In this paper the authors continue their study of the collineation groups of division
ring planes (The collineation groups of division ring planes I. Jordan division
algebras, J. Reine and Angew. Math. vol. 216, 1964). Some of the results obtained
for finite dimensional Jordan division algebras are extended to a special class of
infinite dimensional algebras.
As is wellknown the study of the collineation group of a projective plane
$\pi $ coordinatized
by an algebra
$\mathcal{\mathcal{R}}$
can be reduced to the stutiy of the autotopism group of
$\mathcal{\mathcal{R}}$ or the group of autotopic
collineations of
$\pi $,
$\mathcal{\mathscr{H}}\left(\pi \right)$. The
pair
$\left(a,b\right)$,
$a,b\in \mathcal{\mathcal{R}}$,
is defined to be admissible if and only if there exists an element
$\alpha $ in
$\mathcal{\mathscr{H}}\left(\pi \right)$ with
$\left(1,1\right)\alpha =\left(a,b\right)$. Modulo the
automorphism group of
$\mathcal{\mathcal{R}}$,
the determination of
$\mathcal{\mathscr{H}}\left(\pi \right)$
is equivalent to the determination of all admissible pairs
$\left(a,b\right)$ and coset
representatives
${\phi}_{a,b}\in \mathcal{\mathscr{H}}\left(\pi \right)$ such
that
$\left(1,1\right){\phi}_{a,b}=\left(a,b\right)$. With either
the assumption
$\mathcal{\mathcal{R}}$
algebraic over its center, or the assumptions characteristic of
$\mathcal{\mathcal{R}}$ not equal to 0 and
the centers of
$\mathcal{\mathcal{R}}$ and
${\mathcal{\mathcal{R}}}^{\prime}$ (the algebra of all
elements of
$\mathcal{\mathcal{R}}$ algebraic
over the center of
$\mathcal{\mathcal{R}}$) equal,
the admissible pairs
$\left(a,b\right)$
are determined. Use is made of Kleinfeld’s result on the middle
nucleus of Jordan rings (Middle nucleus = center in a simple Jordan
ring, to appear.) We also prove and use the result that the algebra
$\mathcal{\mathcal{E}}$ consisting of all right
multiplications
${R}_{f}$ is
commutative, where
$f$ is in
the subalgebra generated by
$a$
and
${a}^{1}$
over the base field.
