In this article, we investigate a unique prime factorization property for infinite tensor product factors. We provide several examples of type II and III factors which satisfy this property, ...including all free product factors with diffuse free product components. In the type III setting, this is the first classification result for infinite tensor product non-amenable factors. Our proof is based on Popa's intertwining techniques and a characterization of relative amenability on the continuous cores.
In order to avoid the formation of defects in additive manufacturing (AM) by powder bed fusion (PBF) process, it is crucial to understand the relationship between the quality of powder bed and a ...powder spreading process. In this study, the influences of conditions of powder raking process on the densities and homogeneity of powder bed have been examined by computer simulation using Discrete Element Method (DEM) and experiment of powder bed formation using a blade-type spreader for Ti-6Al-4V powder by way of example. The results were analyzed with a special focus on the effects of the relative size of powder particles with respect to the gap between the blade and the build platform. It has been clearly shown that the gap needs to be larger than the upper bound of the distribution of powder particles diameter to obtain a high-density powder bed. The DEM simulation indicated that the blade can sweep even powder particles that are not in direct contact with the blade when the powder particles are in a close-packed tetragonal configuration. This is the probable reason for the experimental fact that powder particles smaller than half of the gap are suitable for PBF.
Fires can cause serious damage when widespread. This study focuses on two types of fires, forest and factory fires, and investigates how toxic substances released affect the environment. Forest ...fires, caused by anthropogenic factors such as arson and natural factors such as lightning strikes, are widespread and severely damage ecosystems. Factory fires, caused by malfunctioning electrical equipment, mishandling of chemicals, or carelessness with fire, cause serious damage to the neighborhoods and environment. In addition, fires may release chemical substances present in factories. Mercury release is a common phenomenon in both fires. Fires burn mercury-containing vegetation and organic matter, thereby releasing mercury into the atmosphere. The released mercury is deposited in the soil, rivers, and other water bodies, accumulates in ecosystems (bioaccumulation), and moves up the food chain to higher organisms. Mercury poisoning can adversely affect the nervous and immune systems, causing developmental problems and impaired learning abilities, particularly in fetuses and young children. To reduce the damage caused owing to mercury release, fire prevention, early response, prevention of mercury spillage, and the proper management and storage of mercury are important. Currently, the international community has started working on the proper handling of mercury among nations by concluding the Minamata Convention.
We investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras
M
=
B
⋊
Γ
arising from arbitrary actions
Γ
↷
B
of bi-exact discrete groups (e.g. free groups) on ...amenable von Neumann algebras. We prove a spectral gap rigidity result for the central sequence algebra
N
′
∩
M
ω
of any nonamenable von Neumann subalgebra with normal expectation
N
⊂
M
. We use this result to show that for any strongly ergodic essentially free nonsingular action
Γ
↷
(
X
,
μ
)
of any bi-exact countable discrete group on a standard probability space, the corresponding group measure space factor
L
∞
(
X
)
⋊
Γ
has no nontrivial central sequence. Using recent results of Boutonnet et al. (Local spectral gap in simple Lie groups and applications,
2015
), we construct, for every
0
<
λ
≤
1
, a type
III
λ
strongly ergodic essentially free nonsingular action
F
∞
↷
(
X
λ
,
μ
λ
)
of the free group
F
∞
on a standard probability space so that the corresponding group measure space type
III
λ
factor
L
∞
(
X
λ
,
μ
λ
)
⋊
F
∞
has no nontrivial central sequence by our main result. In particular, we obtain the first examples of group measure space type
III
factors with no nontrivial central sequence.
Let $M$ be a $\text{II}_{1}$ factor and let ${\mathcal{F}}(M)$ denote the fundamental group of $M$. In this article, we study the following property of $M$: for any $\text{II}_{1}$ factor $B$, we ...have ${\mathcal{F}}(M\,\overline{\otimes }\,B)={\mathcal{F}}(M){\mathcal{F}}(B)$. We prove that for any subgroup $G\leqslant \mathbb{R}_{+}^{\ast }$ which is realized as a fundamental group of a $\text{II}_{1}$ factor, there exists a $\text{II}_{1}$ factor $M$ which satisfies this property and whose fundamental group is $G$. Using this, we deduce that if $G,H\leqslant \mathbb{R}_{+}^{\ast }$ are realized as fundamental groups of $\text{II}_{1}$ factors, then so are groups $G\cdot H$ and $G\cap H$.
Let (HR,Ut) be any strongly continuous orthogonal representation of R on a real (separable) Hilbert space HR. For any q∈(−1,1), we denote by Γq(HR,Ut)′′ the q‐deformed Araki–Woods algebra introduced ...by Shlyakhtenko and Hiai. In this paper, we prove that Γq(HR,Ut)′′ has trivial bicentralizer if it is a type III1 factor. In particular, we obtain that Γq(HR,Ut)′′ always admits a maximal abelian subalgebra that is the range of a faithful normal conditional expectation. Moreover, using Śniady's work, we derive that Γq(HR,Ut)′′ is a full factor provided that the weakly mixing part of (HR,Ut) is nonzero.
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