Francesco Fidaleo
Let $M$, $N$, $R$ be $W^{*}$--algebras, with $R$ unitally embedded in both $M$ and $N$. by using Reduction Theory, we extend the previous description of the $W^{*}$--tensor product $M\bar\otimes_{R}N$ over the common $W^{*}$--subalgebra $R$ and its predual $(M\bar\otimes_{R}N)_{*}$ to the $σ$--finite case.
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We show that every free semigroup algebras has a (strongly) unique Banach space predual. We also provide a new simpler proof that a weak*-closed unital operator operator algebra containing a weak* dense subalgebra of compact operators has a unique Banach space predual.
Let M_i be a von Neumann algebra, and B_i be a maximal injective von Neumann subalgebra of M_i, i=1,2. If M_1 has separable predual and the center of B_1 is atomic, e.g., B_1 is a factor, then B_1\tensor B_2 is a maximal injective von Neuamnn subalgebra of M_1\tensor M_2. This partly answers a question of Popa
In this paper, we exhibit strongly singular maximal abelian subalgebras living inside certain k-folded tensors of von Neumann group factors. The two classes of groups under consideration are the free groups of rank greater than 2 and the free product of m>2 groups, all finite and having the same order p>2 (m>p-1). We prove actually more, namely that the unique trace preserving conditional expectations from these group factors onto their Laplacian subalgebras are asymptotic homomorphisms, which in turn implies that the subalgebras are strongly singular masas.
In this paper, we present a new class of strongly singular maximal abelian subalgebras living inside the k-folded tensor product of the free group factor (on N>1 generators). A. Sinclair and R. Smith introduced the class of strongly singular maximal abelian von Neumann algebras (MASA) of type II_1 factors. One of their examples was the Laplacian subalgebra of the free group factor, known to be a singularMASA. Using the techniques presented by A. Sinclair and R.Smith, we show that the Laplacian subalgebra of the k-folded tensor product of the free group factor with itself is a strongly singular…
Let $\M$ be a von Neumann algebra acting on a Hilbert space $\H$, and $\N$ be a singular von Neumann subalgebra of $\M.$ If $\N\tensor\B(\K)$ is singular in $\M\tensor\B(\K)$ for any Hilbert space $\K$, we say $\N$ is \emph{completely singular} in $\M$. We prove that if $\N$ is a singular abelian von Neumann subalgebra or if $\N$ is a singular subfactor of a type $II_1$ factor $\M$, then $\N$ is completely singular in $\M$. For any type $II_1$ factor $\M$, we construct a singular von Neumann subalgebra $\N$ of $\M$ ($\N\neq \M$) such that $\N\tensor\B(l^2(\mathbb{N}))$ is regular (hence not si…
We extend Ueda's peak set theorem for subdiagonal subalgebras of tracial finite von Neumann algebras, to sigma-finite von Neumann algebras (that is, von Neumann algebras with a faithful state; which includes those on a separable Hilbert space, or with separable predual.) To achieve this extension completely new strategies had to be invented at certain key points, ultimately resulting in a more operator algebraic proof of the result. Ueda showed in the case of finite von Neumann algebras that his peak set theorem is the fountainhead of many other very elegant results, like the uniqueness of the…
We prove a basic result about tensor products of a $\text{II}_1$ factor with a finite von Neumann algebra and use it to answer, affirmatively, a question asked by S. Popa about maximal injective factors.
In this article, we proved the following results. Let $L(F(n_i))$ be the free group factor on $n_i$ generators and $λ(g_{i})$ be one of standard generators of $L(F(n_i))$ for $1\le i\le N$. Let $\A_i$ be the abelian von Neumann subalgebra of $L(F(n_i))$ generated by $λ(g_{i})$. Then the abelian von Neumann subalgebra $\otimes_{i=1}^N\A_i$ is a maximal injective von Neumann subalgebra of $\otimes_{i=1}^N L(F(n_i))$. When $N$ is equal to infinity, we obtained McDuff factors that contain maximal injective abelian von Neumann subalgebras.
Let $\mathbb{G}$ be a free (unitary or orthogonal) quantum group. We prove that for any non-amenable subfactor $N\subset L^\infty(\mathbb{G})$, which is an image of a faithful normal conditional expectation, and for any $σ$-finite factor $B$, the tensor product $N \otimes B$ has no Cartan subalgebras. This generalizes our previous work that provides the same result when $B$ is finite. In the proof, we establish Ozawa--Popa and Popa--Vaes's weakly compact action on the continuous core of $N \otimes B$ as the one relative to B, by using an operator valued weight to B and the central weak amenabi…
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 arbitr…
Invariant von Neumann subalgebras of crossed products and tensor products
该文研究可数离散群保迹作用下交叉积与张量积的不变冯·诺依曼子代数，在特定假设下分类其结构，并构造了自由群的相关保测作用实例。
Let M be a factor of type III with separable predual and with normal states phi_1,...,phi_k, omega with omega faithful. Let A be a finite dimensional C*-subalgebra of M. Then it is shown that there is a unitary operator u in M such that phi_i Ad u =omega on A for i=1,...,k. This follows from an embedding result of a finite dimensional C*-algebra with a faithful state into M with finitely many given states. We also give similar embedding results of C*-algebras and von Neumann algebras with faithful states into M. Another similar result for a factor of type II_1 instead of type III holds.
Given two correspondences X and Y, we show that (under mild hypotheses) the Cuntz-Pimsner algebra of the tensor product of X and Y embeds as a certain subalgebra of the tensor product of the Cuntz-Pimsner algebra of X and the Cuntz=Pimsner algebra of Y. Furthermore, this subalgebra can be described in a natural way in terms of the gauge actions on the Cuntz-Pimsner algebras. We explore implications for graph algebras, crossed products by the integers, and crossed products by completely positive maps. We also give a new proof of a result of Kaliszewski and Quigg related to coactions on correspo…
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 (w…
The main result of this paper is a generalization of Popa's free independence result for subalgebras of ultraproduct ${\rm II_1}$ factors [Po95] to the framework of ultraproduct von Neumann algebras $(M^ω, φ^ω)$ where $(M, φ)$ is a $σ$-finite von Neumann algebra endowed with a faithful normal state satisfying $(M^φ)' \cap M = \mathbf{C} 1$. More precisely, we show that whenever $P_1, P_2 \subset M^ω$ are von Neumann subalgebras with separable predual that are globally invariant under the modular automorphism group $(σ_t^{φ^ω})$, there exists a unitary $v \in \mathcal U((M^ω)^{φ^ω})$ such that …
Let $M$ be a factor with separable predual and $G$ a compact group of automorphisms of $M$ whose action is minimal, i.e. $M^{G^\prime}\cap M = C$, where $M^G$ denotes the $G$-fixed point subalgebra. Then every intemediate von Neumann algebra $M^G\subset N\subset M$ has the form $N=M^H$ for some closed subgroup $H$ of $G$. An extension of this result to the case of actions of compact Kac algebras on factors is also presented. No assumptions are made on the existence of a normal conditional expectation onto $N$.
Let $(M, φ) = (M_1, φ_1) \ast (M_2, φ_2)$ be the free product of any $σ$-finite von Neumann algebras endowed with any faithful normal states. We show that whenever $Q \subset M$ is a von Neumann subalgebra with separable predual such that both $Q$ and $Q \cap M_1$ are the ranges of faithful normal conditional expectations and such that both the intersection $Q \cap M_1$ and the central sequence algebra $Q' \cap M^ω$ are diffuse (e.g. $Q$ is amenable), then $Q$ must sit inside $M_1$. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles…
Angles Between Intermediate Operator Subalgebras
本文引入了中间算子子代数之间的内角概念，并展示了其在非单位C*代数中的稳定性。
We investigate spectral properties of the tensor products of two quantum channels defined on matrix algebras. This leads to the important question of when an arbitrary subalgebra can split into the tensor product of two subalgebras. We show that for two unital quantum channels $\mathcal{E}_1$ and $\mathcal{E}_2$ the multiplicative domain of $\mathcal{E}_1\otimes\mathcal{E}_2$ splits into the tensor product of the individual multiplicative domains. Consequently, we fully describe the fixed points and peripheral eigen operators of the tensor product of channels. Through a structure theorem of ma…
Frobenius subalgebra lattices in tensor categories
本文将Watatani的中间子因子有限性定理推广到更广泛的单子范畴中，通过引入环境自对偶的概念，刻画了阿贝尔单子范畴中弗罗贝尼乌斯子代数偏序格子的子格，并在弱正性框架下证明了任何连通弗罗贝尼乌斯代数的相干子格都是有限的，从而显著扩展了Watatani定理。
Relations amongst the distances between $C^{*}$-subalgebras and some canonically associated operator algebras
本文证明两个$C^*$-子代数的Christensen距离和Kadison-Kastler距离分别等于其包络von Neumann代数及与任意单位交换$C^*$-代数的最小张量积代数之间的距离。
We define centrally large subalgebras of simple unital C*-algebras, strengthening the definition of large subalgebras in previous work. We prove that if A is any infinite dimensional simple separable unital C*-algebra which contains a centrally large subalgebra with stable rank one, then A has stable rank one. We also prove that large subalgebras of crossed product type are automatically centrally large. We use these results to prove that if X is a compact metric space which has a surjective continuous map to the Cantor set, and h is a minimal homeomorphism of X, then C* (Z, X, h) has stable r…
The boson representation of the sp(4,R) algebra and two distinct deformations of it, are considered, as well as the compact and noncompact subalgebras of each. The initial as well as the deformed representations act in the same Fock space. One of the deformed representation is based on the standard q-deformation of the boson creation and annihilation operators. The subalgebras of sp(4,R) (compact u(2) and three representations of the noncompact u(1,1) are also deformed and are contained in this deformed algebra. They are reducible in the action spaces of sp(4,R) and decompose into irreducible …
We show that outer approximately represenbtable actions of a finite cyclic group on UCT Kirchberg algebras satisfy a certain quasi-freeness type property if the corresponding crossed products satisfy the UCT and absorb a suitable UHF algebra tensorially. More concretely, we prove that for such an action there exists an inverse semigroup of homogeneous partial isometries that generates the ambient C*-algebra and whose idempotent semilattice generates a Cartan subalgebra. We prove a similar result for actions of finite cyclic groups with the Rokhlin property on UCT Kirchberg algebras absorbing a…
On stability of distance under some tensor products and some calculations
本文研究了某些张量积下的距离稳定性，并对交叉积算子代数的子代数间距离进行了具体计算。
Conditioning on Subalgebras - Entropy Duality and Generalized Quantum Stein's Lemma in von Neumann Algebras
该论文将条件熵对偶性推广至任意冯·诺依曼代数子代数，并建立无限维量子斯坦引理，为子代数相对熵提供复合假设检验中的操作解释。
Let X be an infinite compact metric space, α: X \to X a minimal homeomorphism, u the unitary implementing αin the transformation group C*-algebra, and S a class of separable nuclear C*-algebras that contains all unital hereditary C*-subalgebras of C*-algebras in S. Motivated by the success of tracial approximation by finite dimensional C*-algebras as an abstract characterization of classifiable C*-algebras and the idea that classification results for C*-algebras tensored with UHF algebras can be used to derive classification results up to tensoring with the Jiang-Su algebra Z, we prove that th…
We define the notion of strong spectral invariance for a dense Frechet subalgebra A of a Banach algebra B. We show that if A is strongly spectral invariant in a C*-algebra B, and G is a compactly generated polynomial growth Type R Lie group, not necessarily connected, then the smooth crossed product G\rtimes A is spectral invariant in the C*-crossed product G\rtimes B. Examples of such groups are given by finitely generated polynomial growth discrete groups, compact or connected nilpotent Lie groups, the group of Euclidean motions on the plane, the Mautner group, or any closed subgroup of one …
An algebra extension $A \| B$ is right depth two in this paper if its tensor-square is $A$-$B$-isomorphic to a direct summand of any (not necessarily finite) direct sum of $A$ with itself. For example, normal subgroups of infinite groups, infinitely generated Hopf-Galois extensions and infinite dimensional algebras are depth two in this extended sense. The added generality loses some duality results obtained in the finite theory math.RA/0108067 but extends the main theorem of depth two theory, as for example in math.RA/0107064. That is, a right depth two extension has right bialgebroid T = (A …
Splitting of Tensor Products and Intermediate Factor Theorem: Continuous Version
该研究针对离散群作用下的$C^*$-代数，给出中间子代数为张量积的抽象条件，推广经典分裂结果，证明拓扑版中间因子定理，引入一致刚性作用概念并给出多类满足条件的不变中间代数包含关系。
We define and systematically study nonassociative C*-algebras as C*-algebras internal to a topological tensor category. We also offer a concrete approach to these C*-algebras, as G-invariant, norm closed *-subalgebras of bounded operators on a G-Hilbert space, with deformed composition product. Our central results are those of stabilization and Takai duality for (twisted) crossed products in this context.
In a 1991 paper, R. Mercer asserted that a Cartan bimodule isomorphism between Cartan bimodule algebras A_1 and A_2 extends uniquely to a normal *-isomorphism of the von Neumann algebras generated by A_1 and A_2 [13, Corollary 4.3]. Mercer's argument relied upon the Spectral Theorem for Bimodules of Muhly, Saito and Solel [15, Theorem 2.5]. Unfortunately, the arguments in the literature supporting [15, Theorem 2.5] contain gaps, and hence Mercer's proof is incomplete. In this paper, we use the outline in [16, Remark 2.17] to give a proof of Mercer's Theorem under the additional hypothesis that…
For a discrete group $G$, we develop a `$G$-balanced tensor product' of two coactions $(A,δ)$ and $(B,ε)$, which takes place on a certain subalgebra of the maximal tensor product $A\otimes_{\max} B$. Our motivation for this is that we are able to prove that given two actions of $G$, the dual coaction on the crossed product of the maximal-tensor-product action is isomorphic to the $G$-balanced tensor product of the dual coactions. In turn, our motivation for this is to give an analogue, for coaction functors, of a crossed-product functor originated by Baum, Guentner, and Willett, and further de…
Equivariant $\mathcal{D}$-stability for Actions of Tensor Categories
本文为酉张量范畴在 C$^*$-代数上的作用引入等变 $\mathcal{D}$-稳定性概念，证明其等价于 $\mathcal{D}$ 到 Kirchberg 中心序列代数某子代数的单位嵌入，并据此证明一大类 AF-作用具有 $\mathcal{Z}$-稳定性。
We demonstrate von Neumann algebra arising from an icc group $Γ$ in Chifan's, Ioana's, and Kida's class of poly-$\mathcal{C}_\text{rss} $, such as a poly-hyperbolic group with no amenable factors in its composition series, satisfies the following rigidity phenomenon discovered in DHI16 (see also CdSS17): every tensor decomposition of the II$_1$ factor $L(Γ) $ must arise from direct product decomposition of $Γ$ by groups which are poly-$ \mathcal{C}_\text{rss}$. Through heavy usage and developments of the techniques in CdSS15, we improve the second author's and their collaborator's work in CKP1…
A gauge-invariant C*-system is obtained as the fixed point subalgebra of the infinite tensor product of full matrix algebras under the tensor product unitary action of a compact group. In the paper, thermodynamics is studied on such systems and the chemical potential theory developed by Araki, Haag, Kastler and Takesaki is used. As a generalization of quantum spin system, the equivalence of the KMS condition, the Gibbs condition and the variational principle is shown for translation-invariant states. The entropy density of extremal equilibrium states is also investigated in relation to macrosc…
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