MEASURABLE -SEMIGROUPS ARE CONTINUOUS

Let $G$ be a second countable locally compact Hausdorff topological group and $P$ be a closed subsemigroup of $G$ containing the identity element $e\in G$ . Assume that the interior of $P$ is dense in $P$ . Let $\unicode[STIX]{x1D6FC}=\{{\unicode[STIX]{x1D6FC}_{x}\}}_{x\in P}$ be a semigroup of unital normal $\ast$ -endomorphisms of a von Neumann algebra $M$ with separable predual satisfying a natural measurability hypothesis. We show that $\unicode[STIX]{x1D6FC}$ is an $E_{0}$ -semigroup over $P$ on $M$ .

BIDUAL OCTAHEDRAL RENORMINGS AND STRONG REGULARITY IN BANACH SPACES

We prove that every separable Banach space containing an isomorphic copy of $\ell _{1}$ can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.

On dynamical systems preserving weights

The canonical unitary representation of a locally compact separable group arising from an ergodic action of the group on a von Neumann algebra with separable predual preserving a faithful normal semifinite (infinite) weight is weak mixing. On the contrary, there exists a non-ergodic automorphism of a von Neumann algebra preserving a faithful normal semifinite trace such that the spectral measure and the spectral multiplicity of the induced action are respectively the Haar measure (on the unit circle) and $\infty$. Despite not even being ergodic, this automorphism has the same spectral data as that of a Bernoulli shift.

Asymptotic structure of free product von Neumann algebras

Let (M, ϕ) = (M1, ϕ1) * (M2, ϕ2) be the free product of any σ-finite von Neumann algebras endowed with any faithful normal states. We show that whenever Q ⊂ M is a von Neumann subalgebra with separable predual such that both Q and Q ∩ M1 are the ranges of faithful normal conditional expectations and such that both the intersection Q ∩ M1 and the central sequence algebra Q′ ∩ Mω are diffuse (e.g. Q is amenable), then Q must sit inside M1. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion M1 ⊂ M in arbitrary free product von Neumann algebras.

On completely singular von Neumann subalgebras

Let ℳ be a von Neumann algebra acting on a Hilbert space $\mathcal{H}$ and let $\mathcal{N}$ be a von Neumann subalgebra of ℳ. If $\mathcal{N}\operatorname{\bar{\otimes}}\mathcal{B}(\mathcal{K})$ is singular in $\mathcal{M}\operatorname{\bar{\otimes}}\mathcal{B}(\mathcal{K})$ for every Hilbert space $\mathcal{K}$, $\mathcal{N}$ is said to be completely singular in ℳ. We prove that if $\mathcal{N}$ is a singular abelian von Neumann subalgebra or if $\mathcal{N}$ is a singular subfactor of a type-II1 factor ℳ, then $\mathcal{N}$ is completely singular in ℳ. $\mathcal{H}$ is separable, we prove that $\mathcal{N}$ is completely singular in ℳ if and only if, for every θ∈Aut($\mathcal{N}$′) such that θ(X)=X for all X ∈ ℳ′, θ(Y)=Y for all Y∈$\mathcal{N}$′. As the first application, we prove that if ℳ is separable (with separable predual) and $\mathcal{N}$ is completely singular in ℳ, then $\mathcal{N}\operatorname{\bar{\otimes}}\mathcal{L}$ is completely singular in $\mathcal{M}\operatorname{\bar{\otimes}}\mathcal{L}$ for every separable von Neumann algebra $\mathcal{L}$. As the second application, we prove that if $\mathcal{N}$1 is a singular subfactor of a type-II1 factor ℳ1 and $\mathcal{N}$2 is a completely singular von Neumann subalgebra of ℳ2, then $\mathcal{N}_1\operatorname{\bar{\otimes}}\mathcal{N}_2$ is completely singular in $\mathcal{M}_1\operatorname{\bar{\otimes}}\mathcal{M}_2$.

Hochschild cohomology of II1 factors with Cartan maximal abelian subalgebras

In this paper we prove that, for a type-II1 factor N with a Cartan maximal abelian subalgebra, the Hochschild cohomology groups Hn(N,N)=0 for all n≥1. This generalizes the result of Sinclair and Smith, who proved this for all N having a separable predual.

Completely bounded isomorphisms of injective von Neumann algebras

Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaceswhere H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).

Nuclearity and Banach spaces

SummaryLet E be a nuclear space provided with a topology different from the weak topology. Let {Ai: i ∈ I} be a fundamental system of equicontinuous subsets of the topological dual E' of E. If {Fi: i ∈ I} is a family of infinite dimensional Banach spaces with separable predual, there is a fundamental system {Bi: i ∈ I} of weakly closed absolutely convex equicontinuous subsets of E'such that is norm-isomorphic to Fi, for each i ∈ I. Other results related with the one above are also given.