ar X iv : 0 90 5 . 29 11 v 2 [ m at h . FA ] 2 5 Ju l 2 00 9 UNITARY LOOP GROUPS AND FACTORIZATION

We discuss a refinement of triangular factorization for unitary matrix-valued functions on S. 0. Introduction The purpose of this paper is to generalize Theorems 0.1 and 0.2 below to general simply connected compact Lie groups. Throughout the paper K̇ is a simply connected compact Lie group, Ġ is the complexification, and LfinK̇ (LfinĠ) denotes the group consisting of functions S → K̇ (Ġ, respectively) having finite Fourier series, relative to some matrix representation, with pointwise multiplication. For example, for ζ ∈ C and n ∈ Z, the function S → SU(2) : z → a(ζ) ( 1 ζz −ζ̄z 1 ) , where a(ζ) = (1 + |ζ|), is in LfinSU(2). Also, if f(z) = ∑ fnz , then f = ∑ f̄nz . Thus if f ∈ H(∆), then f ∈ H(∆), where ∆ is the open unit disk, and ∆ is the open unit disk at ∞. The following two theorems are from [3]. Theorem 0.1. Suppose that k1 ∈ LfinSU(2). The following are equivalent: (a1) k1 is of the form k1(z) = (