By Benz W.

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We have obtained a contradiction. We will now show that an elementary j-algebra contains no proper j-ideals. iR c H, whence it follows that H = G. It follows from Lemma 1 below that the only j-algebras that do not have j-ideals are elementary. Note that the commutator of an elementary j-algebra G is equal to R + Z, while the orthogonal complernent to it is jR [in the sense of the scalar product k(x, y) = Re hex, y)]. Thus, in an elementary j-algebra the dimension of the orthogonal complement to the commutator is equal to 1.

In this section we will deal primarily with the bounded holomorphic hulls of Lie-group orbits. We will first consider this situation for the case of the unit disk Izl < 1. Here there are three types of one-parameter subgroups: hyperbolic, parabolic, and compact subgroups. , coincides with the orbit itself. The bounded holomorphic hull of orbits of a parabolic subgroup is a set of the form Imz > b, where b is a positive constant. A compact subgroup is conjugate to the following group of transformations of the disk Izl < 1: (4) 42 THE GEOMETRY OF CLASSICAL DOMAINS The orbits of this group have the form Izl = r, where 0 ~ r < 1.

Bounded H%l1lorphic Hulls Let f» be a bounded domain in C" and let X be some set in f». The bounded holomorphic hull O( X) is the set of all Z E f» such that /¢(z)/ ~ sup/¢(z)/ (1) zeX where ¢(z) is any function that is regular and bounded in f». In this section we will deal primarily with the bounded holomorphic hulls of Lie-group orbits. We will first consider this situation for the case of the unit disk Izl < 1. Here there are three types of one-parameter subgroups: hyperbolic, parabolic, and compact subgroups.

### A Beckman-Quarles type theorem for finite desarguesian planes by Benz W.

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