ kF −1 kkFr − F k,

and we can make kFr − F k as small as we like by choosing r close enough
to 1.

Remark 2.1. The above proof also works for some other Banach algebras of
smooth functions contained in the polydisc algebra, for example, if N ∈ N,
d
d
the Banach algebra ∂ −N A(D ) of all functions f ∈ A(D ) whose complex
d
partial derivatives of all orders up to N belong to A(D ), with the norm
X
∂ α1 +···+αd f
1
kf k∂ −N A(Dd ) :=
sup
(z1 , · · · , zd ) .
α1 ! . . . αd ! (z1 ,··· ,zd )∈Dd ∂z1α1 . . . ∂zdαd
α1 +···+αd ≤N

In light of Theorem 1.4, it is natural to ask the analogous question also
for the Hardy algebra. Recall that if U is an open set in Cd , then the Hardy
algebra H ∞ (U ) is the Banach algebra of all complex-valued functions on U
that are bounded and holomorphic in U .

FACTORIZATION IN SLn (R)

5

Conjecture 2.2. SLn (H ∞ (U )) = En (H ∞ (U )) if U is the polydisc Dd or
open unit ball U = Bd := {(z1 · · · , zd ) ∈ Cd : |z1 |2 + · · · + |zd |2 < 1}.
References
[1] Michael Artin. Algebra. Prentice Hall, Englewood Cliffs, NJ, 1991.
[2] Paul M. Cohn. On the structure of the GL2 of a ring. Institut des Hautes Études
Scientifiques. Publications Mathématiques, No. 30, 5-53, 1966.
[3] Michael Gromov. Oka’s principle for holomorphic sections of elliptic bundles. Journal
of the American Mathematical Society, 2:851-897, no. 4, 1989.
[4] Björn Ivarsson and Frank Kutzschebauch. Holomorphic factorization of mappings
into SLn (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 diagonal 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.
E-mail address: amol.sasane@math.lu.se

