Let \((H_{\mathbf{R}}, U_t)\) be any strongly continuous orthogonal representation of \(\mathbf{R}\) on a real (separable) Hilbert space \(H_{\mathbf{R}}\). For any \(q\in (-1,1)\), we denote by ...\(\Gamma_q(H_{\mathbf{R}},U_t)^{\prime\prime}\) the \(q\)-deformed Araki-Woods algebra introduced by Shlyakhtenko and Hiai. In this paper, we prove that \(\Gamma_q(H_{\mathbf{R}},U_t)^{\prime\prime}\) has trivial bicentralizer if it is a type \(\rm III_1\) factor. In particular, we obtain that \(\Gamma_q(H_{\mathbf{R}},U_t)^{\prime\prime}\) always admits a maximal abelian subalgebra that is the range of a faithful normal conditional expectation. Moreover, using Sniady's work, we derive that \(\Gamma_q(H_{\mathbf{R}},U_t)^{\prime\prime}\) is a full factor provided that the weakly mixing part of \((H_{\mathbf{R}}, U_t)\) is nonzero.
Let $M$ be a $\rm II_1$ factor and let $\mathcal{F}(M)$ denote the
fundamental group of $M$. In this article, we study the following property of
$M$: for arbitrary $\rm II_1$ factor $B$, we have ...$\mathcal{F}(M
\overline{\otimes} B)=\mathcal{F}(M)\mathcal{F}(B)$. We prove that for any
subgroup $G\leq \mathbb{R}^*_+$ which is realized as a fundamental group of a
$\rm II_1$ factor, there exists a $\rm II_1$ factor $M$ which satisfies this
property and whose fundamental group is $G$. Using this, we deduce that if $G,H
\leq \mathbb{R}^*_+$ are realized as fundamental groups of $\rm II_1$ factors
(with separable predual), then so are groups $G \cdot H$ and $G \cap H$.
We obtain several rigidity results regarding tensor product decompositions of factors. First, we show that any full factor with separable predual has at most countably many tensor product ...decompositions up to stable unitary conjugacy. We use this to show that the class of separable full factors with countable fundamental group is stable under tensor products. Next, we obtain new primeness and unique prime factorization results for crossed products coming from compact actions of higher rank lattices (e.g.\ \(\mathrm{SL}(n,\mathbb{Z}), \: n \geq 3\)) and noncommutative Bernoulli shifts with arbitrary base (not necessarily amenable). Finally, we provide examples of full factors without any prime factorization.
Let \(M\) be a \(\rm II_1\) factor and let \(\mathcal{F}(M)\) denote the fundamental group of \(M\). In this article, we study the following property of \(M\): for arbitrary \(\rm II_1\) factor ...\(B\), we have \(\mathcal{F}(M \overline{\otimes} B)=\mathcal{F}(M)\mathcal{F}(B)\). We prove that for any subgroup \(G\leq \mathbb{R}^*_+\) which is realized as a fundamental group of a \(\rm II_1\) factor, there exists a \(\rm II_1\) factor \(M\) which satisfies this property and whose fundamental group is \(G\). Using this, we deduce that if \(G,H \leq \mathbb{R}^*_+\) are realized as fundamental groups of \(\rm II_1\) factors (with separable predual), then so are groups \(G \cdot H\) and \(G \cap H\).
We investigate factoriality, Connes' type \({\rm III}\) invariants and fullness of arbitrary amalgamated free product von Neumann algebras using Popa's deformation/rigidity theory. Among other ...things, we generalize many previous structural results on amalgamated free product von Neumann algebras and we obtain new examples of full amalgamated free product factors for which we can explicitely compute Connes' type \({\rm III}\) invariants.
In this note, we introduce and study a notion of bi-exactness for creation operators acting on full, symmetric and anti-symmetric Fock spaces. This is a generalization of our previous work, in which ...we studied the case of anti-symmetric Fock spaces. As a result, we obtain new examples of solid actions as well as new proofs for some known solid actions. We also study free wreath product groups in the same context.
Inductive proximity sensors are noncontact sensing devices used to detect the approach of a target by an increase in coil resistance due to eddy current loss. Extending the operating distance of ...these sensors is demanded. In this paper, we propose the use of a magnetoplated wire (MPW) as a sensing coil. The MPW is a copper wire, whose circumference is plated with a magnetic thin film. We analyze the impedance of a proximity sensor using a copper wire (COW) and MPW coils by a finite element method. The use of the MPW results in a decrease in AC resistance due to the proximity effect, an increase in inductance, and the generation of a higher flux than when the COW is used. Therefore, it is possible to increase the quality factor Q of the MPW coil. As a result, the operating distances of the MPW and COW coils are 5.0 and 3.8 mm, respectively. The operating distance of the MPW coil is 1.3-fold that of the COW coil.
The classical Gaussian functor associates to every orthogonal representation of a locally compact group \(G\) a probability measure preserving action of \(G\) called a Gaussian action. In this paper, ...we generalize this construction by associating to every affine isometric action of \(G\) on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine isometric actions. We use the Patterson-Sullivan theory as well as Lyons-Pemantle work on tree-indexed random walks in order to give a precise description of this phase transition for affine isometric actions of groups acting on trees. We also show that every locally compact group without property (T) admits a nonsingular Gaussian that is free, weakly mixing and of stable type \(\mathrm{III}_1\).
Let \( G \) be a countable discrete group and consider a nonsingular Bernoulli shift action \( G \curvearrowright \prod_{g\in G }(\{0,1\},\mu_g)\) with two base points. When \( G \) is exact, under a ...certain finiteness assumption on the measures \(\{\mu_g\}_{g\in G }\), we construct a boundary for the Bernoulli crossed product C\(^*\)-algebra that admits some commutativity and amenability in the sense of Ozawa's bi-exactness. As a consequence, we obtain that any such Bernoulli action is solid. This generalizes solidity of measure preserving Bernoulli actions by Ozawa and Chifan--Ioana, and is the first rigidity result in the non measure preserving case. For the proof, we use anti-symmetric Fock spaces and left creation operators to construct the boundary and therefore the assumption of having two base points is crucial.
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