Order Isomorphisms of Operator Intervals in von Neumann Algebras

Abstract

We give a complete description of order isomorphisms between operator intervals in general von Neumann algebras. For the description, we use Jordan * -isomorphisms and locally measurable operators. Our results generalize several works by L. Molnár and P.Šemrl on type I factors.2010 Mathematics Subject Classification. Primary 47B49, Secondary 46B40, 46L10.

Transformations preserving the norm of means between positive cones of general and commutative $C^*$-algebras

“…For the noncommutative case, we have also useful results from [20] and [21] in relation with the algebra B(H) of all bounded linear operators on a Hilbert space H. It was proved in Theorem 1 in [20] that, if dim H ≥ 2, every order isomorphism of B(H) + is of the form A → CAC * with some invertible bounded linear or conjugate linear operator C on H. In Theorem 1 in [21] the same conclusion was deduced for the order isomorphisms of B(H) −1 + . Recently, in [27], Mori has substantially extended those results (and also the results of Šemrl in [31] which describe all order isomorphisms of general operator intervals in B(H)) to the setting of von Neumann algebras. Proposition 2.12 (Mori [27,Theorem 4.3]).…”

“…Recently, in [27], Mori has substantially extended those results (and also the results of Šemrl in [31] which describe all order isomorphisms of general operator intervals in B(H)) to the setting of von Neumann algebras. Proposition 2.12 (Mori [27,Theorem 4.3]). Let M be a von Neumann algebra without commutative direct summand.…”

Ring isomorphisms of type II∞$_\infty$ locally measurable operator algebras

“…Proof The implication (2)$$\Rightarrow$$(3) is clear. (1)$$\Rightarrow$$(2) What follows is a reproduction of part of [10, Proof of Lemma 2.2]. By decomposing $$M$$ into a direct sum, we may assume that $$M$$ is of one of the three types I, II or III.…”

“…It follows that $$\chi _{(0, 1/n]}(a) \succ p$$ for every $$n\geqslant 1$$. By [10, Lemma 2.3], there exist a strictly decreasing sequence $$(c_n)_{n\geqslant 1}$$ of positive real numbers and a sequence $$(p_n)_{n\geqslant 1}$$ of projections in $$M$$ such that $$c_n\rightarrow 0\,\,(n\rightarrow \infty )$$, $$p_n\leqslant \chi _{(c_{n+1}, c_n]}(a)$$, $$p_n\prec p$$ and $$\tau (p_n)\geqslant 1-3^{-n}$$, $$n\geqslant 1$$. Take a projection $$\tilde{p}_n\in {\mathcal {P}}(M)$$ such that $$p_n\sim \tilde{p}_n \leqslan...$…”

Order isomorphisms on order intervals of atomic JBW-algebras

“…Another major contribution in this research area was made by Mori in [Mor19], who mostly considered order isomorphisms between the effect algebra of (not necessarily atomic) von Neumann algebras without type I 1 direct summand. He showed that the image of 1 2 I is always locally measurably invertible, a concept from the theory of non-commutative integration.…”