Homomorphisms of Jordan Rings of Self-Adjoint Elements

Abstract

In a previous paper [4](¹) we have defined a special Jordan ring to be a a subset of an associative ring which is a subgroup of the additive group and which is closed under the compositions a→a ²and (a, b)→aba. Such systems are also closed under the compositions (a, b) → ab+ba= {a, b} and (a, b, c) → abc+cba. The simplest instances of special Jordan rings are the associative rings themselves. In our previous paper we studied the (Jordan) homomorphisms of these rings. These are the mappings J of associative rings such that

$$ {\left( {a + b} \right)^J} = {a^J} + {b^J},\;{\left( {{a^2}} \right)^J} = {\left( {{a^J}} \right)^2},\;{\left( {aba} \right)^J} = {a^J}{b^J}{a^J} $$

(1)

A second important class of special Jordan rings is obtained as follows. Let $H$ be an associative ring with an involution a → a ^*, that is, a mapping a→a ^* such that

$$ {\left( {a + b} \right)^*} = {a^*} + {b^*},\;{\left( {ab} \right)^*} = {b^*}{a^*},\;{a^{**}} = a $$

(2)

Let $H$ denote the set of self-adjoint elements h = h ^*. Then <Inline></Inline> is a special Jordan ring. In this paper we shall study the homomorphisms of the rings of this type. It is noteworthy that the Jordan rings of this type include those of our former paper(²).