If subspace $A$ is the fixed points of an involution $\sigma$, then is $K(A)$ the fixed points of $-\sigma$?
Yes.
Proof:
-
Observe that $K(fixed(\eta)) \subseteq fixed(-\eta)$ for both $\eta=\pm \sigma$.
1.1. For $\eta=\sigma$, $K(A) \subseteq fixed(-\sigma)$.
1.2. For $\eta=-\sigma$, $K(fixed(-\sigma)) \subseteq A$.
Apply $K$ to latter set equality to get $fixed(-\sigma) \subseteq K(A)$.
Note: I hopefully do not assume either that $fixed(\sigma) \cong fixed(-\sigma)$ or that $fixed(\sigma) \bigoplus fixed(-\sigma)=V^2$.