1. The problem is to understand the concept of the quasinormalizer of a crossed product von Neumann algebra. 2. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. 3. A crossed product von Neumann algebra arises from a von Neumann algebra $M$ and a group $G$ acting on $M$ by automorphisms, denoted $M \rtimes G$. 4. The quasinormalizer of a subalgebra $N$ in a von Neumann algebra $M$ is the set of elements $x \in M$ such that both $xN$ and $Nx$ are contained in a finite sum of $N$-modules, i.e., $xN \subseteq \sum_i N x_i$ and $Nx \subseteq \sum_j y_j N$ for some $x_i,y_j \in M$. 5. This concept generalizes the normalizer and is important in the study of inclusions and decompositions of von Neumann algebras. 6. In the context of crossed products, the quasinormalizer helps analyze how the group action intertwines with the algebra structure. 7. There is no explicit formula to compute the quasinormalizer in general; it depends on the specific algebra and group action. 8. Understanding the quasinormalizer involves operator algebra techniques, module theory, and group actions. Final answer: The quasinormalizer of a crossed product von Neumann algebra $M \rtimes G$ consists of those elements in $M \rtimes G$ that quasi-normalize the subalgebra, meaning they intertwine $M$ and the group action in a controlled finite manner.