Unitary induced channels and Tsirelson's problem

Micha{\l} Banacki; Pawe{\l} Horodecki

REVIEW 2 major objections 6 minor 43 references

Unitary induced channels and Tsirelson's problem

T0 review · 2 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read The paper proves an exact equivalence: Tsirelson's conjecture holds if and only if generalized unitary induced channels in the commuting and tensor models coincide for every finite number of settings and outcomes. Since Tsirelson's conjectu

desk verdict New channel reformulation of Tsirelson's problem, mostly sound but with a repairable gap in the key proof. read the letter →

arxiv 2508.21808 v1 pith:W75GLWKE submitted 2025-08-29 quant-ph math-phmath.MP

Micha{l} Banacki , Pawe{l} Horodecki This is my paper · ORCID

classification quant-phmath-phmath.MP MSC 46L0646L0746L3081P45 PACS 03.67.-a

keywords unitaryinducedchannelsgeneralizedquantumcommutingmodeltensorTsirelson'sproblemBrownalgebrasfreeproductsofC*-algebrasC*-algebra

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

The paper asks whether the difference between the two standard mathematical pictures of bipartite quantum systems—tensor-product versus commuting-operator—can be seen in protocols that only use finite-dimensional ancillas and no measurements on infinite-dimensional spaces. It defines generalized unitary induced channels to capture such protocols and proves a sharp equivalence: the commuting and tensor versions of these channel families coincide for every finite $m,n$ exactly when Tsirelson's conjecture is true. Since Tsirelson's conjecture has been refuted, the paper concludes that the two channel families genuinely differ for some finite $m,n$. The operational payoff is that the long-sought gap between commuting and tensor quantum models can in principle be witnessed by comparing output density matrices of finite-dimensional ancilla systems.

What carries the argument

The load-bearing objects are generalized unitary induced channels and the Brown algebra $U(n)$, the universal unital C*-algebra generated by the entries of an $n \times n$ unitary matrix. The paper's characterizations put a channel family in the commuting model exactly when there is a state on the maximal tensor product of free products of $U(n)$, and in the tensor model exactly when the same matrix elements are produced by a state on the minimal tensor product. A Fourier-like identity between PVM generators and unitary generators lets the proof convert any Bell-type behaviour into these channel matrix elements, and a free-product extension theorem lets it glue separately defined maps into one map on the

What would settle it

Take a known post-quantum behaviour $P \in C_{qc}(m,n) \setminus C_{qa}(m,n)$, build the unitaries $u^{x}_{a'}$ and $v^{y}_{b'}$ via the Fourier relations (3.3)–(3.4), feed them into formula (2.15), and ask whether the resulting family of output matrices is realizable with a minimal-tensor state. If the family is not realizable, the predicted channel gap is concretely demonstrated; if it is, the equivalence chain in Theorem 3.4 breaks at the construction.

Watch

Extended reading notes

Core claim

The central result is Theorem 3.4, an equivalence with Tsirelson's problem at one end and the unitary-induced-channel families at the other. The paper proves that five statements are equivalent: Tsirelson's conjecture; the equality $L_{qc}(m,n)=L_{qa}(m,n)$ for all $m,n$; a matching condition on channel matrix elements; and the two-outcome restrictions of those equalities. The proof represents a commuting-model channel as a state on the maximal tensor product of free products of copies of the Brown algebra $U(n)$, the universal C*-algebra generated by the entries of an $n \times n$ unitary, while a tensor-model channel is represented by a state on the corresponding minimal tensor product. Equality of the channel

Load-bearing premise

The entire translation from Bell behaviours to channel families relies on the free-product extension theorem for completely positive maps; if that theorem does not apply to the free products of Brown algebras used here, the equivalence between Tsirelson's conjecture and the channel-set equalities no longer follows.

Editorial extensions

If this is right

A finite gap exists: for some finite number of settings m and outcomes n, there are generalized unitary induced channels in the commuting model that cannot be reproduced in the tensor model.
The two-outcome case carries the full content: Lqc(m,2)=Lqa(m,2) for all m would have been exactly Tsirelson's conjecture.
An explicit separating family of channels would serve as a concrete counterexample to Tsirelson's conjecture in a channel language, potentially easier to interpret physically than the original correlation construction.
At the level of a single fixed input state the models agree: the swap operation lets either model transform any input state into any target state, so the gap is intrinsically a multi-setting effect.
Because only finite-dimensional ancillas are used, the difference is testable by finite-dimensional state tomography on the outputs, rather than by measurements on the infinite-dimensional resource.

Reading between the lines

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

A concrete route to an explicit witness would be to take a known post-quantum correlation, apply the Fourier recipe in the proof of Theorem 3.4 to build unitary matrices, and test whether the resulting output matrices violate any finite-dimensional criterion for tensor realizability.
Because the paper proves the single-channel models agree for each fixed state, any experimental or numerical search for the gap should focus on multiple settings rather than on a single channel; the minimal separating (m,n) is a natural optimization target.
The equivalence also suggests that tensor-product characterizations of channel families could help certify non-tensor resources in protocols where the untrusted devices act only on finite-dimensional ancillas and the shared infinite-dimensional system is treated as a black box.

Share X Bluesky LinkedIn Reddit HN

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

Quick take: this is a mostly sound translation of the Brown-algebra formulation of Tsirelson's problem into a channel language. The central equivalence is plausible and likely true, but the proof of Theorem 3.4 has a typo in the POVM completion and a normalization step that is under-argued. Both are fixable.

What is actually new: for a fixed ancilla dimension n, and m settings, the paper defines generalized unitary induced channels in the qc and qa models, Lqc(m,n) and Lqa(m,n), and proves Theorem 3.4 that equality for all m,n is equivalent to Tsirelson's conjecture. That is a genuinely new operational reformulation. The earlier Brown algebra equivalences (Theorem 3.1) are from the literature; the contribution is the channel packaging, including the observation that the gap can be witnessed by comparing finite-dimensional input-output states without measuring the infinite-dimensional system. Theorems 2.6-2.10 are clean and useful: they characterize the channel sets in terms of states on maximal/minimal tensor products of Brown algebras and free products.

The main proof direction (1)=>(2) is straightforward and correct: LLP of Brown algebras plus WEP from Tsirelson gives nuclearity, and the inclusion Lqa subset Lqc gives equality. The reverse direction is where the paper gets sloppy. After Eq. (3.12), the completion Mhat_{n|x} as written makes sum Mhat = 1 + M_n, not 1; the upper limit of the sum inside the bracket should be n, not n-1. That's a typo. More importantly, the sentence 'Using normalization ... one can show p=p'' is not enough. You know the two distributions coincide on the (n-1)x(n-1) block, and both sum to one; that does not determine the remaining entries without using the marginals. The missing step is to observe that equality of the channel maps gives equality of states on expressions like M_a tensor 1 and 1 tensor N_b, by expressing 1 as a sum of diagonal unitaries (e.g., 1 = sum_s v_{1s}v_{1s}^*). From that, the full phat matches p. This is a genuine gap in exposition, but it is repairable and does not undermine the likely truth of the theorem.

The use of Boca's free product theorem is fine; the theorem holds as stated, so that's not a hidden weakness.

Who this is for: quantum information people interested in Tsirelson-type gaps and operator algebra people who want a concrete channel picture. It deserves peer review, but the authors should fix the typo and expand the normalization argument.

Recommendation: yes, send it to review, with the expectation of a light revision.

Referee Report

2 major / 6 minor

Summary. The paper introduces (generalized) unitary induced quantum channels in the commuting and tensor frameworks, denoted Lqc(m,n) and Lqa(m,n), and characterizes them through states on maximal/minimal tensor products of free products of Brown algebras (Theorems 2.6--2.10). The main theorem (Theorem 3.4) asserts that Tsirelson's conjecture is equivalent to Lqc(m,n)=Lqa(m,n) for all m,n, and also to the qubit-outcome variant. Combined with MIP*=RE, the authors conclude that Lqc(m,n) and Lqa(m,n) genuinely differ for some m,n. The proof of (1)=>(2) follows standard LLP/WEP arguments; the difficult direction is (5)=>(2), which reconstructs a quantum commuting correlation from the channel equality using Fourier unitaries and Brown-algebra free products.

Significance. If Theorem 3.4 is established, the paper gives a genuinely useful reformulation of Tsirelson's problem in terms of finite-ancilla channel protocols, and it turns the MIP*=RE noncommutative-tensor gap into an explicit separation of two channel sets. The absence of fitted parameters and the reliance on external, independently proved results (LLP/WEP, MIP*=RE) are strengths. However, the proof of the converse direction contains a concrete normalization error and an unjustified 'using normalization' step, so the central equivalence is not fully supported as written. The result is plausible and likely repairable, but the current manuscript does not establish it.

major comments (2)

[Sec. 3, proof of Thm 3.4, after Eq. (3.12)] The completion \hat M_{n|x} = M_{n|x} + (1 - \sum_{a=1}^{n-1} M_{a|x}) gives \sum_a \hat M_{a|x} = 1 + M_{n|x}, not 1. The same problem occurs for \hat N_{n|y}. The intended definition is \hat M_{n|x} = 1 - \sum_{a=1}^{n-1} M_{a|x} (and analogously for N). Since this construction is the core of the (5)=>(2) direction, the error is load-bearing and must be corrected.
[Sec. 3, proof of Thm 3.4, after Eq. (3.11)-(3.12)] Even after the POVM correction, the statement 'Using normalization ... one can show that p(ab|xy)=\hat p(ab|xy)' is not justified. Equality is known only on the (n-1)^2 blocks with a,b<n; the total normalization of the two probability matrices is a single scalar constraint and does not force the boundary entries. One needs to establish the marginal equalities \hat\phi(1\otimes N_{b|y})=\sum_a p(a,b|xy) and \hat\phi(M_{a|x}\otimes 1)=\sum_b p(a,b|xy), which should follow from the equality of the full channel outputs after partial tracing and the Brown-algebra relations, but this argument is omitted. Thus the central implication (5)=>(2) is incomplete as written.

minor comments (6)

[Sec. 2, Thm 2.8] The stated theorem is cited to Boca [5], but that theorem is commonly stated for completely positive maps into B(H), not an arbitrary C*-algebra C. In the two places where it is invoked, the maps to be extended are in fact *-homomorphisms, so the universal property of the full free product suffices. Please state the precise version used or replace the reference.
[Def. 2.4] After Eq. (2.4), 'By Lqc(m,n)' should read 'By Lqa(m,n)'.
[Eq. (1.1)] The chain of inclusions contains a typo: the middle term 'Cq(m,n)' should presumably be 'Cqa(m,n)'.
[Thm 3.4, conditions (3) and (5)] The notation '(\Lambda_{xy}(E_{kj}\otimes E_{sr}))_{kjsr}' is ambiguous. It should be stated explicitly that this compares the (k,j,s,r) output matrix entry of \Lambda_{xy} evaluated on the input matrix unit E_{kj}\otimes E_{sr}, for all k,j,s,r.
[Eq. (3.7)] The equality P_{a|x}=\pi_x(\sum c u_{a'a'})=\pi_x((\sum c u_{a'a'})(\sum c u_{a'a'})^*) is true only because \pi_x(S)=P_{a|x} is a projection; the two preimages S and SS^* are not equal in the Brown algebra. This should be explained to avoid confusion.
[References] Reference [24] is cited as an arXiv preprint; if a published version exists, it should be cited.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the channel-language equivalence is an independent reduction, not a restatement of its inputs.

full rationale

The paper's central claim, Theorem 3.4, is a conditional equivalence: Tsirelson's conjecture holds iff the generalized unitary induced channel sets Lqc(m,n) and Lqa(m,n) coincide. The two directions are proved by reduction, not by definition. Definitions 2.1–2.4 define Lqc/Lqa directly in terms of states on B(H) or B(HA⊗HB) and commuting/algebraic unitaries, with no reference to Tsirelson or Cqc/Cqa. Theorem 2.9 and 2.10 characterize these channel sets via states on free products of Brown algebras with maximal/minimal tensor products; these equivalences are proved from the universal properties of Brown algebras and Boca's free-product extension theorem [5]. The (1)=>(2) direction of Theorem 3.4 uses the standard LLP/WEP chain (Theorem 3.1, external results [21,34,37]) to convert a max-tensor state into a min-tensor state. The converse (5)=>(2) embeds an arbitrary commuting correlation into the relevant channel moments via a Fourier transform (Eqs. 3.3–3.10), then uses the assumed channel equality to obtain a min-tensor state realizing the same correlation; this is a substantive reduction, not a tautology. The final separation Lqc≠Lqa follows from MIP*=RE [24], an external theorem. The only appearance of the authors' own work is the reference [2] in a list of steering examples; it is not load-bearing. There is a possible technical gap in the (5)=>(2) proof (the POVM completion after Eq. (3.12) as written has sum exceeding 1), but a proof gap is a correctness issue, not a circularity. No fitted parameters are used, and no prediction is generated from fitted data.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters appear; the paper is a pure C*-algebra proof. The axioms are standard tensor-product/free-product facts plus known equivalences. No new physical entities are postulated; the main new object is a definitional class of channels, not an invented entity.

assumptions (6)

standard math Universal property of Brown algebra U(n): for any unital C*-algebra A and unitary U in M_n(A), there is a *-homomorphism U(n)->A mapping generators to the entries of U.
Used in Theorems 2.6, 2.7, and 3.4, e.g. the construction of pi_x in Eqs. (3.5)-(3.6).
standard math Boca's free-product theorem (Theorem 2.8): UCP maps from each free factor into a common unital C*-algebra extend to a UCP map on the full free product.
Used in Theorem 2.9 and Theorem 3.4 to lift maps from copies of U(n) to free products; the target is a C*-algebra, not necessarily B(H).
domain assumption Known equivalence chain in Theorem 3.1: Connes embedding problem and Tsirelson's conjecture are equivalent to U(n) tensor-max = tensor-min for some/all n, and to all LLP algebras having WEP.
Bridges the channel problem to the operator-algebraic formulation; cited from [15,23,21,27,35].
domain assumption Brown algebra U(n) has the local lifting property (LLP) [21].
Needed in Theorem 3.4 direction (1)=>(2) to conclude free products of U(n) have LLP and hence WEP under Tsirelson's conjecture.
domain assumption If A has LLP and B has WEP, then A tensor-max B = A tensor-min B [37].
Used in Theorem 3.4 direction (1)=>(2).
domain assumption MIP*=RE [24], i.e. Tsirelson's conjecture is false.
External result used to derive the stated difference Lqc != Lqa.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Unitary induced channels and Tsirelson's problem." pith.science (2026). https://pith.science/paper/W75GLWKE

@misc{pith2026250821808,
  author       = {Pith},
  title        = {Pith review of: Unitary induced channels and Tsirelson's problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W75GLWKE}},
  note         = {Machine review of arXiv:2508.21808}
}

read the original abstract

Motivated by a recent progress concerning quantum commuting and quantum tensor models of composed systems we investigate a notion of (generalized) unitary induced quantum channel. Using properties of Brown algebras we provide an equivalent characterization of discussed families in both tensor and commuting paradigms. In particular, we provide an equivalent formulation of Tsirelson's conjecture (Connes' embedding problem) in terms of considered paradigms based on protocols which do not require measurements performed on infinite-dimensional subsystems. As a result we show that there is a difference between quantum commuting and quantum tensor models for generalized unitary induced channels.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 36 canonical work pages

[1]

Avitzour, Free products of C*-algebras, Trans

D. Avitzour, Free products of C*-algebras, Trans. Amer. Math. Soc., 271, 423-435 (1982)

work page 1982
[2]

On steering in the C*-algebraic framework

M. Banacki, On steering in the C*-algebraic framework, arXiv:2306.14344 (2023)

work page Pith review arXiv 2023
[3]

Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras, Springer (2006)

B. Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras, Springer (2006)

work page 2006
[4]

Blackadar, Shape theory for C∗-algebras, Mathematica Scandinavica, 56, 249–275, (1985)

B. Blackadar, Shape theory for C∗-algebras, Mathematica Scandinavica, 56, 249–275, (1985)

work page 1985
[5]

Boca, Free products of completely positive positive maps and spectral sets, J

F. Boca, Free products of completely positive positive maps and spectral sets, J. Funct. Anal., 97, 251-263 (1991)

work page 1991
[6]

L. G. Brown, Ext of certain free product C*-algebras, J. Operator Theory, 6, 135–141 (1981)

work page 1981
[7]

N. P. Brown, N. Ozawa, C*-algebras and finite-dimensional approximation, American Mathematical Society (2008)

work page 2008
[8]

Cleve, L

R. Cleve, L. Liu, V. I. Paulsen, Perfect embezzlement of entanglement, J. Math. Phys., 58, 012204 (2017)

work page 2017

Show all 43 references

[9]

Cleve, B

R. Cleve, B. Collins, L. Liu, V. Paulsen, Constant gap between conventional strategies and those based on C*-dynamics for self-embezzlement, Quantum, 6, 755 (2022). 10 MICHA L BANACKI AND PA WE L HORODECKI

2022
[10]

Coladangelo, A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations, Quantum, 4, 282 (2020)

A. Coladangelo, A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations, Quantum, 4, 282 (2020)

2020
[11]

Coladangelo, J

A. Coladangelo, J. Stark, An inherently infinite-dimensional quantum correlation, Nat. Commun., 11, 3335 (2020)

2020
[12]

Connes, Classification of injective factors

A. Connes, Classification of injective factors. Cases II 1, II ∞, III λ, λ ̸= 1, Ann. of Math., 104, 73–115 (1976)

1976
[13]

Dykema, K

K. Dykema, K. Juschenko, Matrices of unitary moments, Math. Scand., 109, 225–239 (2011)

2011
[14]

Dykema, V

K. Dykema, V. I. Paulsen, J. Prakash, Non-closure of the Set of Quantum Correlations via Graphs, Commun. Math. Phys., 365, 1125–1142 (2019)

2019
[15]

Fritz, Tsirelson’s problem and Kirchberg’s conjecture, Rev

T. Fritz, Tsirelson’s problem and Kirchberg’s conjecture, Rev. Math. Phys., 24, 1250012 (2012)

2012
[16]

L. Gao, S. J. Harris, M. Junge, Quantum Teleportation and Super-Dense Coding in Operator Algebras, International Mathematics Research Notices, 12, 9146–9179 (2021)

2021
[17]

Goldbring, The Connes Embedding Problem: A guided tour, Bull

I. Goldbring, The Connes Embedding Problem: A guided tour, Bull. Amer. Math. Soc., 59, 503-560, (2022)

2022
[18]

Goldbring, B

I. Goldbring, B. Hart, On Tsirelson pairs of C*-algebras, Rev. Math. Phys., 35, 2350016 (2023)

2023
[19]

Haag, Local Quantum Physics, Springer-Verlag (1996)

R. Haag, Local Quantum Physics, Springer-Verlag (1996)

1996
[20]

R. Haag, D. Kastler, An algebraic approach to quantum field theory, J. Math. Phys., 5, 848 (1964)

1964
[21]

S. J. Harris, A non-commutative unitary analogue of Kirchberg’s conjecture, Indiana Univ. Math. J., 68, 503-536 (2019)

2019
[22]

S. J. Harris, V. I. Paulsen, Unitary Correlation Sets, Integr. Equ. Oper. Theory, 89, 125–149 (2017)

2017
[23]

Junge, M

M. Junge, M. Navascues, C. Palazuelos, D. Perez-Garcia, V.B. Scholz, R.F. Werner, Connes embedding problem and Tsirelson’s problem, J. Math. Phys., 52, 012102 (2011)

2011
[24]

Z. Ji, A. Natarajan, T. Vidick, J. Wright, H. Yuen, MIP* = RE, arXiv:2001.04383 (2020)

2001 arXiv
[25]

A. S. Kavruk, Nuclearity related properties in operator systems, J. Oper. Theory, 71, 95-156 (2014)

2014
[26]

Kavruk, V

A. Kavruk, V. I. Paulsen, I. G. Todorov, M. Tomforde, Tensor products of operator systems, J. of Funct. Anal., 261, 267-299 (2011)

2011
[27]

Kirchberg, On nonsemisplit extensions, tensor products and exactness of group C*-algebras, Invent

E. Kirchberg, On nonsemisplit extensions, tensor products and exactness of group C*-algebras, Invent. Math., 112, 449–489 (1993)

1993
[28]

van Luijk, R

L. van Luijk, R. Schwonnek, A. Stottmeister, R. F. Werner, The Schmidt rank for the commuting operator framework, arXiv:2307.11619 (2023)

2023 arXiv
[29]

van Luijk, A

L. van Luijk, A. Stottmeister, H. Wilming, Critical Fermions are Universal Embezzlers, arXiv:2406.11747 (2024)

2024 arXiv
[30]

van Luijk, A

L. van Luijk, A. Stottmeister, R. F. Werner, H. Wilming, Embezzlement of entanglement, quantum fields, and the classification of von Neumann algebras, arXiv:2401.07299 (2024)

2024 arXiv
[31]

van Luijk, A

L. van Luijk, A. Stottmeister, R. F. Werner, H. Wilming, Embezzling entanglement from quantum fields, arXiv:2401.07292 (2024)

2024 arXiv
[32]

Musat, M

M. Musat, M. Rørdam, Non-closure of Quantum Correlation Matrices and Factorizable Channels that Require Infinite Dimensional Ancilla (With an Appendix by Narutaka Ozawa),Commun. Math. Phys., 375, 1761–1776 (2020)

2020
[33]

Navascu´ es, D

M. Navascu´ es, D. Perez-Garcia, Quantum steering and space like separation, Phys. Rev. Lett., 109, 160405 (2012)

2012
[34]

Ozawa, About the QWEP conjecture, Int

N. Ozawa, About the QWEP conjecture, Int. J. Math., 15, 501-530 (2004)

2004
[35]

Ozawa, About the Connes embedding conjecture: algebraic approaches, Jpn

N. Ozawa, About the Connes embedding conjecture: algebraic approaches, Jpn. J. Math., 8, 147–183 (2013)

2013
[36]

Pisier, A simple proof of a theorem of Kirchberg and related results on C*-norms, J

G. Pisier, A simple proof of a theorem of Kirchberg and related results on C*-norms, J. operator Theory, 35, 317-335 (1996)

1996
[37]

Pisier, Tensor Products of C*-Algebras and Operator Spaces

G. Pisier, Tensor Products of C*-Algebras and Operator Spaces. The Connes–Kirchberg Problem, Cambridge University Press (2020)

2020
[38]

Paddock, W

C. Paddock, W. Slofstra, Y. Zhao, Y. Zhou, An operator-algebraic formulation of self-testing, arXiv:2301.11291 (2023)

2023 arXiv
[39]

V. B. Scholz, R. F. Werner, Tsirelson’s Problem, arXiv:0812.4305 (2008)

2008 arXiv
[40]

Slofstra, Tsirelson’s problem and an embedding theorem for groups arising from non-local games, J

W. Slofstra, Tsirelson’s problem and an embedding theorem for groups arising from non-local games, J. Amer. Math. Soc., 33, 1-56 (2020)

2020
[41]

Slofstra, The set of quantum correlations is not closed, Forum of Mathematics, 7 (2017)

W. Slofstra, The set of quantum correlations is not closed, Forum of Mathematics, 7 (2017)

2017
[42]

Tsirelson, Bell inequalities and operator algebras, (2006)

B. Tsirelson, Bell inequalities and operator algebras, (2006)

2006
[43]

X. Yan, Z. Yin, L. Li, Approximation of quantum assemblages, Phys. Lett. A, 384, 126082 (2020). International Centre for Theory of Quantum Technologies (ICTQT), University of Gda ´nsk, Jana Ba˙zy´nskiego 1A, 80-309 Gda ´nsk, Poland Institute of Mathematics, F aculty of Mathema...

2020

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Avitzour, Free products of C*-algebras, Trans

D. Avitzour, Free products of C*-algebras, Trans. Amer. Math. Soc., 271, 423-435 (1982)

work page 1982

[2] [2]

On steering in the C*-algebraic framework

M. Banacki, On steering in the C*-algebraic framework, arXiv:2306.14344 (2023)

work page Pith review arXiv 2023

[3] [3]

Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras, Springer (2006)

B. Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras, Springer (2006)

work page 2006

[4] [4]

Blackadar, Shape theory for C∗-algebras, Mathematica Scandinavica, 56, 249–275, (1985)

B. Blackadar, Shape theory for C∗-algebras, Mathematica Scandinavica, 56, 249–275, (1985)

work page 1985

[5] [5]

Boca, Free products of completely positive positive maps and spectral sets, J

F. Boca, Free products of completely positive positive maps and spectral sets, J. Funct. Anal., 97, 251-263 (1991)

work page 1991

[6] [6]

L. G. Brown, Ext of certain free product C*-algebras, J. Operator Theory, 6, 135–141 (1981)

work page 1981

[7] [7]

N. P. Brown, N. Ozawa, C*-algebras and finite-dimensional approximation, American Mathematical Society (2008)

work page 2008

[8] [8]

Cleve, L

R. Cleve, L. Liu, V. I. Paulsen, Perfect embezzlement of entanglement, J. Math. Phys., 58, 012204 (2017)

work page 2017