# Araki's proof of simple connectedness of the restricted orthogonal group

I am trying to understand Araki's proof of the statement that the restricted orthogonal group of a Hilbert space with a unitary structure is simply connected. This proof starts on page 114 of these conference proceedings (the result is stated on page 77, Theorem 6.3 (4)).

I know that there are some alternative proofs of this statement, but I am specifically interested in this version. I will start with some prerequisites, which may or may not be necessary.

Let $V$ be a complex Hilbert space equipped with a real structure, that is, a complex anti-linear involution $\alpha: V \rightarrow V$. Then we define a complex bilinear form $$\beta: V \times V \rightarrow \mathbb{C}, (v,w) \mapsto \langle v, \alpha (w) \rangle.$$ Let $L$ be a Lagrangian subspace with respect to $\beta$. Then we define a so-called unitary structure on $V = L \oplus \alpha(L)$ by \begin{align} J:L \oplus \alpha(L) &\rightarrow L \oplus \alpha(L) \\ (v,w) &\mapsto i(v,-w). \end{align} The restricted orthogonal group is then defined to be $$\operatorname{O}_{J}(V) = \{ U \in \operatorname{U}(V) \mid \| [U,J] \|_{HS} < \infty \},$$ where $\| \cdot \|_{HS}$ is the Hilbert-Schmidt norm. The restricted orthogonal group is a topological (actually Banach-Lie) group, when equipped with the so-called $J$-norm, $$\|U \|_{J} = \|U\| + \|[U,J]\|_{HS},$$ where $\| \cdot \|$ is the usual operator norm. Let $P_{L}$ be the projection onto $L$. The condition that $\|[U,J]\|_{HS} < \infty$ is then equivalent to saying that $\|[U,P_{L}]\|_{HS} < \infty$.

Now, the claim is that (the identity component of) $\operatorname{O}_{J}(V)$ is simply connected. Araki proves this by starting with an arbitrary loop and arguing that it is contractible in $\operatorname{O}_{J}(V)$.

So let $U_{t}$, $0, \leqslant t \leqslant 1$ be a loop in $\operatorname{O}_{J}(V)$. Then we define $P(t) = U_{t}P_{L} U_{t}^{*}$. Araki then claims that $P$ is continuous with respect to the Hilbert-Schmidt norm.

Q1 Why is this so?

My problem here is that it is not clear to me that $U_{t}P_{L}U_{t}^{*}$ has finite Hilbert-Schmidt norm. Because of the fact that the Hilbert-Schmidt operators form an ideal we see that $[U_{t},P_{L}] U_{t}^{*} = U_{t} P_{L} U_{t}^{*} - P_{L}$ is Hilbert-Schmidt. Maybe it is meant that the map $t\mapsto U_{t} P_{L} U_{t}^{*} - P_{L}$ is continuous with respect to the Hilbert-Schmidt norm?

Next, we split the circle into a finite number of intervals $[t_{j},t_{j+1}], j = 1,...,n$ such that $\| P(t_{j}) - P(t) \| <1$. All we require here is that $P$ is norm-continuous.

For any $t \in [t_{j},t_{j+1}]$ we can find an operator $H(t)$ with finite Hilbert-Schmidt norm, such that $$e^{H(t)}P(t_{j})e^{-H(t)} = P(t).$$ The fact that $H(t)$ has finite Hilbert-Schmidt norm implies that $H(t) \in \operatorname{O}_{J}(H)$.

Q2 Why does such $H(t)$ exist? Presumably it uses the condition $\|P(t_{j}) - P(t) \| <1$.

We then define a path $W: [0,1] \rightarrow \operatorname{U}(V)$ by $$W(t) = e^{H(t)} e^{H(t_{j})} ... e^{H(t_{2})}, \text{ for } t \in [t_{j},t_{j+1}].$$ I guess that it is somehow clear that $H(t_{j} + \epsilon)$ can be made arbitrarily close to $0$. Then we use the relation $$e^{H(t_{j+1})} P(t_{j}) = P(t_{j+1}) e^{H(t_{j+})},$$ to prove that $W(1)$ commutes with $P_{L}$.

Next, Araki claims that $\|W(1) - 1\|_{HS} < \infty$, which implies that there exists an operantor $H = -H^{*}$ such that $P_{L}H = HP_{L} = H$, and furthermore $e^{H}P = W(1)P$ and finally $\|H\|_{HS} < \infty$.

Q3 Why does this $H$ exist?

For this step I am really lost.

The proof doesn't end here, but I think these are already way too many questions.