Spaces of UCP maps and subalgebras of von Neumann algebras

Fran\c{c}ois Le Ma\^itre; Issan Patri; Kunal Mukherjee; Pierre Fima

REVIEW 2 major objections 5 minor 69 references

Spaces of UCP maps and subalgebras of von Neumann algebras

T0 review · 2 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Subalgebras admitting state-preserving conditional expectations form a closed Polish space, and in many factors their absence is generic.

desk verdict Solid, well-written operator algebra paper: the Polish-space framework for state-preserving ucp maps is real, the recovery of Haagerup–Winslow is honest, and the new applications (closedness of amenable/Haagerup subalgebras, semicontinuity of Lambda_cb, genericity of no expectation) are genuine, with the main caveat being a black-box use of Isono in Theorem 5.5. read the letter →

arxiv 2607.14358 v1 pith:HG67HYZW submitted 2026-07-15 math.OA

Pierre Fima , François Le Maître , Kunal Mukherjee , Issan Patri This is my paper · ORCID

classification math.OA MSC 46L1054H05

keywords ucpmapsconditionalexpectationsEffros–MaréchaltopologyPolishspacevonNeumannalgebrasamenabilityHaageruppropertyCowling-Haagerupconstant

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 establishes that for a von Neumann algebra equipped with a faithful normal state, the collection of subalgebras that are the ranges of state-preserving conditional expectations forms a closed subspace of the space of all state-preserving unital completely positive maps. Being closed in a Polish space, it inherits a Polish topology, which the paper shows coincides with the Effros–Maréchal topology on subalgebras. On this space, amenability, the Haagerup property, and bounded Cowling–Haagerup constant are closed properties, and the Cowling–Haagerup constant is lower semicontinuous. In the opposite direction, the paper proves that for many purely infinite factors, the subalgebras lacking a state-preserving conditional expectation form an open and dense set, so a generic subalgebra is not the range of such an expectation. The upshot is a clear topological picture of which subalgebras can be "expected onto" and which structural properties are stable under limits.

What carries the argument

The key machinery is the bijection between subalgebras in E_φ and their unique φ-preserving conditional expectations, which are exactly the idempotent state-preserving ucp maps. The map sending a ucp map to its square is continuous in the topology of pointwise strong convergence, so the idempotent ones form a closed set; then each idempotent is identified with the conditional expectation onto its fixed-point algebra. For the density results, the construction replaces a corner of a given subalgebra with a subalgebra of a corner algebra that admits no conditional expectation, using the classical characterization of finite von Neumann algebras by the existence of a masa with no normal condition

What would settle it

A single counterexample to Theorem 5.5: a sequence N_n of subalgebras in E_φ converging to N with each N_n amenable relative to P but N not amenable relative to P. Such a sequence would directly falsify the closedness claim. A second falsifier for Theorem 6.2 would be finding a type III_λ factor with λ<1 and a faithful normal state for which subalgebras lacking the expectation are not dense.

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Extended reading notes

Core claim

The central discovery is that E_φ—the set of von Neumann subalgebras of M that admit a φ-preserving conditional expectation—is closed in the Polish space of state-preserving ucp maps, making it Polish, and that its topology is exactly the restriction of the Effros–Maréchal topology. This is obtained by identifying E_φ with the set of idempotent state-preserving ucp maps, observing that Φ ↦ Φ² is continuous, so idempotency is closed. The paper also shows the complement direction: when M is purely infinite and the centraliser M_φ is infinite-dimensional, the subalgebras without a φ-preserving expectation are open and dense, so the existence of such an expectation is a rare property.

Load-bearing premise

The paper's proof that relatively amenable subalgebras are closed inherits a criterion from the literature that equates relative amenability with a weak-containment condition at the level of continuous cores; the paper does not verify the hypotheses of that criterion, so if it fails for arbitrary type III algebras, that closedness conclusion is not established.

Editorial extensions

If this is right

Amenable subalgebras with φ-preserving expectations form a closed set; a non-amenable factor admitting such an expectation cannot be approximated from inside by amenable subalgebras with expectations.
The Cowling–Haagerup constant is lower semicontinuous on E_φ, and the set of weakly amenable subalgebras is F_σ; moreover the paper constructs examples where the constant is not continuous, so it is genuinely only semicontinuous.
Subalgebras with the relative Haagerup property form a closed set within E_φ, giving a unified proof of closedness for amenable and Haagerup classes.
For purely infinite algebras whose state has infinite-dimensional centraliser—including all type III_λ factors with λ<1 and many type III_1 factors—the generic subalgebra does not admit a φ-preserving conditional expectation.
The Effros–Maréchal topology restricted to E_φ is Polish and coincides with pointwise convergence of the conditional expectations, extending the classical convergence criterion to this full space.

Reading between the lines

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

If the generic-density result extends to all states on type III_1 factors (the generic state has trivial centraliser), then the phenomenon would be even more sweeping: expectation-free subalgebras would be residual for a generic state, not just for states with infinite-dimensional centraliser.
The closedness of relatively amenable subalgebras suggests a tool for detecting relative amenability by approximation: a limit of relatively amenable subalgebras remains relatively amenable, which may be useful in rigidity questions for subfactor lattices.
The dichotomy—closed Polish space of expected subalgebras versus open dense complement—could be read as an obstruction to any continuous parametrization of conditional expectations over the whole space of subalgebras.
A testable extension: determine whether the set Exp(M) in Theorem 6.4 is Borel for non-finite M; the paper leaves this as an open question.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper delivers what the abstract promises: a clean Polish-space framework for state-preserving ucp maps, a closed subspace E_phi of subalgebras admitting a phi-preserving conditional expectation, and a demonstration that the induced topology matches Effros–Maréchal. The authors are explicit that the topology result recovers Haagerup–Winslow; the genuinely new content is the framework itself and the applications: closedness of amenable and Haagerup subalgebras in E_phi, lower semicontinuity and failure of continuity of the Cowling–Haagerup constant, and the open-density results for subalgebras without state-preserving expectations. I checked the main proofs (Prop 3.3, Thm 4.3, Thm 4.7, Thm 6.2, Thm 6.4) and they hold together. The square-map continuity argument, the projection convergence via Lemma 2.3, and the approximation in Thm 4.7 are all sound. The self-citation to [FMMP24] is legitimate; the Polishness of Maréchal topology is an independent prior result and the paper says so.

The soft spots are real but not load-bearing. Theorem 5.5, closedness of relatively amenable subalgebras, rests on Isono’s relative-amenability/core criterion ([Iso19, Thm 3.2, A.6]) as a black box, with hypotheses not re-verified for arbitrary type III inclusions. A specialist should check that step before building on it. But the reader and stress-test are right: this theorem is not needed for the paper’s headline results, and Proposition 5.2 (amenable subalgebras) has an independent proof. The examples in 5.15/5.17 use external results (Ozawa, Cowling–Haagerup) but those citations are standard and the logic is transparent. There are also quite a few typos and minor notational slips; they don’t obscure the arguments.

This is a careful, honest paper. The central claims are proven in detail, the literature is engaged with fairly, and the new applications are nontrivial. It deserves a serious referee, and I would send it out as-is rather than desk reject.

Referee Report

2 major / 5 minor

Summary. The paper studies spaces of state-preserving unital completely positive (ucp) maps between von Neumann algebras equipped with faithful normal states. It proves that this space is Polish for the topology of pointwise strong (equivalently 2-norm) convergence, and it identifies the set E_φ of subalgebras admitting a φ-preserving conditional expectation with a closed subspace of the space of idempotent state-preserving ucp maps. The main topological claims are that E_φ is Polish and that its induced 1-Lipschitz topology coincides with the restriction of the Effros–Maréchal topology on S(M), recovering a theorem of Tsukada and Haagerup–Winslow. These tools are then applied to structural classes: amenable, relatively amenable, Haagerup, and weakly amenable subalgebras, with closedness or lower-semicontinuity results. In the final section, the paper shows that for certain type III algebras the subalgebras lacking a state-preserving conditional expectation form an open dense set, and characterizes non-finite von Neumann algebras by denseness of subalgebras without any normal conditional expectation.

Significance. If the proofs are completed, the paper makes a solid contribution: it unifies several natural topologies on UCP spaces, gives a clean Polish topology on E_φ, recovers the Haagerup–Winslow correspondence, and provides new generic-density results for subalgebras lacking conditional expectations. The overall organization is clear, and the main constructions — idempotent UCP maps, projection convergence, approximation arguments — are mostly presented in detail. The use of prior work, especially [FMMP24], is explicit. However, the proof of the key topology-equality theorem (Theorem 4.7) contains an incorrect operator identity, and one relative-amenability result relies on an external black box whose hypotheses are not verified. These issues need to be addressed before the paper can be accepted.

major comments (2)

[§4, Theorem 4.7 (second half)] The proof of the second half of Theorem 4.7 uses the identity E_N^φ(x)=e_N x e_N, stated before Theorem 4.3, to justify the equality sup_{x∈M1} |⟨E_N(x)ξ,ξ⟩−⟨E_Nn(x)ξ,ξ⟩| = sup_{x∈M1} |⟨x e_N ξ,e_N ξ⟩−⟨x e_Nn ξ,e_Nn ξ⟩| for an arbitrary vector ξ. This identity is false on all of L^2(M,φ); the correct relation is e_N x e_N = E_N^φ(x)e_N, i.e. the identity holds only on L^2(N,φ). For example, take M=M_2(C), N the diagonal matrices, φ=tr, ξ=E_{12}, and x=E_{11}. Then E_N(x)ξ=E_{11}E_{12}=E_{12}, so ⟨E_N(x)ξ,ξ⟩=1, while e_N ξ=0, so ⟨x e_N ξ,e_N ξ⟩=0. Since this equality is the step that proves Maréchal convergence from pointwise 2-norm convergence, the proof of the second half of Theorem 4.7 is incomplete. The theorem itself is true (Tsukada/HW98), so this is repairable, but the proof as written needs correction.
[§5.1, Theorem 5.5] The proof of closedness of relatively amenable subalgebras A_P passes through the continuous core and invokes [Iso19, Theorem 3.2 and Appendix Theorem A.6] as black boxes. The manuscript does not state or verify the hypotheses of these results in the stated generality of arbitrary type III inclusions with a faithful normal state φ. If Isono's criterion requires additional assumptions (e.g. factoriality or specific core conditions), the closedness assertion may fail as stated. This theorem is not needed for the main E_φ topology results or for Theorem 6.2, so the issue is secondary, but it should be addressed by either verifying the hypotheses or restricting the statement.

minor comments (5)

[Throughout] There are several typos: 'Tsukuda' should be 'Tsukada' (Theorem 4.7 heading), 'Hany' should be 'any' (Remark 5.11), 'idempodent' should be 'idempotent' (Lemma 4.2), and 'posses' should be 'possess' (Section 4).
[§5.2, Definition 5.6 and Theorem 5.7] Definition 5.6 says 'We say that M has the Haagerup property relative to N', but Theorem 5.7 concerns 'N has the Haagerup property relative to P'. Please clarify which algebra is being tested and make the notation consistent.
[§4, Theorem 4.7 and §5.5] The notation E^ω(fM) appears in the proof of Theorem 5.5, but the paper elsewhere writes E_φ. Please define E^ω (or E_ω) explicitly when it is first used.
[§6, Theorem 6.2] The proof assumes the existence of an increasing sequence q_n∈M_φ with q_n→1 strongly and p_n=1−q_n nonzero for all n. This is true under the stated hypothesis that M_φ is infinite-dimensional, but a one-sentence justification (e.g. existence of a countable partition of unity in the σ-finite algebra M_φ) would help.
[§3, Remark 4.6] Remark 4.6 says that separate continuity of composition implies Φ↦Φ^2 is Baire class 1 for the pointwise ultraweak topology. This is not immediate and, if used, needs a reference or a proof. Since the remark is non-essential, this is a minor point.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the closedness/Polishness of E_φ and its Effros–Maréchal identification are derived directly from standard idempotent/projection arguments and external selector theorems; self-citations are not load-bearing.

full rationale

The central derivation is not circular. E_φ is defined as the set of subalgebras admitting a φ-preserving conditional expectation, characterized by Takesaki's modular-invariance theorem; Lemma 4.2 independently identifies such conditional expectations with idempotent conservative ucp maps. Theorem 4.3 is proved by showing that the square map is continuous on conservative ucp maps and that idempotents form a closed set, so no part of the proof assumes the theorem being proved. Theorem 4.7 compares the Maréchal topology with the pointwise-strong topology in two directions, using standard strong-convergence-of-projections facts and external dense-selector theorems (HW98, with FMMP24 cited secondarily); it does not presuppose the convergence of the conditional expectations. The self-citations to [FMMP24] concern Polishness of the Maréchal topology, a projection-convergence fact, and a selector theorem; these are prior independent results and the main arguments do not reduce to assuming the current conclusions. The black-box use of [Iso19] in Theorem 5.5 is a verification/correctness risk about whether the core-criterion hypotheses hold, not a circular derivation, and Theorem 5.5 is not load-bearing for the headline E_φ-topology or genericity results. No fitted parameters, no renamed known results, and no author-imported uniqueness theorem force the central claims.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new physical or algebraic entities. The central claims rest on standard Takesaki theory, the authors' prior FMMP24 Polishness proof, and several external theorems (Iso19, Oza12, CH89, Con73) used as black boxes. The most consequential external dependency is Isono's core criterion in Theorem 5.5.

assumptions (7)

standard math Takesaki's conditional expectation theorem: for N⊂M and nfs φ, a φ-preserving conditional expectation M→N exists iff σ_t^φ(N)=N for all t.
Used throughout, e.g., in the definition of E_φ and in the proof of Theorem 4.7; the paper relies on the theorem but does not prove it.
standard math The Effros-Maréchal topology on S(M) is Polish and admits the continuous selection/approximation results of [FMMP24] used in Remark 4.4 and Theorem 4.7.
The paper cites [FMMP24] for the full proof; it is the authors' own prior work, but it is an independent preprint with a proof.
standard math Takesaki's masa theorem: every non-finite von Neumann algebra contains a masa A with no normal conditional expectation M→A.
Used in Lemma 6.1 and repeatedly in Theorem 6.2/6.4 to choose subalgebras A_n in non-finite corners.
domain assumption Isono's relative amenability/core criterion ([Iso19], Thm 3.2 and Appendix Thm A.6): (N,E_N)⋖_M P iff its continuous core satisfies the corresponding weak containment condition.
Load-bearing in Theorem 5.5; the paper does not reprove this and invokes it for general (possibly type III) von Neumann algebras.
standard math Ozawa's result [Oza12, Cor 4]: if Λ is non-amenable and Σ non-trivial, then Λcb(Σ≀Λ)=∞.
Used in Proposition 5.15 to produce discontinuity of Λcb; cited, not proved.
standard math Cowling-Haagerup product formula [CH89]: Λcb of a direct product is the product of the Λcb of the factors.
Used in Proposition 5.17 to build a sequence Γ≤n with Λcb→∞ while Γ≤n→Γ; cited, not proved.
domain assumption Connes' classification fact [Con73]: for type III_λ factors, 0≤λ<1, every faithful normal state has infinite-dimensional centralizer; for type III_1, faithful normal states with infinite-dimensional centralizer satisfy Theorem 6.2 hypotheses.
Used in Remark 6.3 to extend Theorem 6.2 to type III_λ factors.

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Cite this review

Pith. "Pith review of Spaces of UCP maps and subalgebras of von Neumann algebras." pith.science (2026). https://pith.science/paper/HG67HYZW

@misc{pith2026260714358,
  author       = {Pith},
  title        = {Pith review of: Spaces of UCP maps and subalgebras of von Neumann algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HG67HYZW}},
  note         = {Machine review of arXiv:2607.14358}
}

abstract

In this paper, we establish that several natural topologies on the space of state-preserving unital completely positive maps coincide and that make it a Polish space. We then focus on the subspace of state-preserving conditional expectations and analyse its topology in detail, recovering the Haagerup-Winslow result that it aligns with the Effros-Mar\'echal topology on the space of von Neumann subalgebras. This correspondence is then applied to structural classes of subalgebras, including amenable, Haagerup and weakly amenable subalgebras. Among other consequences, we demonstrate the closedness of amenable subalgebras admitting state-preserving conditional expectations and analyze the semicontinuity and failure of continuity of the Cowling-Haagerup constant as a function on subalgebras. Finally, we investigate the space of von Neumann subalgebras that are not the image of state preserving conditional expectations for a fixed faithful normal state. For several important classes of von Neumann algebras, such as type ${\rm III}_\lambda$ factors with $0 \le \lambda < 1$ and type ${\rm III}_1$ factors with a state whose centraliser is infinite dimensional, we show that the subalgebras lacking state-preserving conditional expectations form an open and dense subset. Thus, in these settings, the generic subalgebra is not the range of state preserving conditional expectation.

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Reference graph

Works this paper leans on

69 extracted references · 4 canonical work pages

[1]

1950 , journal =

Minimal Identities for Algebras , author =. 1950 , journal =

1950
[2]

Journal of Functional Analysis , volume =

On Approximate Factorizations of Completely Positive Maps , author =. Journal of Functional Analysis , volume =
[3]

Forrest Stinespring , journal =

W. Forrest Stinespring , journal =. Positive Functions on C*-Algebras , urldate =
[4]

Arveson , title =

William B. Arveson , title =. Acta Mathematica , number =. 1969 , doi =

1969
[5]

Bannon, Jon P and Cameron, Jan and Mukherjee, Kunal , year =. On. International Mathematics Research Notices , volume =
[6]

An Embedding Theorem for the

Beer, Gerald , year =. An Embedding Theorem for the. Michigan Mathematical Journal , volume =
[7]

Beer, Gerald , year =. A. Proceedings of the American Mathematical Society , volume =
[8]

1993 , series =

Topologies on Closed and Closed Convex Sets , author =. 1993 , series =

1993

Show all 69 references

[9]

2010 , url =

An Introduction to II_1 factors , author =. 2010 , url =

2010
[10]

, year =

Conway, John B. , year =. A
[11]

2003 , journal =

Partitions of Unity , author =. 2003 , journal =

2003
[12]

Brown and Narutaka Ozawa , title =

Nathanial P. Brown and Narutaka Ozawa , title =. 2008 , isbn =

2008
[13]

1983 , journal =

Group factors of the Haagerup type , author =. 1983 , journal =

1983
[14]

Connes , journal =

A. Connes , journal =. Classification of Injective Factors Cases II _1 , II _ , III _ , 1 , urldate =
[15]

and Jones, V

Connes, A. and Jones, V. , title =. Bulletin of the London Mathematical Society , volume =. doi:https://doi.org/10.1112/blms/17.1.57 , url =. https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/blms/17.1.57 , year =

doi
[16]

International Journal of Mathematics , volume =

FIMA, PIERRE , title =. International Journal of Mathematics , volume =. 2010 , doi =

2010
[17]

, year =

Effros, Edward G. , year =. The. Pacific Journal of Mathematics , volume =
[18]

Haagerup, Uffe and Winsl. The. 1998 , journal =

1998
[19]

Haagerup, Uffe and Winsl. The. 2000 , month = mar, journal =

2000
[20]

Equivalence of Normal States on von

Haagerup, Uffe and St. Equivalence of Normal States on von. 1990 , journal =

1990
[21]

Jolissaint, Paul , year = 2005, journal =

2005
[22]

The linear SL2(

Jolissaint, Paul and Valette, Alain , year = 2023, number =. The linear SL2(. doi:10.48550/arXiv:2311.02683 , archiveprefix =

2023 doi arXiv
[23]

1995 , series =

Classical Descriptive Set Theory , author =. 1995 , series =. doi:10.1007/978-1-4612-4190-4 , isbn =

1995 doi
[24]

, year =

Kelley, John L. , year =. General
[25]

Splitting in Orbit Equivalence, Treeable Groups, and the

Kida, Yoshikata , year = 2017, volume =. Splitting in Orbit Equivalence, Treeable Groups, and the. Hyperbolic

2017
[26]

, year =

Kuratowski, K. , year =. Topology :
[27]

Orbita Mathematicae , volume =

On Dense Orbits in the Space of Subequivalence Relations , author =. Orbita Mathematicae , volume =
[28]

1992 , publisher =

Introduction to Operator Algebras , author =. 1992 , publisher =

1992
[29]

Topologie et Structure Bor\'elienne Sur l'ensemble Des Alg\`ebres de von

Mar. Topologie et Structure Bor\'elienne Sur l'ensemble Des Alg\`ebres de von. 1973 , journal =

1973
[30]

Continuous

Michael, Ernest , year =. Continuous. Annals of Mathematics , volume =
[31]

1951 , journal =

Topologies on Spaces of Subsets , author =. 1951 , journal =

1951
[32]

Masas and Bimodule Decomposition of

Mukherjee, Kunal , year =. Masas and Bimodule Decomposition of. The Quarterly Journal of Mathematics , volume =
[33]

, year =

Nielsen, Ole A. , year =. Borel. American Journal of Mathematics , volume =
[35]

doi:10.48550/arXiv.2503.16017 , archiveprefix =

On Weakly Amenable Groupoids , author =. doi:10.48550/arXiv.2503.16017 , archiveprefix =

doi arXiv
[36]

Popa, Sorin and Vaes, Stefaan , TITLE =. Invent. math. , VOLUME =. 2010 , PAGES =. doi:10.4007/annals.2010.172.713 , URL =

2010 doi
[37]

Completely

Paulsen, Vern , year =. Completely
[38]

1989 , series =

Analysis Now , author =. 1989 , series =. doi:10.1007/978-1-4612-1007-8 , isbn =

1989 doi
[39]

Rudin, Walter , year =
[40]

Geometric

Sakai, Katsuro , year =. Geometric. doi:10.1007/978-4-431-54397-8 , isbn =

doi
[41]

Proceedings of the Japan Academy , number =

Sh. Proceedings of the Japan Academy , number =. 1957 , doi =

1957
[42]

Srivastava, S. M. , year =. A Course on. doi:10.1007/978-3-642-85473-6 , isbn =

doi
[43]

Lectures on von

Stratila, Serban-Valentin and Zsido, Laszlo , year =. Lectures on von
[44]

, TITLE =

Takesaki, M. , TITLE =. 2002 , PAGES =

2002
[45]

Theory of

Takesaki, Masamichi , editor =. Theory of. doi:10.1007/978-3-662-10451-4 , isbn =

doi
[46]

Conditional Expectations in von

Takesaki, Masamichi , year =. Conditional Expectations in von. Journal of Functional Analysis , volume =. doi:10.1016/0022-1236(72)90004-3 , langid =

doi
[47]

Tsukada, Makoto , year =. The. Proceedings of the American Mathematical Society , volume =
[48]

2024 , eprint=

Michael's selection theorem and applications to the Mar\'echal topology , author=. 2024 , eprint=

2024
[49]

Marrakchi, Amine and Vaes, Stefaan , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2024 , PAGES =. doi:10.1515/crelle-2024-0007 , URL =

2024 doi
[50]

, TITLE =

Connes, A. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1976 , NUMBER =. doi:10.2307/1971057 , URL =

1976 doi
[51]

Caspers, Martijn and Skalski, Adam , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2015 , NUMBER =. doi:10.1093/imrn/rnu252 , URL =

2015 doi
[52]

Haagerup, Uffe , TITLE =. Invent. Math. , FJOURNAL =. 1978/79 , NUMBER =. doi:10.1007/BF01410082 , URL =

1978 doi
[53]

Cowling, Michael and Haagerup, Uffe , TITLE =. Invent. Math. , FJOURNAL =. 1989 , NUMBER =. doi:10.1007/BF01393695 , URL =

1989 doi
[54]

Ozawa, Narutaka , TITLE =. Kyoto J. Math. , FJOURNAL =. 2012 , NUMBER =. doi:10.1215/21562261-1550985 , URL =

2012 doi
[55]

Amrutam, Tattwamasi and Hartman, Yair and Oppelmayer, Hanna , TITLE =. J. Funct. Anal. , FJOURNAL =. 2025 , NUMBER =. doi:10.1016/j.jfa.2024.110718 , URL =

2025
[56]

Elek, G\'abor and Szab\'o, Endre , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2011 , NUMBER =. doi:10.1090/S0002-9939-2011-11222-X , URL =

2011 doi
[57]

Haagerup, Uffe , TITLE =. J. Lie Theory , FJOURNAL =. 2016 , NUMBER =

2016
[58]

Connes, Alain , TITLE =. Ann. Sci. \'Ecole Norm. Sup. (4) , FJOURNAL =. 1973 , PAGES =

1973
[59]

Journal of Functional Analysis , volume =

Takesaki, Masamichi , title =. Journal of Functional Analysis , volume =
[60]

Jones, Vaughan F. R. , title =. Inventiones Mathematicae , volume =
[61]

Christopher , title =

Lance, E. Christopher , title =. Inventiones Mathematicae , volume =
[62]

Journal of the American Mathematical Society , volume =

Junge, Marius and Xu, Quanhua , title =. Journal of the American Mathematical Society , volume =
[63]

Annals of Mathematics , volume =

Popa, Sorin , title =. Annals of Mathematics , volume =
[64]

Journal de Mathématiques Pures et Appliquées , volume =

Houdayer, Cyril and Vaes, Stefaan , title =. Journal de Mathématiques Pures et Appliquées , volume =. 2013 , doi =

2013
[65]

Compositio Mathematica , volume =

Houdayer, Cyril and Ueda, Yoshimichi , title =. Compositio Mathematica , volume =. 2016 , doi =

2016
[66]

Journal of the European Mathematical Society , volume =

Isono, Yusuke , title =. Journal of the European Mathematical Society , volume =. 2017 , doi =

2017
[67]

Advances in Mathematics , volume =

Caspers, Martijn and Klisse, Mario and Skalski, Adam and Vos, Gerrit and Wasilewski, Mateusz , title =. Advances in Mathematics , volume =. 2023 , pages =. doi:10.1016/j.aim.2023.109017 , issn =

2023
[68]

Journal of Functional Analysis , volume =

Isono, Yusuke , title =. Journal of Functional Analysis , volume =. 2019 , doi =. 1712.00925 , archivePrefix =

2019 arXiv
[69]

1986 , url =

Popa, Sorin , title =. 1986 , url =

1986
[70]

doi:10.48550/arXiv.2407.10905 , archiveprefix =

Noncommutative Topological Boundaries and Amenable Invariant Random Intermediate Subalgebras , author =. doi:10.48550/arXiv.2407.10905 , archiveprefix =. 2407.10905 , primaryclass =

doi arXiv

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

1950 , journal =

Minimal Identities for Algebras , author =. 1950 , journal =

1950

[2] [2]

Journal of Functional Analysis , volume =

On Approximate Factorizations of Completely Positive Maps , author =. Journal of Functional Analysis , volume =

[3] [3]

Forrest Stinespring , journal =

W. Forrest Stinespring , journal =. Positive Functions on C*-Algebras , urldate =

[4] [4]

Arveson , title =

William B. Arveson , title =. Acta Mathematica , number =. 1969 , doi =

1969

[5] [5]

Bannon, Jon P and Cameron, Jan and Mukherjee, Kunal , year =. On. International Mathematics Research Notices , volume =

[6] [6]

An Embedding Theorem for the

Beer, Gerald , year =. An Embedding Theorem for the. Michigan Mathematical Journal , volume =

[7] [7]

Beer, Gerald , year =. A. Proceedings of the American Mathematical Society , volume =

[8] [8]

1993 , series =

Topologies on Closed and Closed Convex Sets , author =. 1993 , series =

1993