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Non-commutative Stein's Method: Applications to Free Probability and Sums of Non-commutative Variables

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Pith's one-line read A non-commutative Stein method bounds semicircular approximation error in total variation and Wasserstein distance.

desk verdict A genuinely novel dual free Stein equation built on the free Ornstein-Uhlenbeck semigroup, but the main proofs rest on an invalid application of the Cauchy integral formula to non-holomorphic test functions. read the letter →

arxiv 2411.16103 v2 pith:LHUC2GHY submitted 2024-11-25 math.PR

Mario Díaz , Arturo Jaramillo This is my paper · ORCID
classification math.PR MSC 60F0546L5360G5060B10
keywords freeprobabilityStein'smethodsemicirculardistributionBerry-EsseentheoremtotalvariationdistanceWassersteinOrnstein-Uhlenbecksemigroupdependencygraphs
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 develops a Stein-type method for the semicircular distribution in free probability, using the free Ornstein-Uhlenbeck semigroup to interpolate between a candidate measure and the semicircle law. The method yields quantitative estimates for two distances: total variation and the non-commutative Wasserstein (Kantorovich-Rubinstein) distance, between the free convolution of many small centered measures and the semicircle. In the homogeneous, freely independent case the estimates are of order $n^{-1/2}$ in total variation and $n^{-(q-1)/2}$ in Wasserstein distance when the summands match the semicircle's moments up to order $q$. The bounds also accommodate weak dependence, encoded by dependency graphs, with the maximum degree of the graph entering polynomially into the error.

What carries the argument

The central object is the operator $L_{\boxplus}[g](x,y) = -xg(x) + (g(y)-g(x))/(y-x)$, which plays the role of the classical Stein operator for the semicircle, together with the dual free Stein equation built from the free Ornstein-Uhlenbeck semigroup $P^*_\theta[\mu] = D_{e^{-\theta}}[\mu] \boxplus D_{\sqrt{1-e^{-2\theta}}}[s]$. The semigroup interpolates between $\mu$ at $\theta=0$ and the semicircle at $\theta=\infty$, and the difference $\langle s,h\rangle-\langle\mu,h\rangle$ is written as an integral over $\theta$ of $\langle P^*_\theta[\mu]\otimes P^*_\theta[\mu], L_{\boxplus}[Dh]\rangle$. The cancellations then rely on a non-commutative Taylor expansion (Lemma 4) with $\Delta(a,r)=2s[(z-a)r]-r^2$, applied to the resolvent $g(x)=(z-x)^{-2}$. This decomposition is what converts the third moments of the summands into the error bound.

What would settle it

Choose a smooth, compactly supported, non-holomorphic test function, such as a bump function, and compute both sides of the identity $\langle s,h\rangle - \langle\nu_n,h\rangle = \int_0^\infty \langle P^*_\theta[\nu_n]\otimes P^*_\theta[\nu_n] - s\otimes s, L_{\boxplus}[Dh]\rangle\,d\theta$ for an explicit inhomogeneous family $\nu_n$; if the right-hand side is not the value given by the Cauchy substitution used in the paper, then the derivation of the bound (1.2) does not go through for this $h$. Equivalently, the bound's universal constant can be checked against a direct numerical evaluation of $d_{TV}$ for simple summands.

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Extended reading notes

Core claim

The paper's central claim is that the semicircular Stein equation can be replaced by a dual free Stein equation: for a probability measure $\mu$, $\langle s,h\rangle - \langle \mu,h\rangle = \langle S^*_{\boxplus}[\mu], L_{\boxplus}[Dh]\rangle$, where $S^*_{\boxplus}[\mu] = \int_0^\infty (P^*_\theta[\mu]\otimes P^*_\theta[\mu] - s\otimes s)\,d\theta$ and $P^*_\theta[\mu] = D_{e^{-\theta}}[\mu] \boxplus D_{\sqrt{1-e^{-2\theta}}}[s]$ is the free Ornstein-Uhlenbeck semigroup. The operator $L_{\boxplus}[g](x,y) = -xg(x) + (g(y)-g(x))/(y-x)$ characterizes the standard semicircle in the sense that $\mu=s$ if and only if $\langle \mu\otimes\mu, L_{\boxplus}[f]\rangle=0$ for all $f\in C^1$. Using this identity, the paper proves that centered, uniformly bounded summands with total variance one and dependency graph of maximum degree $D(E_n)$ satisfy $d_{TV}(\mu_{1,n}\boxplus\cdots\boxplus\mu_{n,n}, s) \le C D(E_n)^2 \sum_k m_3[\mu_{k,n}]$, and with moment-matching rank $q$, $d_W(\nu_n,s)\le C D(E_n)^{q+1}\sum_k m_{q+1}[\mu_{k,n}]$. The proof transfers the classical Stein cancellation to the resolvent $g(x)=(z-x)^{-2}$ through a Cauchy integral representation of the test function $h$, which is the step that carries the main technical assumption.

Load-bearing premise

The total-variation proof assumes that every $C^1$ test function $h$, and its derivative, can be written through the Cauchy integral formula on a rectangle around the support of the measures; this representation is valid only for holomorphic functions, and the paper does not state or prove that the $C^1$ functions defining the distance are holomorphic.

Editorial extensions

If this is right

  • The free Berry-Esseen theorem holds under total variation with rate $n^{-1/2}$ in the homogeneous, freely independent case, matching the classical Stein-method rate.
  • Moment matching up to order $q$ accelerates the Wasserstein convergence to $n^{-(q-1)/2}$, showing that the third-moment improvement is a special case of a general hierarchy.
  • Weak dependence enters the error only through the maximum degree $D(E_n)$, so the same bounds cover partial freeness, including sparse dependence structures among the summands.
  • The dual free Stein equation gives a probabilistic route to free limit theorems that does not rely on analytic Cauchy-transform or R-transform computations.

Reading between the lines

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

  • Editorial inference: the same interpolation should apply to non-unit-variance semicircle targets by conjugating $L_{\boxplus}$ with a dilation; this would mend the comparison in Theorem 4 between blocks $\mu_V$ and $g[\mu_V]$ when the block variance is not one.
  • Editorial inference: the dependency-graph setup suggests the rate estimates should transfer to random-matrix ensembles with sparse interaction structures, where asymptotic freeness replaces exact freeness; the paper does not state this application.
  • Editorial inference: one could test the sharpness of the exponents by constructing summands with near-zero third moment but large fourth moment, to see whether the Wasserstein bound can be improved beyond the stated power of $D(E_n)$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes a non-commutative analogue of Stein's method for the semicircular distribution, built on the free Ornstein-Uhlenbeck semigroup and a 'dual free Stein equation' acting on measures. The main applications are quantitative free Berry-Esseen theorems: Theorem 3 claims a total-variation bound for sums of weakly dependent (in the free sense) centered variables, with rate n^{-1/2} in the homogeneous case, and Theorem 4 claims an improved Wasserstein bound under moment-matching conditions, with rate n^{-(q-1)/2}. The proofs interpolate between the summands and semicircular variables via the free Ornstein-Uhlenbeck semigroup, reduce the test function to a resolvent via the Cauchy integral formula, and then perform cancellation estimates using free independence and moment-matching assumptions.

Significance. If the results were correct, they would constitute a substantial contribution: a Stein-method proof of a free Berry-Esseen theorem in total variation for weakly dependent summands, and a Wasserstein refinement under moment matching, are both natural and potentially influential results. The paper is clearly organized and the dual free Stein equation idea is elegant. However, the central proof step—the application of the Cauchy integral formula to arbitrary C^1 test functions—is invalid, and the Wasserstein proof contains a variance mismatch that is not merely cosmetic. The paper also relies on an unproved expansion (4.13) and on an unpublished reference for a key Wasserstein subadditivity lemma. These are load-bearing gaps, so the central claims are not established in the present manuscript.

major comments (4)
  1. [Section 6.1, after Eq. (6.1)] The proof applies the Cauchy integral formula to an arbitrary bounded C^1 test function h, writing h(x)=(2πi)^{-1}∮_R h(z)/(z-x) dz and Dh(x)=(2πi)^{-1}∮_R h(z)/(z-x)^2 dz over the rectangle with vertices (±6,±1), and then uses this representation to replace L⊞[Dh] by expressions involving the resolvent g(x)=(z-x)^{-2}. This formula is valid only for h holomorphic in a neighborhood of the contour; the class of C^1 bounded functions defining the total-variation dual includes compactly supported smooth bumps that have no holomorphic extension, and the supremum defining d_TV is over measurable functions. Since the subsequent symmetrization identity, the introduction of g, and all cancellation estimates for Y_{θ,z} depend on this representation, the proof of Theorem 3 does not establish the claimed bound.
  2. [Theorems 3 and 4; Corollaries 1 and 2] The theorem statements omit hypotheses that the proof uses and that appear only in the corollaries. Theorem 3 asserts the bound for any sequence ξ_n with dependency graph, but the proof uses that the summands are centered (to obtain τ[η_θ^V g(F^V_{θ,n})]=0), that the total variance is one (to identify the limit as the standard semicircle s), and that the supports are uniformly bounded (to obtain Supp ν_n ⊂ [-3,3] and Supp P_θ[ν_n] ⊂ [-5,5] via superconvergence). Without these assumptions the statement is false as written, for example for uncentered summands or for summands whose variances do not sum to one. Theorem 4 is stated without defining q and without listing the normalization and moment-matching hypotheses used in its proof.
  3. [Section 6.2, Eqs. (6.4)-(6.6)] Proposition 4 and the operator L⊞ characterize the standard semicircular distribution s with variance one. In the proof of Theorem 4, the same operator is applied to ⟨µV,h⟩ - ⟨g[µV],h⟩, where g[µV] is the semicircular distribution with the variance of µV, which is not equal to one in general. The needed rescaling of L⊞ and P_θ for non-unit variance is not supplied, so equation (6.6) does not follow from Proposition 4. Moreover, the decomposition s = ⊞_{V∈J} g[µV] used in (6.4) requires the variances of the blocks to sum to one, a normalization that is absent from Theorem 4.
  4. [Section 4.5, Eq. (4.13); Section 7, Lemma 4] The expansion (a Δ(a,r))^j = ∑_{α∈I} (f^1_{j,α} Q^1_{j,α}(z) Υ^1_{j,α}(a,r) + f^2_{j,α} Q^2_{j,α}(z) Υ^2_{j,α}(a,r)) is asserted with 'universal constants' and polynomials Q that are never defined, and it is used in the proof of Theorem 4 to separate moments of ξ_θ^V from terms involving g(ψ_θ^V) and to show independence of θ. Lemma 4, on which the proof of Theorem 3 also relies, is not actually proved: the displayed argument assumes the identity for q and reduces the induction to the q=1 case, which is only asserted to follow from a direct computation and is not carried out. These are load-bearing gaps for the moment-matching Wasserstein result.
minor comments (5)
  1. [Section 4.4, Lemma 2] Lemma 2 states that S*_⊞[µ] is a signed measure, but the sentence 'The fact that S*_⊞[µ] is a well-defined probability measure' conflicts with that; moreover, the proof only bounds the pairing with Lipschitz functions and does not construct the signed measure.
  2. [Section 3.2, Lemma 3.2] The Wasserstein subadditivity for free convolution in (3.3) is cited to the unpublished preprint [20]; since it is used in (6.4) in the proof of Theorem 4, the authors should either include a proof or provide a published reference.
  3. [Corollary 2] The moment matching rank q is defined using 'mj[ν] = mj[g[ν]]', but no measure ν has been defined in the statement of the corollary; the intended statement is presumably about the µ_{k,n} or about the whole convolution, and this needs to be corrected.
  4. [Throughout] There are numerous typos and notational inconsistencies, including 'In oder' (p.6), 'Lipchitz' (p.19), 'well-possednes' (p.15), the missing bracket in '⟨P∞µn,h⟩' (p.20), and the inconsistent use of P_θ versus P*_θ in Proposition 3.
  5. [Section 6.2, proof of Theorem 4] The proof introduces q, but the theorem statement does not define q; in addition, the hypothesis lim_n D(E_n)^2 Σ m_3[µ_{k,n}] = 0 is not used in the argument, and the relation between the matching rank of individual blocks and the rank of the whole convolution should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central Stein bounds are derived from explicit semigroup identities and moment estimates, and the sole self-citation is an auxiliary parameter-free lemma.

full rationale

The derivation chain of Theorems 3 and 4 is not circular. The dual free Stein identity (4.5)/(6.1) is obtained from an explicit semigroup representation of S*⊞[µ] and the generator computation in Proposition 3; the target inequalities are not inserted as assumptions. The subsequent bounds are ordinary Lindeberg/Stein estimates: Theorem 3 reduces to bounding Yθ,z by C e^{-θ} Σ τ[|ξθ_{k,n}|^3] via the expansion of Lemma 4 and support estimates, and Theorem 4 reduces to bounding the (q+1)-th moment and uses the moment-matching cancellation of Lemma 3. No parameter is fitted to data and then renamed a prediction. The reliance on [20] for Lemma 3.2 is a self-citation, but the lemma is a parameter-free, standalone inequality (dW(⊞γk, ⊞ρk) ≤ Σ dW(γk,ρk)) whose stated assumptions do not include the CLT rates, so under the hard rules it is independent support and not a circular transfer of the conclusion. The main internal weaknesses—the use of the Cauchy integral formula for arbitrary C^1 h in Section 6.1, the unproved in-house expansion (4.13), and the variance scaling gap for g[µV] in Theorem 4—are correctness and rigor gaps, not instances of the claimed result being equivalent to its inputs.

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

No physical or empirical entities are postulated. The dual free Stein solution S*⊞ and the semigroup Pθ are new mathematical definitions, not entities needing independent falsifiable evidence. The load-bearing postulates are instead the unstated analytic assumptions listed above.

assumptions (5)
  • domain assumption Every probability measure in the statements is determined by its moments, and polynomial test functions can be approximated by the allowed C^1 class without changing the distances.
    Proposition 2 and the passage to total variation require moment determinacy and density of C^1 bounded functions in the appropriate topology; this is not fully stated.
  • domain assumption The Bercovici-Voiculescu superconvergence theorem extends to the weakly dependent block-sum convolutions ν_n, yielding supp ν_n contained in [-3,3] for large n.
    Section 6.1 uses this to justify support truncation to [-5,5]; the paper asserts the extension without proof for dependency-graph sums.
  • ad hoc to paper An arbitrary C^1 bounded test function h satisfies the Cauchy integral formula h(x) = (2πi)^{-1} ∮ h(z)/(z-x) dz over a rectangle.
    Section 6.1; this is false as stated and is the critical unsupported premise of both main proofs.
  • ad hoc to paper The Stein operator L⊞, which characterizes only the unit-variance semicircle, can be used to compare any block µV with its own semicircular approximation g[µV] of arbitrary variance.
    Section 6.2 applies Proposition 4 to the difference with g[µV] without rescaling; Proposition 4's target is the standard semicircle s.
  • ad hoc to paper The expansion (a Δ(a,r))^j = Σ (f Q Υ_1 + f Q Υ_2) in (4.13) with universal constants and polynomials exists for all j.
    Section 4.5 asserts the expansion without proof; it is used to organize the moment-matching terms in Theorem 4.

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Pith. "Pith review of Non-commutative Stein's Method: Applications to Free Probability and Sums of Non-commutative Variables." pith.science (2026). https://pith.science/paper/LHUC2GHY

@misc{pith2026241116103,
  author       = {Pith},
  title        = {Pith review of: Non-commutative Stein's Method: Applications to Free Probability and Sums of Non-commutative Variables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LHUC2GHY}},
  note         = {Machine review of arXiv:2411.16103}
}
read the original abstract

We present a straightforward formulation of Stein's method for the semicircular distribution, specifically designed for the analysis of non-commutative random variables. Our approach employs a non-commutative version of Stein's heuristic, interpolating between the target and approximating distributions via the free Ornstein-Uhlenbeck semigroup. A key application of this work is to provide a new perspective for obtaining precise estimates of accuracy in the semicircular approximation for sums of weakly dependent variables, measured under the total variation metric. We leverage the simplicity of our arguments to achieve robust convergence results, including: (i) A Berry-Esseen theorem under the total variation distance and (ii) Enhancements in rates of decay under the non-commutative Wasserstein distance towards the semicircular distribution, given adequate high-order moment matching conditions.

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Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [20]

    Jaramillo and J

    A. Jaramillo and J. Vazquez-Becerra. Distributional compariso n for non-commutative, infinitely divisible probability measures. Arxiv

    work page

  2. [1]

    Universality of free random variables: atoms for non-commutative rational functions

    Octavio Arizmendi, Guillaume C´ ebron, Roland Speicher, and Sheng Yin. Universality of free random variables: atoms for non-commutative rational functions. Adv. Math., 443:Paper No. 109595, 73, 2024

    work page 2024

  3. [2]

    Robert B. Ash. Probability and measure theory . Harcourt/Academic Press, Burlington, MA, second edition, 2000. With contributions by Catherine Dol´ eans-Dade

    work page 2000

  4. [3]

    Bercovici and D

    H. Bercovici and D. Voiculescu. Superconvergence to the cent ral limit and failure of the Cram´ er theorem for free random variables. Probab. Theory Related Fields , 103(2):215–222, 1995. NON-COMMUTATIVE STEIN METHOD 31

    work page 1995

  5. [4]

    Free convolution of measure s with unbounded support

    Hari Bercovici and Dan Voiculescu. Free convolution of measure s with unbounded support. Indiana Univ. Math. J. , 42(3):733–773, 1993

    work page 1993

  6. [5]

    Andrew C. Berry. The accuracy of the Gaussian approximation t o the sum of independent variates. Trans. Amer. Math. Soc. , 49:122–136, 1941

    work page 1941

  7. [6]

    Biane and D

    P. Biane and D. Voiculescu. A free probability analogue of the Wass erstein metric on the trace-state space. Geom. Funct. Anal. , 11(6):1125–1138, 2001

    work page 2001

  8. [7]

    Bobkov, Gennadiy P

    Sergey G. Bobkov, Gennadiy P. Chistyakov, and Friedrich G¨ otz e. Rate of convergence and Edgeworth- type expansion in the entropic central limit theorem. Ann. Probab., 41(4):2479–2512, 2013

    work page 2013

Show all 31 references
  1. [8]

    Bobkov, Gennadiy P

    Sergey G. Bobkov, Gennadiy P. Chistyakov, and Friedrich G¨ otz e. Berry-Esseen bounds in the entropic central limit theorem. Probab. Theory Related Fields , 159(3-4):435–478, 2014

    2014

  2. [9]

    Poisson convergence on the free Poisson alge bra

    Solesne Bourguin. Poisson convergence on the free Poisson alge bra. Bernoulli, 21(4):2139–2156, 2015

    2015

  3. [10]

    Vector-valued semicircular limits on the free P oisson chaos

    Solesne Bourguin. Vector-valued semicircular limits on the free P oisson chaos. Electron. Commun. Probab., 21:Paper No. 55, 11, 2016

    2016

  4. [11]

    Free quantitative fourth moment theorems on Wigner space

    Solesne Bourguin and Simon Campese. Free quantitative fourth moment theorems on Wigner space. Int. Math. Res. Not. IMRN , (16):4969–4990, 2018

    2018

  5. [12]

    Semicircular limits on the f ree Poisson chaos: counterexamples to a transfer principle

    Solesne Bourguin and Giovanni Peccati. Semicircular limits on the f ree Poisson chaos: counterexamples to a transfer principle. J. Funct. Anal. , 267(4):963–997, 2014

    2014

  6. [13]

    A quantitative fourth moment theorem in fre e probability theory

    Guillaume C´ ebron. A quantitative fourth moment theorem in fre e probability theory. Adv. Math. , 380:Paper No. 107579, 27, 2021

    2021

  7. [14]

    A note on existence of free stein kernels

    Guillaume C´ ebron, Max Fathi, and Tobias Mai. A note on existence of free stein kernels. Proceed- ings of the American Mathematical Society , 148(4):1583–1594, 2020. Article electronically published on November 19, 2019

    2020

  8. [15]

    Normal approximation by Stein’s method, volume 2

    Louis HY Chen, Larry Goldstein, and Qi-Man Shao. Normal approximation by Stein’s method, volume 2. Springer, 2011

    2011

  9. [16]

    G. P. Chistyakov and F. G¨ otze. Limit theorems in free probabilit y theory. I. Ann. Probab., 36(1):54–90, 2008

    2008

  10. [17]

    G. P. Chistyakov and F. G¨ otze. Asymptotic expansions in the C LT in free probability. Probab. Theory Related Fields, 157(1-2):107–156, 2013

    2013

  11. [18]

    On the Liapounoff limit of error in the theor y of probability

    Carl-Gustav Esseen. On the Liapounoff limit of error in the theor y of probability. Ark. Mat. Astr. Fys. , 28A(9):19, 1942

    1942

  12. [19]

    The rate of convergence for spectra of GUE and LUE matrix ensembles

    Friedrich G¨ otze and Alexander Tikhomirov. The rate of convergence for spectra of GUE and LUE matrix ensembles. Cent. Eur. J. Math. , 3(4):666–704, 2005

    2005

  13. [21]

    Wigner chaos and the fourth mo- ment

    Todd Kemp, Ivan Nourdin, Giovanni Peccati, and Roland Speiche r. Wigner chaos and the fourth mo- ment. Ann. Probab., 40(4):1577–1635, 2012

    2012

  14. [22]

    Free zero bias and infinite divisibility

    Todd Kemp Larru Goldstein. Free zero bias and infinite divisibility. 2 024

  15. [23]

    J. W. Lindeberg. Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung. Math. Z. , 15(1):211–225, 1922

    1922

  16. [24]

    Addition of freely independent random variables

    Hans Maassen. Addition of freely independent random variables . J. Funct. Anal. , 106(2):409–438, 1992

    1992

  17. [25]

    Lectures on the combinatorics of free probability , volume 335 of London Mathematical Society Lecture Note Series

    Alexandru Nica and Roland Speicher. Lectures on the combinatorics of free probability , volume 335 of London Mathematical Society Lecture Note Series . Cambridge University Press, Cambridge, 2006

    2006

  18. [26]

    Normal approximations with Malliavin calculus: from Stein ’s method to universality

    Ivan Nourdin and Giovanni Peccati. Normal approximations with Malliavin calculus: from Stein ’s method to universality . Number 192. Cambridge University Press, 2012

    2012

  19. [27]

    Quantitative es timates for operator-valued and in- finitesimal boolean and monotone limit theorems

    Pei-Lun Tseng Octavio Arizmendi, Marwa Banna. Quantitative es timates for operator-valued and in- finitesimal boolean and monotone limit theorems. 2022

    2022

  20. [28]

    Random geometric graphs, volume 5 of Oxford Studies in Probability

    Mathew Penrose. Random geometric graphs, volume 5 of Oxford Studies in Probability. Oxford University Press, Oxford, 2003

    2003

  21. [29]

    Faster convergence in the free central limit t heorem

    Mauricio Salazar. Faster convergence in the free central limit t heorem. Complex Anal. Oper. Theory , 17(5):Paper No. 66, 10, 2023

    2023

  22. [30]

    A bound for the error in the normal approximatio n to the distribution of a sum of dependent random variables

    Charles Stein. A bound for the error in the normal approximatio n to the distribution of a sum of dependent random variables. In Proceedings of the sixth Berkeley symposium on mathematica l statistics 32 MARIO D ´IAZ† AND ARTURO JARAMILLO and probability, volume 2: Probability...

    1972

  23. [31]

    Symmetries of some reduced free product C∗ -algebras

    Dan Voiculescu. Symmetries of some reduced free product C∗ -algebras. In Operator algebras and their connections with topology and ergodic theory (Bu¸ steni, 19 83), volume 1132 of Lecture Notes in Math. , pages 556–588. Springer, Berlin, 1985. Mario D´ıaz† : Instituto de Inve...

    1985

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