In the first and fourth authors' paper in 2017, it was shown that there exists a BSE-algebra of type I isomorphic to no C*-algebras, which solved negatively a question posed by the fourth author and O. Hatori. However, this result suggests a further investigation of commutative Banach algebra of type I. In the first part of the paper, we classify type I algebras into six families by means of BSE, BED, and Tauberian. It is shown that a Banach algebra of type I is isomorphic to a Segal algebra in some commutative C*-algebra if and only if it is Tauberian. In the second part, we give concrete examples of type I algebras to show that all of six families mentioned above are nonempty.

Let $\mathbb{P}^1(\overline{\mathbb{Q}})$ be the projective line over $\overline{\mathbb{Q}}$ and $H$ the Weil height on $\mathbb{P}^1(\overline{\mathbb{Q}})$. A classical result in algebraic number theory, so called Kronecker's theorem, states that $H(1,x)=1$ if and only if $x\in\overline{\mathbb{Q}}$ is 0 or a root of unity. In [4], Talamanca introduced some height functions on $M_n(\overline{\mathbb{Q}})$. The purpose of this paper is to show analogues of Kronecker's theorem for these heights: We determine height one matrices relative to these heights.

Tanabe [5] gives a geometrical proof of the existence of $\lim\frac{p(1,t)-1}{t}$ for a natural norm $p$ on ${\mathbb R}^2$. Following his idea, a shorter proof is given.