Automorphisms of spectral lattices of unbounded positive operators

Abstract

We introduce and study spectral order on unbounded operators. Main result of this note characterizes spectral order automorphism of the lattice of positive (possibly unbounded) self-adjoint operators.

Automorphisms of spectral lattices of positive contractions on von Neumann algebras

“…Content of Molnar-Šemrl theorem is that any bijective transformation of positive contractions on a Hilbert space that preserves spectral order in both direction is a composition of the above mentioned two transformations. In [14] we generalized this result to unbounded operators. As a consequence, any spectral lattice automorphism of Hilbert space positive contractions automatically preserves scalar operators.…”

“…In our initial paper [14] we characterized preservers of positive (possibly unbounded) operators acting on a Hilbert space. In the present paper we focused on spectral order preservers in general context of von Neumann algebras rather than in setting of particular full algebras of operators acting on Hilbert spaces.…”

Spectral order isomorphisms and AW$AW$*‐factors

“…The spectral order has been especially studied on von Neumann algebras (see, for example, [3, 8] and references therein). Nevertheless, it has been considered also in contexts of $$AW$$*‐algebra, unbounded operators, and further structures [7, 10, 12, 20, 24]. From the physical point of view, the spectral order is significant in a categorical formulation of quantum theory [4, 9, 25].…”

“…positive elements in the unit ball) in $$AW$$*‐factors of Type I has the form $$\varphi (x)=\Theta _\tau (f(x))$$, where $$f:[0,1]\rightarrow [0,1]$$ is a strictly increasing bijection, τ is an isomorphism between projection lattices, and $$\Theta _\tau$$ is defined by $$E^{\Theta _\tau (x)}_\lambda =\tau \big (E^{x}_\lambda \big )$$ for all $$\lambda \in \mathbb {R}$$. Let us remark that isomorphisms of the above form were studied for the first time by Turilova in [24]. As each $$AW$$*‐factor of Type I is precisely a von Neumann algebra of all bounded operators on a Hilbert space, our result is a direct extension of a complete description of spectral order automorphisms of the set of all effects on a Hilbert space found by Molnár, Nagy, and Šemrl [16, 17].…”

Multidimensional spectral order for selfadjoint operators

“…For instance, automorphisms of partially ordered set (E(H), ), where E(H) denote the set of all positive bounded operators in the closed unit ball of B(H), were described in [22,21]. In turn, automorphisms of subsets of positive selfadjoint operators were investigated in [30,17]. Spectral order is also an important ingredient of the topos formulation of quantum theory (see for example [9,10,31,12]).…”