A Quantitative Logarithm Near the Identity in SU(2)

Put $s=\sin (\pi \mathrm{/}6)$. Since $0<\pi \mathrm{/}6<\pi \mathrm{/}2$, [F8] and [F9] give $s>0$. By [F10], $s=\cos (\pi \mathrm{/}3)=1-2s^2$, so $(2s-1)(s+1)=0$ and positivity gives $s=1\mathrm{/}2$. By [F1], $D$ is unitarily diagonalizable with eigenvalues ${\lambda }_1,{\lambda }_2$ on the unit circle. Since $\det D=1$, ${\lambda }_1{\lambda }_2=1$. Choose principal angular representatives ${\theta }_j\in (-\pi ,\pi ]$ using [F7]. The norm bound and [F5] give $2\mathrm{\mid }\sin ({\theta }_j\mathrm{/}2)\mathrm{\mid }=\mathrm{\mid }{\lambda }_j-1\mathrm{\mid }\le \rho \le 1\mathrm{/}2$, hence $\mathrm{\mid }\sin ({\theta }_j\mathrm{/}2)\mathrm{\mid }\le 1\mathrm{/}4<1\mathrm{/}2=\sin (\pi \mathrm{/}6)$. Oddness and strict monotonicity in [F9] force $\mathrm{\mid }{\theta }_j\mathrm{\mid }<\pi \mathrm{/}3$. Now [F6] gives ${\theta }_1+{\theta }_2\in 2\pi \mathbb{Z}$; the bound on their sum forces ${\theta }_2=-{\theta }_1$.

For $\mathrm{\mid }z\mathrm{\mid }<1$, the scalar power series $L(z)={\sum }_{j\ge 1}(-1{)}^{j+1}{z}^j\mathrm{/}j$ converges absolutely by comparison with ${\sum }_{j\ge 1}\mathrm{\mid }z{\mathrm{\mid }}^j$, so its disc of convergence contains the unit disk. By [F11], $L^{\mathrm{\prime }}(z)={\sum }_{j\ge 1}(-1{)}^{j+1}{z}^{j-1}={\sum }_{k\ge 0}(-z{)}^k=\frac{1}{1+z},$ where the last equality is the finite geometric-sum identity followed by a limit. Since $1+z\mathrm{\ne }0$ on the unit disk, [F12] gives ${\left(\frac{e^{L(z)}}{1+z}\right)}^{\mathrm{\prime }}=\frac{e^{L(z)}{L}^{\mathrm{\prime }}(z)(1+z)-e^{L(z)}}{(1+z{)}^2}=0.$ The open unit disk is convex, hence a connected open domain, so [F12] makes this quotient constant there. Its value at $0$ is $1$, and therefore $e^{L(z)}=1+z$. For $z=e^{it}-1$ with $\mathrm{\mid }t\mathrm{\mid }<\pi \mathrm{/}3$, [F5], [F9], and [F10] show $\mathrm{\mid }z\mathrm{\mid }=2\mathrm{\mid }\sin (t\mathrm{/}2)\mathrm{\mid }<1$. Thus [F6] gives $L(e^{it}-1)-it\in 2\pi i\mathbb{Z}$. This difference is continuous in $t$ by [F13], [F5], and [F8]. Dividing it by $2\pi i$ gives a continuous integer-valued real function. If it took distinct values at two points, restrict to the closed interval between them; [F14] would force it to take a noninteger value between them; hence it is constant and, at $t=0$, zero. Substitution of $z={\lambda }_j-1$ now shows the matrix series diagonalizes to $i{\theta }_1,i{\theta }_2$, so exponentiating it gives $D$.

By [F2], the norm of the series is at most ${\sum }_{j\ge 1}{\rho }^j\mathrm{/}j$. The scalar identity from step 2.1 at $z=-\rho$ and [F4] identify this sum with $-\log (1-\rho )$. Since $1\mathrm{/}j\le 1$ and ${\sum }_{j\ge 1}{\rho }^j=\rho \mathrm{/}(1-\rho )\le 2\rho$ for $0\le \rho \le 1\mathrm{/}2$, the claimed bound follows. The eigenvalues of $\log D$ are $i{\theta }_1,-i{\theta }_1$, so it is skew-Hermitian and has trace zero.