Fixed points Cross products

Question

Let $(A,G,\alpha)$ be a $C^*$-dynamical system with $G$ amenable (So that I need to consider just cross-product since reduced and full are the same). Let $\theta$ be an automorphism of $A \rtimes G$. Is it always true that $A$ is in the fixed point set of $\theta$? I have the $C*$ algebra $C(X)\rtimes_{\phi}\mathbb{Z}$ where $X$ is a compact Hausdorff space and $\phi $ a homeomorphism on $X$.

Answer 1

Here’s an example arising from a non-trivial homeomorphism $\phi$ that acts non-trivially both on $C(X)$ and on the unitaries corresponding to elements of $\mathbb{Z}$.

Let $X = \mathbb{T}$ and let $\phi : \mathbb{T} \to \mathbb{T}$ be rotation by $2\pi\alpha$ radians, so that $C(X) \rtimes_\phi \mathbb{Z}$ is the rotation algebra/noncommutative $2$-torus generated by unitaries $U$ (corresponding to the Fourier mode $(z \mapsto z) \in C(X)$) and $V$ (corresponding to the automorphism $\phi$) satisfying the commutation relation $$VU = e^{-2\pi i \alpha}UV.$$ Then the flip automorphism $\theta : C(X) \rtimes_\phi \mathbb{Z} \to C(X) \rtimes_\phi \mathbb{Z}$ defined on generators by $$\theta(U) := U^\ast, \quad \theta(V) := V^\ast$$ doesn’t fix $U \in C(X) \subset C(X) \rtimes_\phi \mathbb{Z}$. Indeed, $\theta$ restricts to the $\ast$-automorphism of $C(X)$ corresponding to reflection in the real axis on $X = \mathbb{T} \subset \mathbb{C}$.

Answer 2

Not necessarily. If $\phi=\operatorname{id}_X:X\to X$, then $C(X)\rtimes_\phi\mathbb Z\cong C(X)\otimes C(\mathbb T)$. Now let $\psi$ be a non-trivial homeomorphism of $X$ (i.e. $\psi\neq\operatorname{id}_X$). Then $C(X)\subset C(X)\otimes C(\mathbb T)$ is not in the fixed point set of the automorphism $\theta=\psi\otimes 1\in\operatorname{Aut}(C(X)\otimes C(\mathbb T))$.