1. K1(A)
Let A be a ring. Let GLn(A) be the general linear group over A. Let φn : GLn(A) → GLn+1(A) be the homomorphism a 7→ a ⊕ 1. Then for n ≤ m, we obtain a group homomor- phism
φn,m: GLn(A) → GLm(A)
given by φn,m = φm−1◦· · ·◦φn. Hence we obtain a directed system of groups (GLn(A), φn,m).
The directed limit of the directed system is denoted by GL(A) = lim−→
n
GLn(A).
A matrix in GL(A) is called elementary if it coincides with the identity matrix except a single off-diagonal entry. The subgroup E(A) generated by elementary matrices is the commutator subgroup of GL(A).1
Definition 1.1. The quotient group K1(A) = GL(A)/E(A) is called the Whitehead group.
Let us assume further that A is a commutative ring. Denote A∗ the multiplicative group consisting of units of A. Then
det : GL(A) → A∗
is defined. (We can consider det : GLn(A) → A∗ and let n → ∞.) Then det : GL(A) is a group homomorphism whose kernel is denoted by SL(A). Hence we find
GL(A)/SL(A) ∼= A∗.
We see that E(A) is a subgroup of SL(A). The homomorphism det gives us a direct sum decomposition
K1(A) = A∗⊕ (SL(A)/E(A)) .
We denote SL(A)/E(A) by SK1(A). In this note, we are going to study K1(A) or SK1(A) when A is a commutative (complex or real) unital Banach algebra. Notice that if A is a commutative Banach-algebra, GLn(A) becomes a topological group.
Notation: We shall use e(n)ij (λ) to denote the n × n elementary matrix whose (ij)-entry is λ where λ ∈ A. (Sometimes, for simplicity, when n is specified, we denote it by eij(λ).) Lemma 1.1. The group En(A) is an open, path-connected subgroup of SLn(A).
To do this, we need the following lemma.
Lemma 1.2. Let a be an n × n matrix over A such that In+ a has determinant 1. If each entry of a satisfies kaijk < 1/(n−1), then In+ a can be expressed as a product of n2+ 5n − 6 elementary matrices, each of which depends continuousy on a.
Proof. To decompose In+ a into product of elementary matrices, we need to do the row operations for In+ a at first. The conditions given above will allow us to do the elementary row operations.
Since ka11k < 1/(n − 1) < 1, 1 + a1 is invertible with k(1 + a11)−1k < (n − 1)/(n − 2).
Set b11 = (1 + a11)−1. Multiplying In+ a to the left by en1(−b11an1) · · · e21(−b11a21), we obtain a new matrix of the form In+ a0 such that a0k1 = 0 for all 2 ≤ k ≤ n, where a0kj = akj− b11ak1a1j, for all k, j. For all k, j, by the triangle inequality, and by the normed inequality (for Banach algebra),
ka0kjk ≤ kakjk + kb11kkak1kka1jk
1The group generated by n × n elementary matrices over A is denoted by En(A).
1
2
≤ 1
n − 1+ n − 1 n − 2· 1
n − 1· 1 n − 2
= 1
n − 2 < 1 n − 1.
Inductively, elementary row operations give us a diagonal matrix. The diagonal matrix can also be composed into a product of elementary matrices. Hence we can inductively decompose In+ a into a product of elementary matrices. The number of operations can be calculated; we leave it to the reader as an exercise. Moreover, we see that the each elementary matrix which appears in the decomposition of In+ a depends continuously on
a.
For each elementary matrix eij(λ) ∈ En(A), we define a path c(t) = eij(tλ) in En(A).
Then c(t) is a continuous path connecting c(0) = In and c(1) = eij(λ). Using Lemma 1.2, for any given b in En(A), we decompose b into a product of elementary matrices. Since each elementary matrix depends continuously in b, we can define a path c(t) in En(A) connecting Inand b. This shows that En(A) is path connected. Moreover, Lemma 1.2 also implies that En(A) is open in SLn(A). Since En(A) is a normal subgroup of SLn(A), we obtain that En(A) is also closed. Hence En(A) is the identity component of SLn(A). Therefore the quotient group SLn(A)/En(A) can be identified with the group π0(SLn(A)), i.e. we have a natural identification
SLn(A)/En(A) ∼= π0(SLn(A)).
Taking n → ∞, we obtain
SL(A)/E(A) = lim−→
n
SLn(A)/En(A) ∼= lim−→
n
π0(SLn(A)).
Equipping SL(A) with the direct limit topology, we identify π0(SL(A)) = lim−→
n
π0(SLn(A)).
Hence we obtain the following natural identification:
SK1(A) ∼= π0(SL(A)).
Corollary 1.1. The group K1(A) splits into the direct sum of A∗and SK1(A) ∼= π0(SL(A)).
Let us consider the case when A = C(X; R) for some compact Hausdorff space X. We know that
SLn(C(X; R)) ∼= C(X; SLn(R)).
Since C(X; SLn(R)) contains C(X; SO(n)) as deformation retact,
π0(SLn(C(X; R))) ∼= π0(C(X; SLn(R))) ∼= π0(C(X; SO(n))).
Since we know that π0(C(X; SO(n))) is the space [X, SO(n)]2 of homotopy maps from X into SO(n), by taking n → ∞, and Corollary 1.1, we obtain:
Corollary 1.2. The group K1(C(X, R)) splits as a direct sum of the group C(X, R×) and the group [X, SO].
Similarly, we have
Corollary 1.3. The group K1(C(X, C)) has the following direct sum decomposition:
K1(C(X, C)) = C(X, C×) ⊕ [X, SU ].
2[X, Y ] is another notation for π0(C(X, Y )).
