L. J. Bunce

The Mackey-Gleason problem

Let A be a von Neumann algebra with no direct summand of Type I 2 , and let P(A) be its lattice of projections. Let X be a Banach space. Let m:P(A)→X be a bounded function such that m(p+q)=m(p)+m(q) whenever p and q are orthogonal projections. The main theorem states that m has a unique extension to a bounded linear operator from A to X. In particular, each bounded complex-valued finitely additive quantum measure on P(A) has a unique extension to a bounded linear functional on A

The alternative Dunford-Pettis property in C*-algebras and von Neumann preduals

A Banach space X is said to have the alternative Dunford-Pettis property if, whenever a sequence x n → x weakly in X with ||x n || → ||x||, we have p n (x n )→ 0 for each weakly null sequence (ρ n ) in X*. We show that a C*-algebra has the alternative Dunford-Pettis property if and only if every one of its irreducible representations is finite dimensional so that, for C*-algebras, the alternative and the usual Dunford-Pettis properties coincide as was conjectured by Freedman. We further show that the predual of a von Neumann algebra has the alternative Dunford-Pettis property if and only if the von Neumann algebra is of type I.

Images of contractive projections on operator algebras

It is shown that if P is a weak ∗ -continuous contractive projection on a JBW ∗ -triple M, then P( M)is of type I or semifinite, respectively, if M is of the corresponding type. We also show that P( M)has no infinite spin part if M is a type I von Neumann algebra.

Real contractive projections on commutative C\(^*\)-algebras

Complex (Banach) Jordan triples occurred in the study of bounded symmetric domains in several complex variables and in the study of contractive projections on (complex) C∗-algebras. These spaces are equipped with a ternary product, the Jordan triple product, and are essentially geometric objects in that the linear isometries between them are exactly the linear maps preserving the triple product [13].

Extension of Jauch-Piron states on Jordan algebras

A state ρ on a JW -algebra or von Neumann algebra M is said to be a Jauch–Piron state if whenever e and f are projections in M with ρ( e ) = ρ( f ) = 0 then ρ( e ∨ f ) = 0.

The alternative Dunford-Pettis property, conjugations and real forms of C*-algebras

Let be a conjugation, alias a conjugate linear isometry of order 2, on a complex Banach space and let be the real form of of -fixed points. In contrast to the Dunford-Pettis property, the alternative Dunford-Pettis property need not lift from to . If is a C*-algebra it is shown that has the alternative Dunford-Pettis property if and only if does and an analogous result is shown when is the dual space of a C*-algebra. One consequence is that both Dunford-Pettis properties coincide on all real forms of C*-algebras.

Universally reversible JC*-triples and operator spaces

We prove that the vast majority of JC*-triples satisfy the condition of universal reversibility. Our characterisation is that a JC*-triple is universally reversible if and only if it has no triple homomorphisms onto Hilbert spaces of dimension greater than two nor onto spin factors of dimension greater than four. We establish corresponding characterisations in the cases of JW*-triples and of TROs (regarded as JC*-triples). We show that the distinct natural operator space structures on a universally reversible JC*-triple E are in bijective correspondence with a distinguished class of ideals in its universal TRO, identify the Shilov boundaries of these operator spaces and prove that E has a unique natural operator space structure precisely when E contains no ideal isometric to a nonabelian TRO. We deduce some decomposition and completely contractive properties of triple homomorphisms on TROs.

Spin spaces and positive decomposition of linear maps on ordered Banach spaces

Spin factors and generalizations are used to revisit positive generation of B(E, F), where E and F are ordered Banach spaces. Interior points of B(E, F)+ are discussed and in many cases it is seen that positive generation of B(E, F) is controlled by spin structure in F when F is a JBW-algebra.