= Near-identity Sobolev loop factorization
{title2=$X=X_-X_+$, $X_+(0)=I$}
For matrix-valued circle loops in $H^s$, $s>1/2$, use a submultiplicative algebra norm with contractive Fourier projections. If $\|X-I\|<1$, solve $Y_+=I+y_+$ by $y_+=-P_{>0}((X-I)(I+y_+))$, and solve $V_-=I+v_-$ by $v_-=-P_{<0}((I+v_-)(X-I))$. Both are contractions. Then $XY_+$ has only nonpositive modes, $V_-X$ has only nonnegative modes, and their compatible products equal a constant matrix. Invertibility of the Hardy compression $I+P_{\geq0}(X-I)$ makes that constant invertible. This yields the stated normalized factors; uniqueness follows because the two Fourier subalgebras intersect in constants.
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