Pith. sign in

REVIEW 1 major objections 4 minor 56 references

Weak relative Dixmier property and Popa's intertwining technique for type III subfactors

T0 review · 1 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Averaging unitary orbits hits the relative commutant, no trace needed

desk verdict Significant and likely-true generalization of the weak relative Dixmier property to operator valued weights, but the written proof of Theorem A has a genuine gap in the semifinite case that needs a separate argument. read the letter →

arxiv 2508.17592 v2 pith:SGGWUMN5 submitted 2025-08-25 math.OA

Yusuke Isono This is my paper · ORCID
classification math.OA MSC 46L1046L3646L40
keywords weakrelativeDixmierpropertyoperatorvaluedweightstypeIIIfactorsintertwining-by-bimodulessolidityGaloiscorrespondencesemigroupcrossedproducts
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves a localized averaging theorem for inclusions of von Neumann algebras: whenever a faithful normal semifinite operator valued weight exists from a larger algebra to a subalgebra, every positive element of finite weight can be moved by averages of unitary conjugations from the subalgebra into the relative commutant. The result extends the weak relative Dixmier property from the conditional-expectation setting to the more flexible operator valued weight setting, where no expectation need exist. The paper then uses this elementwise criterion to reformulate the intertwining-by-bimodules technique without any trace or finite subalgebra assumption, to obtain relative solidity for crossed products by biexact groups, and to prove a Galois correspondence for crossed products by totally disconnected groups, resolving a question about intermediate subfactors.

What carries the argument

The engine is the weak Dixmier semigroup $\mathrm{DSG}(\mathrm{Ad}_A)$, the compact convex semigroup of completely positive maps obtained by closing convex combinations of unitary conjugations $u\cdot u^*$ with $u\in\mathcal U(A)$ in the point-$\sigma$-weak topology. Inside it, the proof isolates a minimal idempotent $\Phi$ whose restriction to $A$ is the expectation onto the center, then shows $\Phi$ acts like the identity on its own image inside the finite-weight domain $m_{E_A}$; a parallelogram-law argument forces that image to consist of elements commuting with $A$. The operator valued weight $E_A$ plays the role of keeping the finite-weight domain invariant under averaging, substituting for the trace or conditional expectation that earlier arguments required.

What would settle it

Take any genuinely type III subalgebra $A$ inside a larger von Neumann algebra $M$ with a faithful normal semifinite operator valued weight that is not a conditional expectation, choose a positive $x$ with $E_A(x)<\infty$, and compute whether the $\sigma$-weak closure of $\mathrm{conv}\{uxu^*\mid u\in\mathcal U(A)\}$ intersects $A'\cap M$; if it misses $A'\cap M$, Theorem A is false, and the paper predicts this cannot happen.

Watch

Extended reading notes

Core claim

The central claim is Theorem A: for an inclusion $A\subset M$ with a faithful normal semifinite operator valued weight $E_A:M\to A$, every positive $x\in M$ with $E_A(x)<\infty$ satisfies the weak relative Dixmier property in the strong sense that the $\sigma$-weak closure of the convex hull $K(x,A)=\mathrm{conv}^{\mathrm{weak}}\{uxu^*\mid u\in \mathcal U(A)\}$ intersects $A'\cap M$. In its precise form, the paper shows $m_{E_A}\subset D(A\subset M)$: every element on which the weight is finite lies in the weak Dixmier subspace, so for every averaging map one can find nonempty fixed points of the convex hull of its unitary orbit. In particular, there is a ucp map $\Psi$, obtained as a point-$\sigma$-weak limit of convex combinations of conjugations by unitaries of $A$, with $\Psi(m_{E_A})\subset A'\cap M$. This elementwise, weight-finite version is the engine that makes the later applications possible.

Load-bearing premise

The proof's load-bearing premise is that a type III algebra admits a faithful normal expectation onto its center that can be obtained as a limit of unitary conjugations; if that deep fact fails for the subalgebra in question, or the extension of such an expectation to the whole algebra cannot be made minimal, the main theorem is not established.

Editorial extensions

If this is right

  • Popa's intertwining criterion $A\preceq_M B$ is characterized, for type III algebras or a properly infinite $B$, by the absence of a ucp map on the basic construction that annihilates the Jones projection, with no tracial assumption needed.
  • Bernoulli crossed products by actions with finite stabilizers are solid relative to the base inclusion, giving a spectral gap rigidity theorem for type III factors.
  • Crossed products by biexact group actions on amenable von Neumann algebras are solid relative to the base algebra, even when the base is type III.
  • For properly outer actions of totally disconnected groups, every intermediate subfactor between $B$ and $B\rtimes G$ is of the form $B\rtimes H$ for a closed subgroup $H\le G$, settling a previously open question.
  • Corollary B gives a dichotomy: if every nonzero positive element of $A'\cap M$ has infinite $E_A$-value, then every positive element of finite weight can be averaged to $0$ inside the convex hull of its unitary orbit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the elementwise formulation suggests a quantitative version: the speed at which convex combinations of conjugations approach the relative commutant may be controlled by the value of $E_A(x)$, which could give uniform approximation bounds for whole classes of elements.
  • Since the intertwining reformulation needs no finiteness of $A$ or $B$, it should extend rigidity arguments to free Araki-Woods factors and other non-tracial constructions where earlier criteria stalled.
  • The Galois correspondence for totally disconnected groups may hold under weaker proper-outerness hypotheses than the one stated; the exact boundary of the condition is a testable extension.
  • The same averaging machinery may yield a relative bicentralizer computation for inclusions with operator valued weights, paralleling the conditional-expectation bicentralizer results.
  • A concrete search for counterexamples could target the basic construction of a genuinely type III subfactor: if any positive element of finite weight has a convex hull of unitarily conjugated elements missing the relative commutant, Theorem A would be false.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. This paper proves a localized weak relative Dixmier theorem for inclusions A⊂M admitting a faithful normal semifinite operator valued weight E_A:M→A: every positive x∈M with E_A(x)<∞ can be averaged, via the σ-weak closure of the convex hull of {uxu*:u∈U(A)}, into A'∩M. The proof is organized around the notion of weak Dixmier subspaces, decomposes A into semifinite and type III parts, and applies Marrakchi's compact semigroup machinery. The theorem is then used to derive several applications: a Popa intertwining criterion without tracial assumptions, relative solidity theorems for type III crossed products, and a Galois correspondence for crossed products by totally disconnected groups, resolving a question of Boutonnet and Brothier.

Significance. If fully established, the main theorem is a substantial generalization of Marrakchi's weak relative Dixmier theorem from conditional expectations to operator valued weights, and the applications are broad. The paper is well structured, states precise elementwise conditions (Theorems 3.3 and A), and makes honest and explicit use of published deep results ([Ma19], [Ma23, Theorem 5.2]). The proofs are largely self-contained after those inputs, with no hidden parameter fitting or circularity. However, the central proof has a gap in the semifinite properly infinite case that must be repaired before the applications can be regarded as consequences of the stated argument.

major comments (1)
  1. [§3.4 (Proof of Theorem 3.3)] The proof does not cover the properly infinite semifinite summand. It invokes Propositions 3.11 and 3.13 to conclude m_{E_{A_z}}⊂D(Ad_{A_z}) and m_{E_{A_{z⊥}}}⊂D(Ad_{A_{z⊥}}). Proposition 3.11, in the second bullet, is applied with φ=Tr_{A_z}∘E_{A_z} and yields only m_φ⊂D(Ad_{A_z}). For a properly infinite A_z, m_{E_{A_z}} is not contained in m_φ: for A=B(H), M=A, E_A=id, and x=1, one has x∈m_{E_A} but Tr(1)=∞. Corollary 3.5(2) gives only norm closedness of D(α), not σ-weak closedness, so the missing inclusion cannot be obtained by approximation in the manner suggested by the phrase 'standard Hilbert space methods' in the introduction. Since Theorem 3.3 is the engine for Theorem A, Corollary B, and the applications in Sections 4–7, this is a load-bearing gap; a separate argument for the properly infinite semifinite case (or a reduction to the finite case) is required.
minor comments (4)
  1. [§3.3 (Proposition 3.13)] The first paragraph uses the notation `\tilde E`∈D(Ad_A:U(A)↷A) for what should be an element of DSG(Ad_A:U(A)↷M) extending E; D(α) is a subspace of the algebra, not a set of maps. Please correct the notation and explain why replacing Φ by Φ∘\tilde E yields Φ|_A=E.
  2. [Lemma 3.8(1)] The displayed equality pK(x,Ad_A)p=K(pxp,Ad_{Ap})=K(pxp,Ad_{Ap}) contains a duplicated right-hand side; one occurrence should refer to the reduced action K(pxp,Ad_{A_p}).
  3. [§3.2 (Lemma 3.10)] The line 'As in the proof of [HI15a, Lemma 4.4], we can embed K(d,α)→L^2(M,φ) and the range is closed in the L^2-norm topology' is not evident from the text; since it is used to justify existence of the minimizer, a brief proof or a more precise reference to this closedness statement would improve readability.
  4. [§5.3 (Lemma 5.11)] The proof refers to 'reduced word in (B_0)_{a*}LZ' without defining the notation (B_0)_{a*} and LZ; please define these or add a reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem A is proved from Marrakchi's external theorems and elementary convex-semigroup arguments; self-citations are used only as published tools.

full rationale

Walking the derivation chain, I find no step in which an alleged prediction is identical to an input by construction. Theorem A/Theorem 3.3 is not obtained by assuming m_EA⊂D; the proof decomposes A and invokes Proposition 3.11 (an invariant-weight Hilbert-space argument) and Proposition 3.13. Proposition 3.13 is the load-bearing type III step: it takes E:A→Z(A) from Theorem 2.2, whose type III case is Marrakchi's external published theorem [Ma23, Theorem 5.2], extends it to a minimal idempotent Φ∈DSG(A⊂M), and proves Φ(M0)⊂A'∩M by a Schwarz-inequality/star-isomorphism argument. The conclusion m_EA⊂D is then read off from Lemma 3.12 and the definition of D; it is not fed back into the proof. The applications (Theorems C–F) use Theorem A together with previously published intertwining criteria ([Is19, HI15a]) as tools; these are self-citations but they are not what forces the main result, and the cited results do not depend on the present paper. I also checked the places where one might suspect circularity—Corollary B, Theorem 4.2(3)⇒(1), Theorem 4.14, and the Galois claim—and in each case the cited external theorem is used as a black box, not as a renamed version of the target. The only concern I noted is a possible correctness gap in the semifinite case of Theorem 3.3 (Proposition 3.11 gives m_φ⊂D for φ=Tr_A∘E_A, whereas the proof needs m_EA⊂D; these differ for properly infinite A). That is an internal proof gap, not circularity, so it does not change the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters and no invented entities: the paper is pure mathematics. The inputs that the reader did not pay for upstream are the theorems of Marrakchi ([Ma19] and [Ma23, Theorem 5.2]) that enter through Theorem 2.2, the [Ma23] relative bicentralizer machinery behind Lemma 5.4, and the Ding-Peterson biexactness framework behind Theorem E. All are cited openly; the risk is depth of dependency, not hidden assumptions.

assumptions (5)
  • standard math Ellis' lemma: the compact semigroup DSG(A ⊂ M) of ucp maps contains a minimal idempotent, and minimal idempotents are interrelated by Choi-Effros isomorphisms (part of [Ma19]).
    Stated in Section 2.2 following [Ma19, Remark 3.7 and Corollary 3.8] and used in Proposition 3.13 to select the minimal idempotent Φ with Φ|_A = E.
  • domain assumption [Ma19, MAIN THEOREM] (Theorem 2.2, first part): for A ⊂ M with a faithful normal conditional expectation, a conditional expectation M → A' ∩ M lies in DSG(A ⊂ M).
    Quoted as Theorem 2.2; the case A = M of it gives the central expectation E: A → Z(A) used at the start of Proposition 3.13, and it builds all relative-commutant expectations in Theorems 4.2, 4.14, and Lemma 7.1.
  • domain assumption [Ma23, Theorem 5.2] (Theorem 2.2, second part): for the type III summand A_1 of A, a faithful normal φ-preserving conditional expectation onto the center lies in DSG(Ad_A).
    The paper states 'If A is of type III, then it is proved in [Ma23, Theorem 5.2]'. This is the load-bearing external input for the type III case of Theorem A and propagates to Theorem C, Theorem 4.14, and Theorem F.
  • domain assumption [Ma23, Theorem 9.17 and Section 6]: the relative bicentralizer of an inclusion tensored with a type III_1 factor equals the algebraic bicentralizer, giving B(A⊗N ⊂ M⊗N, ψ) = (A'∩M)⊗C1_N.
    Lemma 5.4 adapts this with 'only minor modifications'; it is the input for Lemma 5.8 and therefore for the spectral gap rigidity theorem (Theorem 5.6) and Theorem D.
  • domain assumption Ding-Peterson biexactness: M is biexact relative to a boundary piece X if the inclusion M → S_X(M) is M-nuclear, and biexact groups give biexact group von Neumann algebras ([DP23]).
    Section 6.1 adopts [DP23, Definitions 6.1, Theorem 6.2, Propositions 8.3 and 6.14] as the framework in which Theorem E and Theorem 6.5 are proved.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Weak relative Dixmier property and Popa's intertwining technique for type III subfactors." pith.science (2026). https://pith.science/paper/SGGWUMN5

@misc{pith2026250817592,
  author       = {Pith},
  title        = {Pith review of: Weak relative Dixmier property and Popa's intertwining technique for type III subfactors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SGGWUMN5}},
  note         = {Machine review of arXiv:2508.17592}
}
abstract

Let \( A \subset M \) be an inclusion of von Neumann algebras equipped with a faithful normal semifinite operator valued weight \( E \colon M \to A \). We prove that every positive element \( x \in M \) with \( E(x) < \infty \) satisfies the weak Dixmier property relative to \( A \): the \( \sigma \)-weak closure of the convex hull of its unitary orbit under \( \mathcal{U}(A) \) intersects the relative commutant \( A' \cap M \). This extends Marrakchi's result for the case of conditional expectations. We apply this result to obtain new structural theorems for type III factors, including a reformulation of Popa's intertwining criterion without tracial assumptions, an extension of Ozawa's relative solidity theorem to the type III setting, and a Galois-type correspondence for crossed products by totally disconnected groups. The last result resolves a question posed by Boutonnet and Brothier regarding the structure of intermediate subfactors.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

56 extracted references · 55 canonical work pages

  1. [1]

    Ando and U

    H. Ando and U. Haagerup, Ultraproducts of von Neumann algebras. J. Funct. Anal. 266 (2014), 6842--6913

    work page 2014

  2. [2]

    H. Ando, U. Haagerup, C. Houdayer, and A. Marrakchi, Structure of bicentralizer algebras and inclusions of type III factors. Math. Ann. 376 (2020), no. 3-4, 1145--1194

    work page 2020

  3. [3]

    Bannon, A

    J. Bannon, A. Marrakchi, and N. Ozawa, Full factors and co-amenable inclusions. Comm. Math. Phys. 378 (2020), no. 2, 1107--1121

    work page 2020

  4. [4]

    Boutonnet and A

    R. Boutonnet and A. Brothier, Crossed-products by locally compact groups intermediate subfactors. J. Operator Theory 79 (2018), no. 1, 101--137

    work page 2018

  5. [5]

    Boutonnet and C

    R. Boutonnet and C. Houdayer, Amenable absorption in amalgamated free product von Neumann algebras. Kyoto J. Math. 58 (2018), 583--593

    work page 2018

  6. [6]

    N. P. Brown and N. Ozawa, C^ -algebras and finite-dimensional approximations . Graduate Studies in Mathematics, 88. American Mathematical Society, Providence, RI, 2008

    work page 2008

  7. [7]

    Chifan and C

    I. Chifan and C. Houdayer, Bass-Serre rigidity results in von Neumann algebras, Duke Math.\ J.\ 153 (2010), 23--54

    work page 2010

  8. [8]

    Chifan and A

    I. Chifan and A. Ioana, Ergodic subequivalence relations induced by a Bernoulli action. Geom. Funct. Anal. 20 (2010), no. 1, 53--67

    work page 2010

Show all 56 references
  1. [9]

    Choda, A Galois correspondence in a von Neumann algebra

    H. Choda, A Galois correspondence in a von Neumann algebra. Tohoku Math. J. 30(1978), 491--504

    1978

  2. [10]

    Choi and E

    M.-D. Choi and E. G. Effros, Injectivity and operator spaces. J. Funct. Anal. 24 (1977) 156--209

    1977

  3. [11]

    Connes, Une classification des facteurs de type III

    A. Connes, Une classification des facteurs de type III . Ann. Sci. \'Ecole Norm. Sup. (4) 6 (1973), 133--252

    1973

  4. [12]

    Connes, Classification of injective factors

    A. Connes, Classification of injective factors. Ann. of Math. (2) 104 (1976), no. 1, 73--115

    1976

  5. [13]

    Connes and M

    A. Connes and M. Takesaki, The flow of weights on factors of type III . T\^ o hoku Math. J. (2) 29 (1977), no. 4, 473--575

    1977

  6. [14]

    C. Ding, J. Peterson, and S. K. Elayavalli, Properly proximal von Neumann algebras. Duke Math. J. 172 (2023), no. 15, 2821--2894

    2023

  7. [15]

    Ding and J

    C. Ding and J. Peterson, Biexact von Neumann algebras. Preprint 2023. arXiv:2309.10161

    2023 arXiv

  8. [16]

    Dixmier, Les anneaux d'op\' erateurs de classe finie

    J. Dixmier, Les anneaux d'op\' erateurs de classe finie. Ann. Sci. \'Ecole Norm. Sup. (3) 66 (1949), 209--261

    1949

  9. [17]

    Haagerup, Operator valued weights in von Neumann algebras, I

    U. Haagerup, Operator valued weights in von Neumann algebras, I. J. Funct. Anal. 32 (1979), 175--206

    1979

  10. [18]

    Haagerup, Operator valued weights in von Neumann algebras, II

    U. Haagerup, Operator valued weights in von Neumann algebras, II. J. Funct. Anal. 33 (1979), 339--361

    1979

  11. [19]

    Haagerup Connes' bicentralizer problem and uniqueness of the injective factor of type III_1

    U. Haagerup Connes' bicentralizer problem and uniqueness of the injective factor of type III_1 . Acta Math. 158 (1987), no. 1-2, 95--148

    1987

  12. [20]

    Houdayer and Y

    C. Houdayer and Y. Isono, Unique prime factorization and bicentralizer problem for a class of type III factors. Adv. Math. 305 (2017), 402--455

    2017

  13. [21]

    Houdayer and Y

    C. Houdayer and Y. Isono, Bi-exact groups, strongly ergodic actions and group measure space type III factors with no central sequence. Comm. Math. Phys. 348 (2016), no. 3, 991--1015

    2016

  14. [22]

    Houdayer and Y

    C. Houdayer and Y. Isono, Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras. Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), 1495--1532

    2020

  15. [23]

    Houdayer and E

    C. Houdayer and E. Ricard, Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors. Adv.\ Math.\ 228 (2011), 764--802

    2011

  16. [24]

    Houdayer, D

    C. Houdayer, D. Shlyakhtenko, and S. Vaes, Classification of a family of non almost periodic free Araki-Woods factors. J. Eur. Math. Soc. (JEMS) 21 (2019), 3113--3142

    2019

  17. [25]

    Houdayer and S

    C. Houdayer and S. Vaes, Type III factors with unique Cartan decomposition. J. Math. Pures Appl. (9) 100 (2013), no. 4, 564--590

    2013

  18. [26]

    Herman and M

    R. Herman and M. Takesaki, States and Automorphism Groups of Operator Algebras. Comm. Math. Phys. 19 (1970), 142--160

    1970

  19. [27]

    Ioana, Rigidity for von Neumann algebras

    A. Ioana, Rigidity for von Neumann algebras. Proceedings of the International Congress of Mathematicians-Rio de Janeiro 2018. Vol. III. Invited lectures, 1639-1672, World Sci. Publ., Hackensack, NJ, 2018

    2018

  20. [28]

    Isono, Weak Exactness for C^* -algebras and Application to Condition (AO)

    Y. Isono, Weak Exactness for C^* -algebras and Application to Condition (AO) . J. Funct. Anal.\ 264 (2013), 964--998

    2013

  21. [29]

    Isono, Some prime factorization results for free quantum group factors

    Y. Isono, Some prime factorization results for free quantum group factors. J. Reine Angew. Math. 722 (2017), 215--250

    2017

  22. [30]

    Isono, Unique prime factorization for infinite tensor product factors

    Y. Isono, Unique prime factorization for infinite tensor product factors. J. Funct. Anal. 276 (2019), 2245--2278

    2019

  23. [31]

    Isono, Unitary conjugacy for type III subfactors and W ^* -superrigidity

    Y. Isono, Unitary conjugacy for type III subfactors and W ^* -superrigidity. J. Eur. Math. Soc. (JEMS) 24 (2022), 1679--1721

    2022

  24. [32]

    Isono and A

    Y. Isono and A. Marrakchi, Tensor product decompositions and rigidity of full factors. Ann. Sci. \'Ec. Norm. Sup e r. (4) 55 (2022), 109--139

    2022

  25. [33]

    Izumi, R

    M. Izumi, R. Longo, S. Popa, A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras. J.\ Funct.\ Anal. 155 (1998), 25--63

    1998

  26. [34]

    Krieger, On ergodic flows and the isomorphism of factors

    W. Krieger, On ergodic flows and the isomorphism of factors. Math. Ann. 223 (1976), no. 1, 19--70

    1976

  27. [35]

    Marrakchi, Solidity of type III Bernoulli crossed products

    A. Marrakchi, Solidity of type III Bernoulli crossed products. Comm. Math. Phys. 350 (2017), no. 3, 897--916

    2017

  28. [36]

    Marrakchi, On the weak relative Dixmier property

    A. Marrakchi, On the weak relative Dixmier property. Proc. London Math. Soc. 122 (1) (2019), 118--123

    2019

  29. [37]

    Marrakchi, Kadison's problem for type III subfactors and the bicentralizer conjecture

    A. Marrakchi, Kadison's problem for type III subfactors and the bicentralizer conjecture. Invent. Math. 239 (2025), 79--163

    2025

  30. [38]

    Ozawa, Solid von Neumann algebras

    N. Ozawa, Solid von Neumann algebras. Acta Math. 192 (2004), 111--117

    2004

  31. [39]

    Ozawa, A Kurosh type theorem for type II_1 factors

    N. Ozawa, A Kurosh type theorem for type II_1 factors. Int. Math. Res. Not. (2006), Art. ID 97560, 21 pp

    2006

  32. [40]

    Ozawa and S

    N. Ozawa and S. Popa, Some prime factorization results for type II_1 factors, Invent.\ Math.\ 156 (2004), 223--234

    2004

  33. [41]

    Ocneanu, Actions of discrete amenable groups on von Neumann algebras

    A. Ocneanu, Actions of discrete amenable groups on von Neumann algebras. Lecture Notes in Mathematics, 1138 . Springer-Verlag, Berlin, 1985. iv+115 pp

    1985

  34. [42]

    Popa, On a problem of R

    S. Popa, On a problem of R. V. Kadison on maximal abelian -subalgebras in factors. Invent. Math. 65 (1981), no. 2, 269--281

    1981

  35. [43]

    Popa, Classification of subfactors and their endomorphisms

    S. Popa, Classification of subfactors and their endomorphisms. CBMS Regional Conf. Ser. in Math., 86

  36. [44]

    Popa, On the relative Dixmier property for inclusions of C ^* -algebras

    S. Popa, On the relative Dixmier property for inclusions of C ^* -algebras. J. Funct. Anal. 171 (2000) 139--154

    2000

  37. [45]

    Popa, On a class of type II_1 factors with Betti numbers invariants

    S. Popa, On a class of type II_1 factors with Betti numbers invariants. Ann.\ of Math.\ 163 (2006), 809--899

    2006

  38. [46]

    Popa, Strong rigidity of II_1 factors arising from malleable actions of w-rigid groups I

    S. Popa, Strong rigidity of II_1 factors arising from malleable actions of w-rigid groups I . Invent.\ Math.\ 165 (2006), 369--408

    2006

  39. [47]

    Popa, Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups

    S. Popa, Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups. Invent. Math. 170 (2007), no. 2, 243--295

    2007

  40. [48]

    Popa, On the superrigidity of malleable actions with spectral gap

    S. Popa, On the superrigidity of malleable actions with spectral gap. J. Amer. Math. Soc. 21 (2008), no. 4, 981--1000

    2008

  41. [49]

    Popa, Deformation and rigidity for group actions and von Neumann algebras

    S. Popa, Deformation and rigidity for group actions and von Neumann algebras. In Proceedings of the International Congress of Mathematicians (Madrid, 2006), Vol. I, European Mathematical Society Publishing House, 2007, p. 445--477

    2006

  42. [50]

    Schwartz, Two finite, non-hyperfinite, non-isomorphic factors

    J. Schwartz, Two finite, non-hyperfinite, non-isomorphic factors. Comm. Pure Appl. Math. 16 (1963) 19--26

    1963

  43. [51]

    Str a til a and L

    S. Str a til a and L. Zsid a , The commutation theorem for tensor products over von Neumann algebras. J. Funct. Anal. 165 (2), 293--346 (1999)

    1999

  44. [52]

    Takesaki, Theory of operator algebras II

    M. Takesaki, Theory of operator algebras II . Encyclopedia of Mathematical Sciences, 125. Operator Algebras and Non-commutative Geometry, 5 . Springer-Verlag, Berlin, 2002

    2002

  45. [53]

    Ueda, Some analysis of amalgamated free products of von Neumann algebras in the non-tracial setting

    Y. Ueda, Some analysis of amalgamated free products of von Neumann algebras in the non-tracial setting. J.\ Lond.\ Math.\ Soc.\ (2) 88 (2013), no. 1, 25--48

    2013

  46. [54]

    Ueda, A free product pair rigidity result in von Neumann algebras

    Y. Ueda, A free product pair rigidity result in von Neumann algebras. J. Noncommut. Geom. 13 (2019), no. 2, 587--607

    2019

  47. [55]

    Vaes, Rigidity for von Neumann algebras and their invariants

    S. Vaes, Rigidity for von Neumann algebras and their invariants. Proceedings of the ICM (Hyderabad, India, 2010), Vol. III, Hindustan Book Agency (2010), 1624--1650

    2010

  48. [56]

    Vaes and P

    S. Vaes and P. Verraedt, Classification of type III Bernoulli crossed products. Adv. Math. 281 (2015), 296--332

    2015

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.