Exercise 8, § 6

Let $(A,\mathscr{B})$ be a natural basal algebra such that A is of finite rank.

a) For every $x\in A$, the element

is central.

In the remainder of this exercise, suppose in addition that $(A,\mathscr{B})$ is unital.

b) For all $b,b'$ in $\mathscr{B}$, there exist $b_1$ and $b_2$ in $\mathscr{B}$ such that the coefficient of $b'$ in $bb_1$ and in $b_2b$ is nonzero.

c) For every $x\in \mathscr{B}$, the matrix representing in the basis $\mathscr{B}$ the mapping $y\mapsto xy$ from A to A is a real matrix all of whose coefficients are positive. We denote by pf the unique homomorphism from A to $\mathbf{R}$ such that pf($b$) is the spectral radius of this matrix when $b\in \mathscr{B}$.

d) For every $b\in \mathscr{B}$, the real number $\alpha =$ pf($b$) is an algebraic integer such that $\alpha \geqslant 1$; for every conjugate $\alpha '\in \mathbf{C}$ of $\alpha$, one has $|\alpha '|\leqslant \alpha$.

e) Let $a_0\in A$ be the sum of the elements of $\mathscr{B}$. All the coefficients of the matrix representing the mapping $f:y\mapsto ya_0$ in the basis $\mathscr{B}$ are strictly positive. There exists an element $r\in \mathbf{C}\otimes A$ whose all coefficients in the basis $(1\otimes b)_{b\in\mathscr{B}}$ are strictly positive such that $f_{\mathbf{C}}(r) =\varrho (f_{\mathbf{C}})r$.

f) For every $x\in A$, there exists a unique complex number $\delta (x)$ such that $xr=$ $\delta (x)r$ in $\mathbf{C}\otimes A$. The mapping $\delta : A\rightarrow \mathbf{C}$ is a character of A.

g) One has $\delta (x) =$ pf($x$) for every $x\in A$. In particular, the mapping $x\mapsto$ pf($x$) is a character of A.

h) The mapping pf is the unique character of A such that pf($b$)$\geqslant 0$ for every $b\in \mathscr{B}$; moreover, one has pf($b$)$>0$ for every $b\in \mathscr{B}$.

i) For every $y\in A$, one has $ry=\delta (y)r$ in $\mathbf{C}\otimes A$.

j) The element

$r_0=\sum_{b\in\mathscr{B}}$ pf($b$)$b$

has the properties requested in question i).