  First, we consider some fundamental properties including dual spaces, complex
interpolations of $\alpha$-modulation spaces $M^{s,\alpha}_{p,q}$ with $0<p,q
\le \infty$. Next, necessary and sufficient conditions for the scaling property
and the inclusions between $\alpha_1$-modulation and $\alpha_2$-modulation
spaces are obtained. Finally, we give some criteria for $\alpha$-modulation
spaces constituting multiplication algebra. As a by-product, we show that there
exists an $\alpha$-modulation space which is not an interpolation space between
modulation and Besov spaces.
