### The basic construction from the conditional expectation on the quantum double of a finite group

Let $G$ be a finite group and $H$ a subgroup. Denote by $D(G;H)$ (or $D\left(G\right)$) the crossed product of $C\left(G\right)$ and $\u2102H$ (or $\u2102G$) with respect to the adjoint action of the latter on the former. Consider the algebra $\langle D\left(G\right),e\rangle $ generated by $D\left(G\right)$ and $e$, where we regard $E$ as an idempotent operator $e$ on $D\left(G\right)$ for a certain conditional expectation $E$ of $D\left(G\right)$ onto $D(G;H)$. Let us call $\langle D\left(G\right),e\rangle $ the basic construction from the conditional expectation $E:D\left(G\right)\to D(G;H)$. The paper constructs a crossed product algebra $C(G/H\times G)\u22ca\u2102G$, and proves that there is an algebra isomorphism between $\langle D\left(G\right),e\rangle $ and $C(G/H\times G)\u22ca\u2102G$.