A relationship between two almost f-algebra products

Let A be an uniformly complete almost f-algebra. Then \( \Pi(A) = \{ab: a,b \in A\} \) is a positively generated ordered vector subspace of A with \( \Sigma (A) = \{a^2 : a \in A\} \) as a positive cone. If \( T : \Pi(A) \rightarrow A \) is a positive linear operator, we put \( \rho : A \rightarrow {\cal L}_b(A) \) the linear operator defined by \( \rho(a) = T_a \) with \( T_a(b) = T(ab) \) for all \( b \in A ({\cal L}_b(A) \) is the algebra of all order bounded linear operators of A). Let \( {\cal L}^T_b(A) \) denote the range of \( \rho \) and let's define a new product \( * \) by putting \( a * b = T(ab) \) for all \( a, b \in A \). It is easily checked that if \( a \wedge b = 0 \) then \( a * b = T(ab) = 0 \), this shows that if it happens that the product \( * \) is associative then A is an almost f- algebra with respect to this new product. It turns out that a necessarily and sufficient condition in order that \( * \) be an associative product is that \( {\cal L}^T_b(A) \) is a commutative subalgebra of \( {\cal L}_b(A) \). We find necessarily and sufficient conditions on T in order that \( * \) is an almost f-algebra (respect.; d-algebra, f-algebra) product. Such conditions are described in terms of the algebraic and order structure of the algebra \( {\cal L}^T_b(A) \).¶The converse problem is also studied. More precisely, let A be an uniformly complete almost f-algebra and assume that \( * \) is another almost f-algebra product on A. The aim is to find sufficient conditions in order that there exist \( T:\Pi(A)\rightarrow A \) such that \( a*b=T(ab) \) for all \( a,b\in A \). It will be showed that a sufficient condition is that A is a d-algebra with respect to the initial product. An example is produced which shows that the condition "A is a d-algebra with respect to the initial product" can not be weakened.

Authors and Affiliations

1. Département des classes préparatoires, Institut Préparatoire aux Etudes, Scientifiques et Techniques, BP 51, 2070-La Marsa, Tunisia, e-mail: Karim.Boulabiar@ipest.rnu.tn, , , , , , TN

   K. Boulabiar

Received November 8, 1999; accepted in final form February 14, 2000.

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