Let $ A_\theta$ be an irrational rotation algebra and $B_\theta$ be the AF-algebra defined by Effros and Shen. Then we have monomorphisms of $A_\theta$ into $B_\theta$. In the present paper we will show that there are automorphisms of $A_\theta$ which can not be extended to any automorphism of $B_\theta$ for any monomorphism of $A_\theta$ into $ B_\theta$.

In recent years, the theory of Sobolev spaces over the abstract Wiener space was constructed in the study of stochastic differential equations ([9, 11, 15]). In the present paper, we prove an interpolation theorem on Sobolev spaces over the abstract Wiener space.

Let $B$ be a unital $C^*$-algebra which is strongly Morita equivalent to an irrational rotation $C^*$-algebra. Then Rieffel showed that it is isomorphic to $A_\theta\otimes M_n$ where $A_\theta$ is an irrational rotation $C^*$-algebra and $M_n$ is the $n\times n$ matrix algebra over $C$. In the present paper we will show that for any automorphism $\alpha$ of $A_\theta\otimes M_n$ there are unitary elements $w\in A_\theta\otimes M_n$, $W\in M_n$ and an automorphism $\beta$ of $ A_\theta$ such that $\alpha=\mathrm{Ad}(w)\circ(\beta\otimes\mathrm{Ad}(W))$.

In this paper we introduce a new partial order $\leqq$ on the set of all classical knots. We show for example that every nontrivial knot $\geqq$ the trefoil knot.