Completely order bounded maps on non-commutative $${\varvec{L_p}}$$-spaces

We define norms on $$L_p({\mathcal {M}}) \otimes M_n$$ L p ( M ) ⊗ M n where $${\mathcal {M}}$$ M is a von Neumann algebra and $$M_n$$ M n is the space of complex $$n \times n$$ n × n matrices. We show that a linear map $$T: L_p({\mathcal {M}}) \rightarrow L_q({\mathcal {N}})$$ T : L p ( M ) → L q ( N ) is decomposable if $${\mathcal {N}}$$ N is an injective von Neumann algebra, the maps $$T \otimes Id_{M_n}$$ T ⊗ I d M n have a common upper bound with respect to our defined norms, and $$p = \infty $$ p = ∞ or $$q = 1$$ q = 1 . For $$2p< q < \infty $$ 2 p < q < ∞ we give an example of a map $$T$$ T with uniformly bounded maps $$T \otimes Id_{M_n}$$ T ⊗ I d M n which is not decomposable.

Von Neumann algebras as complemented subspaces of $B({\mathscr{H}})$

Let [Formula: see text] be a von Neumann algebra of type II1 which is also a complemented subspace of [Formula: see text]. We establish an algebraic criterion, which ensures that [Formula: see text] is an injective von Neumann algebra. As a corollary we show that if [Formula: see text] is a complemented factor of type II1 on a Hilbert space [Formula: see text], then [Formula: see text] is injective if its fundamental group is nontrivial.

Further results on the relative entropy

Given any subset ℬ, containing the identity (1), of ℬ (ℋ) (the bounded operators on some Hilbert space ℋ), and given two states σ and ρ on ℬ(ℋ), a definition was given in [3] of entℬ (σℬ|ρ|ℬ) - ‘the entropy of σ relative to ρ given the information in ℬ’. It was shown that, for ℬ an injective von Neumann algebra, the resulting relative entropy agreed with those of Umegaki, Araki, Pusz and Woronowicz, and Uhlmann. The purpose of this paper is to explore this definition further. After some technical preliminaries in Section 2, in Section 3 a new characterization of entℬ(ℋ) (σ|ρ) for σ and ρ normal states will be given. In Section 4 it will be shown that under fairly general circumstances the relative entropy on algebras can be used for statistical inference. This is important for applications of the relative entropy. I shall given the briefest sketches of how I see these applications being made in the measurement problem in quantum theory and in a ‘many worlds’ interpretation. The vigilant reader will notice that the scheme proposed in Section 4 for modelling measurements subject to given compatibility requirements differs slightly from that proposed in the introduction to [3]. The reason for this is outlined in Section 5, where an explicit computation is made of the relative entropy for the simplest non-trivial case in which ℬ is not an algebra; when ℬ = {1, P, Q} for P and Q projections subject to certain conditions.

On Approximately Finite-Dimensional Von Neuman Algebras, II

An intrinsic characterization is given of those von Neumann algebras which are injective objects in the category of C*-algebras with completely positive maps. For countably generated von Neumann algebras several such characterizations have been given, so it is in fact enough to observe that an injective von Neumann algebra is generated by an upward directed collection of injective countably generated sub von Neumann algebras. The present work also shows that three of the intrinsic characterizations known in the countably generated case hold in general.