On Some Automorphisms of the Set of Effects on Hilbert Space

The set of all effects on a Hilbert space has an affine structure (it is a convex set) as well as a multiplicative structure (it can be equipped with the so-called Jordan triple product). In this paper we describe the corresponding automorphisms of that set. Let H be a complex Hilbert space. Denote by B(H) the C∗-algebra of all bounded linear operators on H. The operator interval [0, I] of all positive operators in B(H) which are bounded by the identity I is called the Hilbert space effect algebra. This has important applications in quantum mechanics. The effect algebra [0, I] can be equipped with several algebraic operations. For example, one can define a partial addition on it. Namely, if A,B ∈ [0, I] and A + B ∈ [0, I], then one can set A ⊕ B = A + B. This structure has been investigated in several papers (see [3, 5, 6] and the references therein). Moreover, on [0, I] there is a natural partial ordering ≤ which comes from the usual ordering between the self-adjoint operators on H and one can also define the operation of the so-called orthocomplementation by ⊥ : A 7→ I − A. The set of all effects on H equipped with this ordering and orthocomplementation has been studied for example in [4]. Next, [0, I] is clearly a convex subset of the linear space B(H). So, one can consider the operation of convex combinations on it. The set of all effects with this structure has been investigated in [6], for instance. Finally, as for a mutliplicative operation on [0, I], note that in general A,B ∈ [0, I] does not imply that AB ∈ [0, I]. However, we all the time have ABA ∈ [0, I]. This multiplication which is a nonassociative operation and sometimes called Jordan triple product also appears in infinite dimensional holomorphy as well as in connection with the geometrical properties of C∗-algebras. Because of the importance of effect algebras, it is a natural problem to study the isomorphisms of the mentioned structures. The aim of this paper is to contribute to these investigations. Date: February 8, 2008. 1991 Mathematics Subject Classification. Primary: 81Q10, 47N50, 46N50.