Let $A\subset M$ be a MASA in a $\mathrm{II}_{1}$ factor $M$. We describe the von Neumann subalge... more Let $A\subset M$ be a MASA in a $\mathrm{II}_{1}$ factor $M$. We describe the von Neumann subalgebra of $M$ generated by $A$ and its normalizer $\mathcal N(A)$ as the set $N_q^w(A)$ consisting of those elements $m\in M$ for which the bimodule $\smash{\overline{AmA}}$ is discrete. We prove that two MASAs $A$ and $B$ are conjugate by a unitary $u\in N^{w}_{q}(A)$ iff $A$ is discrete over $B$ and $B$ is discrete over $A$ in the sense defined by Feldman and Moore [5]. As a consequence, we show that $A$ is a Cartan subalgebra of $M$ iff for any MASA $B\subset M$, $B=uAu^{*}$ for some $u\in M$ exactly when $A$ is discrete over $B$ and $B$ is discrete over $A$.
Journal of the Institute of Mathematics of Jussieu, 2015
We prove that, under rather general conditions, the 1-cohomology of a von Neumann algebra$M$with ... more We prove that, under rather general conditions, the 1-cohomology of a von Neumann algebra$M$with values in a Banach$M$-bimodule satisfying a combination of smoothness and operatorial conditions vanishes. For instance, we show that, if$M$acts normally on a Hilbert space${\mathcal{H}}$and${\mathcal{B}}_{0}\subset {\mathcal{B}}({\mathcal{H}})$is a norm closed$M$-bimodule such that any$T\in {\mathcal{B}}_{0}$issmooth(i.e., the left and right multiplications of$T$by$x\in M$are continuous from the unit ball of$M$with the$s^{\ast }$-topology to${\mathcal{B}}_{0}$with its norm), then any derivation of$M$into${\mathcal{B}}_{0}$is inner. The compact operators are smooth over any$M\subset {\mathcal{B}}({\mathcal{H}})$, but there is a large variety of non-compact smooth elements as well.
Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006
We present some recent rigidity results for von Neumann algebras (II 1 factors) and equivalence r... more We present some recent rigidity results for von Neumann algebras (II 1 factors) and equivalence relations arising from measure preserving actions of groups on probability spaces which satisfy a combination of deformation and rigidity properties. This includes strong rigidity results for factors with calculation of their fundamental group and cocycle superrigidity for actions with applications to orbit equivalence ergodic theory.
Plate heat exchangers are very important equipments used in industrial applications. The paper pr... more Plate heat exchangers are very important equipments used in industrial applications. The paper presents an analysis related to the influence of the number of plates on the performance of a heat exchanger. 3D models are made for eight cases and using finite element method are performed numerical simulations of fluid flow distribution. Numerical results are presented for 1 pass-1pass counter-flow plate heat exchanger.
Researches Concerning the Improving of the Cutting Inserts Durability Using Titanium Deposition
Advanced Materials Research, 2014
It is known that in the cutting processes, due to the parameters variation of the working regimes... more It is known that in the cutting processes, due to the parameters variation of the working regimes, to the unevenness of the cutting depth, to the physico-mechanical characteristics of the processed material, to the existence or nonexistence of the coolant oiling, can appear more or less an accentuated wear of the cutting tools. All this factors have a dominant and negative influence on the durability, fact that impose being necessary taking into account some measures to increase the values of the cutting tools and/or of theirs cutting plates durability. In this sense are known various coating by deposition in vacuum researches (vacuum thermal evaporation and condensation from the vapor phase, ionic plating, vapour chemical deposition at low pressure, and so on) of some different filler materials, in thin layers, having protection role for the cutting plates, to increase the edge hardness to the superficial layer and of the tool locating and clearance surfaces and respectively, of th...
Research on the Main Parameters of the Turbo-Blower Axes TiN Covering Process
Applied Mechanics and Materials, 2014
In this paper we will elaborate a statistic analysis on the main parameters which influence the t... more In this paper we will elaborate a statistic analysis on the main parameters which influence the tribologic depositions process of thin TiN layers. The deposition of tribologic layers will be executed on the axes of the supercharging turbines at the automobiles made of C120, the maximum allowed thickness being of 1÷1,5μm. The main stages of the TiN coating operation are washing and drying conditions which we relate to adhesion layer deposited titanium and titanium coating process, the process consists of important elements of the TiN coating (Electron heating time-Argon plasma, electron temperature, plasma heating argon for Bias voltage ramp., Bias voltage temperature ramp., temperature ion cleaning time for coating titanium pressure nitrogen nitrogen temperature) relative to the thickness of the deposited titanium. The operation of coating with a layer of TiN will be done on a DREVA 400 machine at S.C. “Rulmenti” S.A. Barlad. The statistical analysis of the parameters will be done u...
Proceedings of the National Academy of Sciences, 2004
We present here a shorter version of the proof of our earlier work, showing that the von Neumann ... more We present here a shorter version of the proof of our earlier work, showing that the von Neumann factor associated with the group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathbb{Z}}^{2}SL(2,{\mathbb{Z}})\end{equation*}\end{document} has trivial fundamental group.
Journal of the American Mathematical Society, 2007
We prove that if a countable group Γ \Gamma contains infinite commuting subgroups H , H ′ ⊂ Γ H, ... more We prove that if a countable group Γ \Gamma contains infinite commuting subgroups H , H ′ ⊂ Γ H, H’\subset \Gamma with H H non-amenable and H ′ H’ “weakly normal” in Γ \Gamma , then any measure preserving Γ \Gamma -action on a probability space which satisfies certain malleability, spectral gap and weak mixing conditions (e.g. a Bernoulli Γ \Gamma -action) is cocycle superrigid. If in addition H ′ H’ can be taken non-virtually abelian and Γ ↷ X \Gamma \curvearrowright X is an arbitrary free ergodic action while Λ ↷ Y = T Λ \Lambda \curvearrowright Y=\mathbb {T}^{\Lambda } is a Bernoulli action of an arbitrary infinite conjugacy class group, then any isomorphism of the associated II 1 _{1} factors L ∞ X ⋊ Γ ≃ L ∞ Y ⋊ Λ L^{\infty }X \rtimes \Gamma \simeq L^{\infty }Y \rtimes \Lambda comes from a conjugacy of the actions.
Journal of the American Mathematical Society, 2009
Fundamental group of II 1 factors, fundamental group of II 1 equivalence relations, outer automor... more Fundamental group of II 1 factors, fundamental group of II 1 equivalence relations, outer automorphism group, actions of free groups, rigid actions, deformation/rigidity.
Let N/M be an inclusion of von Neumann algebras with a conditional expectation E: M Ä N satisfyin... more Let N/M be an inclusion of von Neumann algebras with a conditional expectation E: M Ä N satisfying the finite index condition of [PiPo], i.e., there exists c>0 such that E(x) cx, \x # M +. In [Po4] we proved that such inclusions N/M satisfy the relative version of Dixmier's property, namely for any x # M, the norm closure of the convex hull of the averaging elements of x by unitaries in N, C N (x)=co n [uxu* | u unitary element in N], contains elements from the relative commutant of N in M : C N (x) & N$ & M{<. The proof used at a key point the classical result of Dixmier for the single von Neumann algebra N ([D]), showing that for x # N the abovè`a veraging'' sets C N (x) satisfy C N (x) & Z(N){<, where Z(N)= N$ & N is the center of N. In this paper we investigate the C*-algebra version of this result, proving the relative Dixmier property for certain inclusions of C*-algebras B/A, with conditional expectations E: A Ä B satisfying the finite index condition in [PiPo]. As in the von Neumann algebra case treated in [Po4], the proof will depend on the validity of the Dixmier property for the single C*-algebra B. Thus, our result roughly shows that the relative Dixmier property for an inclusion B/A with Ind(B/A)< holds whenever the single C*-algebra B has itself the Dixmier property. Since the notion of``center'' of an algebra doesn't fit so well in the context of C*-algebras (although it can be formally defined in the same way as for von Neumann algebras), for single C*-algebras B one takes the property``C B (x) & C1{<, \x # B'' to be, by definition, the``Dixmier property''.
We prove that II 1 factors M have a unique (up to unitary conjugacy) crossproduct type decomposit... more We prove that II 1 factors M have a unique (up to unitary conjugacy) crossproduct type decomposition around "core subfactors" N ⊂ M satisfying the property HT of ([P1]) and a certain "torsion freeness" condition. In particular, this shows that isomorphism of factors of the form L α (Z 2) ⋊ Γ, for torsion free, non-amenable subgroups Γ ⊂ SL(2, Z) and α = e 2πit , t ∈ Q, implies isomorphism of the corresponding groups Γ.
We introduce the outer conjugacy invariants S(), S s () for cocycle actions of discrete groups G ... more We introduce the outer conjugacy invariants S(), S s () for cocycle actions of discrete groups G on type II 1 factors N, as the set of real numbers t > 0 for which the amplification t of can be perturbed to an action, respectively, to a weakly mixing action. We calculate explicitly S(), S s () and the fundamental group of , F(), in the case G has infinite normal subgroups with the relative property (T) (e.g., when G itself has the property (T) of Kazhdan) and is an action of G on the hyperfinite II 1 factor by Connes-StZrmer Bernoulli shifts of weights {t i } i. Thus, S s () and F() coincide with the multiplicative subgroup S of R * + generated by the ratios {t i /t j } i,j , while S() = Z * + if S = {1} (i.e. when all weights are equal), and S() = R * + otherwise. In fact, we calculate all the "1-cohomology picture" of t , t > 0, and classify the actions (, G) in terms of their weights {t i } i. In particular, we show that any 1-cocycle for (, G) vanishes, modulo scalars, and that two such actions are cocycle conjugate iff they are conjugate. Also, any cocycle action obtained by reducing a Bernoulli ଁ A preliminary version of this paper was circulated as MSRI preprint No. 2001-2005 under the title "A rigidity result for actions of property (T) groups by Bernoulli shifts". The present version of the paper was circulated as a UCLA preprint since November 2001.
We prove that the notion of rigidity (or relative property (T)) for inclusions of finite von Neum... more We prove that the notion of rigidity (or relative property (T)) for inclusions of finite von Neumann algebras defined in [Po1] is equivalent to a weaker property, in which no "continuity constants" are required. The proof is by contradiction and uses infinite products of completely positive maps, regarded as correspondences. The notion of relative property (T) (or rigidity) for inclusions of finite von Neumann algebras with countable decomposable center was introduced in ([P1]) by requiring that one of the following conditions (shown equivalent in [P1]) holds true: (0.1). There exists a normal faithful tracial state τ on N such that: ∀ε > 0, ∃F ′ = F ′ (ε) ⊂ N finite and δ ′ = δ ′ (ε) > 0 such that if H is a Hilbert N-bimodule with a vector ξ ∈ H satisfying the conditions •ξ, ξ − τ ≤ δ ′ , ξ•, ξ − τ ≤ δ ′ and yξ − ξy ≤ δ ′ , ∀y ∈ F ′ , then ∃ξ 0 ∈ H such that ξ 0 − ξ ≤ ε and bξ 0 = ξ 0 b, ∀b ∈ B. (0.2). There exists a normal faithful tracial state τ on N such that: ∀ε > 0, ∃F = F (ε) ⊂ N finite and δ = δ(ε) > 0 such that if φ : N → N is a normal, completely positive (abreviated c.p. in the sequel) map with τ • φ ≤ τ, φ(1) ≤ 1 and φ(x) − x 2 ≤ δ, ∀x ∈ F , then φ(b) − b 2 ≤ ε, ∀b ∈ B, b ≤ 1. (0.3). Condition (0.1) above is satisfied for any normal faithful tracial state τ on N. (0.4). Condition (0.2) above is satisfied for any normal faithful tracial state τ on N. This definition is the operator algebra analogue of the Kazhdan-Margulis relative property (T) for inclusions of groups H ⊂ G ([M]). It is formulated in the same spirit Connes and Jones defined the property (T) for single von Neumann algebras in ([CJ]), starting from Kazhdan's property (T) for groups, by using Hilbert bimodules/c.p. maps, i.e., Connes' correspondences ([C2]). Thus, while in the case H = G the relative property (T) of H ⊂ G amounts to the property (T) of G, in the case B = N and N is a factor the relative property (T) of B ⊂ N in the sense of ([P1]) is equivalent to the property (T) of N in the sense of ([CJ]). But there are in fact two possible ways to define the relative property (T) for inclusions of groups H ⊂ G: one requiring that all representations of G that have an almost
We consider II 1 factors L µ (G) arising from 2-cocyles µ ∈ H 2 (G, T) on groups G containing inf... more We consider II 1 factors L µ (G) arising from 2-cocyles µ ∈ H 2 (G, T) on groups G containing infinite normal subgroups H ⊂ G with the relative property (T) (i.e. G w-rigid). We prove that given any separable II 1 factor M, the set of 2-cocycles µ |H ∈ H 2 (H, T) with the property that L µ (G) is embeddable into M is at most countable. We use this result, the relative property (T) of Z 2 ⊂ Z 2 ⋊ Γ for Γ ⊂ SL(2, Z) non-amenable and the fact that every cocycle µ α ∈ H 2 (Z 2 , T) ≃ T extends to a cocycle on Z 2 ⋊SL(2, Z), to show that the one parameter family of II 1 factors M α (Γ) = L µ α (Z 2 ⋊ Γ), α ∈ T, are mutually non-isomorphic, modulo countable sets, and cannot all be embedded into the same separable II 1 factor. Other examples and applications are discussed.
We prove that any isomorphism θ : M 0 ≃ M of group measure space II 1 factors, M 0 = L ∞ (X 0 , µ... more We prove that any isomorphism θ : M 0 ≃ M of group measure space II 1 factors, M 0 = L ∞ (X 0 , µ 0) ⋊ σ 0 G 0 , M = L ∞ (X, µ) ⋊ σ G, with G 0 an ICC group containing an infinite normal subgroup with the relative property (T) of Kazhdan-Margulis (i.e. G 0 w-rigid) and σ a Bernoulli shift action of some group G, essentially comes from an isomorphism of probability spaces which conjugates the actions with respect to some identification G 0 ≃ G. Moreover, any isomorphism θ of M 0 onto a "corner" pMp of M, for p ∈ M an idempotent, forces p = 1. In particular, all group measure space factors associated with Bernoulli shift actions of w-rigid ICC groups have trivial fundamental group and any isomorphism of such factors comes from an isomorphism of the corresponding groups. This settles a "group measure space version" of Connes rigidity conjecture, shown in fact to hold true in a greater generality than just for ICC property (T) groups. We apply these results to ergodic theory, establishing new strong rigidity and superrigidity results for orbit equivalence relations. 0. Introduction. We continue in this paper the study of rigidity properties of isomorphisms θ : M 0 ≃ M of crossed product II 1 factors initiated in ([Po4]), concentrating here on the "group measure space" case, when the factors M 0 , M involved come from free ergodic measure preserving (m.p.) actions of groups on probability spaces. Similarly to ([Po4]), we assume the "source" factor M 0 comes from an arbitrary free ergodic measure preserving action σ 0 of a "mildly rigid" group G 0 (i.e. having a "large" subgroup with the relative property (T) of Kazhdan-Margulis), while the "target" factor M comes from an action σ satisfying good "deformation+mixing"
Uploads
Papers by Sorin Popa