We show the equivalence of a certain property of topological dynamical systems $\Sigma=(X,G)$ and a particular structure of ideals in the corresponding crossed product $A(\Sigma)=C(X)\underset{\alpha}{\rtimes}G$ where $X$ is a compact set and $G$ is a discrete group. As an application, we give a complete characterization for $A(\Sigma)$ to be simple.

We study the blowing-up conditions of solutions for nonlinear Schrödinger equations with interaction which does not satisfy known Glassey's condition [4]. We also give some remarks on the blowing-up conditions on an exterior domain with a star-shaped complement under the Dirichlet boundary condition and on a complement of a ball under the Neumann boundary condition. Finally, we show global existence of solutions for the equation: $i\dfrac{\partial u}{\partial t}=\Delta u+\left(\dfrac{1}{|x|^2}*|u|^2\right)u$.

Let $\theta$ be an irrational number and $A_\theta$ be the corresponding irrational rotation $C^*$-algebra. Let $\mathbf{K}$ be the $C^*$-algebra of all compact operators on a countably infinite dimensional Hilbert space $H$. Let $\alpha$ be an automorphism of $A_{\theta}\otimes\mathbf{K}$ with $\alpha_*=\mathrm{id}$ on $K_0(A_{\theta}\otimes\mathbf{K})$. If the set of invertible elements in $A_\theta$ is dense in $A_\theta$ or $\alpha$ preserves the canonical dense $*$-subalgebra $F^{\infty}(A_{\theta}\otimes\mathbf{K})$ of $A_{\theta}\otimes\mathbf{K}$, then there are an automorphism $\beta$ of $A_\theta$ and unitary elements $w$ in the double centralizer $M(A_{\theta}\otimes\mathbf{K})$ of $A_{\theta}\otimes\mathbf{K}$ and $W$ in $\mathbf{B}(H)$ such that $\alpha=\mathrm{Ad}(w)\circ(\beta\otimes\mathrm{Ad}(W))$.