C. Martin Edwards

Compact tripotents in bi-dual JB*-triples 1

The set %(C)~ consisting of the partially ordered set °U(G) of tripotents in a JBW*triple C with a greatest element adjoined forms a complete lattice. This paper is mainly concerned with the situation in which C is the second dual ^4** of a complex Banach space A and, more particularly, when A is itself a JB* -triple. A subset %(A)~ of ^l(A**)~ consisting of the set %(A) of tripotents compact relative to A (denned in Section 4) with a greatest element adjoined is studied. It is shown to be an atomic complete lattice with the properties that the infimum of an arbitrary family of elements of %{A)~ is the same whether taken in %C(A)~ or in <%(A**)~ and that every decreasing net of non-zero elements of %(A)~ has a non-zero infimum. The relationship between the complete lattice %(A)~ and the complete lattice %(B)~, where B is a Banach space such that B** is a weak*-closed subtriple of A** is also investigated. When applied to the special case in which A is a C*-algebra the results provide information about the set of compact partial isometries relative to A and are closely related to those recently obtained by Akemann and Pedersen. In particular it is shown that a partial isometry is compact relative to A if and only if, in their terminology, it belongs locally to A. The main results are applied to this and other examples.

On the facial structure of the unit ball in a JB*-triple

Abstract It is shown that every norm-closed face of the closed unit ball A 1 in a JB*-triple A is norm-semi-exposed, thereby completing the description of the facial structure of A 1.

A geometric characterization of structural projections on a JBW∗-triple☆

A structural projection R on a Jordan∗-triple A is a linear projection such that, for all elements a, b and c in A, R{aRbc}={RabRc}. The L-orthogonal complement G◊ of a subset G of a complex Banach space E is the set of elements x in E such that, for all elements y in G, ||x±y||=||x||+||y||. A contractive projection P on E is said to be neutral if the condition that ||Px||=||x|| implies that the elements Px and x coincide, and is said to be a GL-projection if the L-orthogonal complement (PE)◊ of the range PE of P is contained in the kernel ker(P) of P. It is shown that, for a JBW∗-triple A, with predual A∗, a linear projection R on A is structural if and only if it is the adjoint of a neutral GL-projection on A∗, thereby giving a purely geometric characterization of structural projections.

On conditional probability in GL spaces

We investigate the notion of conditional probability and the quantum mechanical concept of state reduction in the context of GL spaces satisfying the Alfsen-Shultz condition.

Faithful Inner Ideals in Jbw*-Triples

The kernel Ker(J) and the annihilator J⊥ of a weak*-closed inner ideal J in a JBW*-triple A consist of the sets of elements a in A for which {J a J} and {J a A} are zero, respectively, and J is said to be faithful if, for every non-zero ideal I in A, I ∩ Ker (J) is non-zero. It is shown that every weak*-closed inner ideal J in A has a unique orthogonal decomposition into a faithful weak*-closed inner ideal f(J) and a weak*-closed ideal f (J)⊥ ∩ J of A. The central structure of f ( J) is investigated and used to show that J has zero annihilator if and only if it coincides with the multiplier of f (J). The results are applied to the cases in which J is the Peirce-two or Peirce-zero space A2(v) or A0(v) corresponding to a tripotent v in A, and to the case in which the JBW*-triple A is a von Neumann algebra.

Involutive and Peirce Gradings in JBW*-Triples

Abstract A Peirce grading (J 0, J 1, J 2) of a Jordan*-triple A consists of subspaces J 0, J 1 and J 2 of A, with direct sum A, which satisfy the conditions that and, for j, k, and l equal to 0, 1, or 2, if j − k + l is equal to 0, 1 or 2 then and, if not then An involutive grading (B +, B −) of A consists of a pair of subtriples of A, with direct sum A, satisfying the conditions Every Peirce grading (J 0, J 1, J 2) of A gives rise to an involutive grading (J 0 ⊕ J 2, J 1) of A. It is shown that, conversely, when A is a JBW*-triple factor and (B +, B −) is an involutive grading of A, either B + is also a JBW*-triple factor or, for each weak*-closed ideal J 0 of B +, with complementary weak*-closed ideal J 2, writing J 1 for B −, (J 0, J 1, J 2) is a Peirce grading of A.

Orthogonal pairs of weak*-closed inner ideals in a JBW*-triple

Pre-symmetric complex Banach spaces have been proposed as models for state spaces of physical systems. A neutral GL-projection on a pre-symmetric space represents an operation on the corresponding system, and has as its range a further pre-symmetric space which represents the state space of the resulting system. Two neutral GL-projections S and T on the pre-symmetric space A* are said to be L-orthogonal if for all elements x in SA* and y in TA*, By studying the algebraic properties of the dual space A of A*, which is a JBW*-triple, it is shown that, provided that the orthogonal neutral GL-projections S and T satisfy a certain geometrical condition, there exists a smallest neutral GL-projection ST majorizing both S and T, and that S, T and ST form a compatible family.

Lattice of tripotents in a JBW*-triple

The complete lattice of tripotents in a JBW*-triple and the unit ball in its predual are respectively proposed as models for the complete lattice of propositions and for the generalized normal state space of a nonassociative, noncommutative physical system. A subsystem of such a system may be defined in terms of either principal ideals in the complete lattice of propositions or norm-closed faces of the generalized state space. It is shown that the two definitions are equivalent and that each subsystem is associative.