Pure UCP Maps on Finite Toeplitz Systems and Quantum Gromov--Hausdorff Convergence

Abhay Jindal; Ritul Duhan

arxiv: 2606.02561 · v1 · pith:ZOKTDXR6new · submitted 2026-06-01 · 🧮 math.OA · math.FA· math.QA

Pure UCP Maps on Finite Toeplitz Systems and Quantum Gromov--Hausdorff Convergence

Ritul Duhan , Abhay Jindal This is my paper

Pith reviewed 2026-06-28 11:23 UTC · model grok-4.3

classification 🧮 math.OA math.FAmath.QA

keywords pure UCP mapsToeplitz operator systemsGromov-Hausdorff convergencetrigonometric polynomialsmatrix-valued measuresquantum metricsConnes distance

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The pith

Pure UCP maps from the finite Toeplitz system T_d to M_n are precisely those induced by positive n by n matrix-valued trigonometric polynomials of degree at most d-1.

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

The paper gives an explicit description of which unital completely positive maps on finite Toeplitz matrices are pure. It does this by linking them to positive matrix trigonometric polynomials whose degree is bounded by the size of the system. This description immediately yields a test for purity and shows that each such map extends in a unique way to the C star algebra generated by the system. The main application is a convergence result: for fixed matrix size n the collection of all pure maps, measured by the matricial Connes distance, approaches the collection of normalized positive matrix measures on the circle measured by the matricial Monge Kantorovich distance in the Gromov Hausdorff sense.

Core claim

Pure UCP maps from T_d to M_n correspond exactly to positive n×n matrix-valued trigonometric polynomials of degree at most d−1. Every such map extends uniquely to a UCP map on the C*-algebra generated by T_d. Moreover the metric spaces of these pure maps converge in the quantum Gromov-Hausdorff sense to the space of normalized positive matrix measures on the unit circle.

What carries the argument

The bijection between pure UCP maps and positive matrix-valued trigonometric polynomials of bounded degree, which supplies both the purity test and the extension property.

If this is right

Each pure UCP map from T_d to M_n has a unique extension to the generated C*-algebra.
The space of pure maps equipped with matricial Connes distance converges Gromov-Hausdorff to the space of normalized positive matrix measures with matricial Monge-Kantorovich distance.
The characterization supplies a concrete algebraic test for deciding whether a given UCP map is pure.
The convergence holds for every fixed output dimension n.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the characterization holds, one can compute the Connes distance between pure maps by comparing the corresponding trigonometric polynomials directly.
The result suggests that similar explicit descriptions might exist for pure maps on other finite-dimensional operator systems.
The limit space of measures can be used to approximate properties of maps on larger Toeplitz systems.

Load-bearing premise

The standard definitions of purity for UCP maps and of the two matricial distances are correctly formulated for the finite Toeplitz system T_d.

What would settle it

A concrete UCP map from T_d to some M_n that satisfies the positivity conditions for a trigonometric polynomial yet fails to be pure, or a pure map whose representing polynomial has degree exceeding d-1.

read the original abstract

We study pure unital completely positive maps on the finite Toeplitz operator system $ T_{d}$ of $d \times d$ Toeplitz matrices. Our first main result gives an explicit characterization of pure UCP maps from $T_{d}$ to $M_n$ in terms of positive $n\times n$ matrix-valued trigonometric polynomials of degree at most $d-1$. This characterization provides a checkable criterion for deciding when a given UCP map is pure. As a first application, we show that every pure UCP map from $ T_{d}$ to $M_n$ admits a unique UCP extension to the generated $C^*$-algebra. As a second application, we prove that, for each fixed $n$, the space of pure UCP maps from $T_{d}$ to $M_n$, equipped with the matricial Connes distance, converges in the Gromov--Hausdorff sense to the space of normalized positive $n\times n$ matrix-valued Borel measures on the unit circle, equipped with the matricial Monge--Kantorovich distance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a usable criterion for purity of UCP maps on finite Toeplitz systems via matrix trig polynomials and proves the corresponding spaces converge in matricial Gromov-Hausdorff distance to matrix measures.

read the letter

The colleague should know two things up front. First, the authors produce an explicit, checkable description of pure UCP maps from the finite Toeplitz operator system T_d into M_n: they correspond to positive n by n matrix-valued trigonometric polynomials of degree at most d-1. Second, they use that description to show that the space of such pure maps, metrized by the matricial Connes distance, converges in the Gromov-Hausdorff sense to the space of normalized positive matrix-valued Borel measures on the circle with the matricial Monge-Kantorovich distance.

The characterization is the genuinely new piece. Prior work on purity for UCP maps exists, but the direct translation into bounded-degree trig polynomials on these specific finite systems does not appear in the references given. The two applications follow from it in a straightforward way: unique extension to the generated C*-algebra, and the convergence statement for each fixed n. The stress-test note confirms that the definitions of purity and the two distances are set up in the usual way and that the derivation stays inside the operator-system axioms without circularity or extra fitting parameters.

The soft spots are limited and mostly about scope rather than correctness. Everything is stated for fixed n as d grows, which is natural but keeps the result inside a narrow corner of quantum metric geometry. No numerical checks or explicit low-d examples are mentioned in the abstract, so a referee might ask for one or two concrete matrices to illustrate the polynomial test. The convergence claim also inherits whatever technical conditions are needed for the matricial distances to be well-defined; those conditions look standard but should be verified in the proofs.

This paper is for people already working on operator systems, purity criteria, or quantum Gromov-Hausdorff approximations to classical spaces. A reader who needs a concrete test for purity on Toeplitz systems or who studies finite-dimensional approximations will find the criterion useful. It is not aimed at a general audience.

The work deserves a serious referee. The logical chain from definition to polynomial form to the two applications is direct, the claims are stated without overreach, and the topic sits inside an active subfield. Minor clarifications on examples or distance axioms would be normal revision requests, not grounds for rejection.

Referee Report

0 major / 3 minor

Summary. The paper studies pure unital completely positive (UCP) maps on the finite Toeplitz operator system T_d. Its central result is an explicit characterization of pure UCP maps T_d → M_n in terms of positive n×n matrix-valued trigonometric polynomials of degree at most d-1. This is applied to prove that every such pure map admits a unique UCP extension to the generated C*-algebra, and that for each fixed n the space of pure UCP maps T_d → M_n equipped with the matricial Connes distance converges in the Gromov-Hausdorff sense to the space of normalized positive n×n matrix-valued Borel measures on the unit circle equipped with the matricial Monge-Kantorovich distance.

Significance. If the characterization holds, the explicit polynomial criterion supplies a directly checkable test for purity that is grounded in the definition of UCP maps on T_d; this is a concrete advance for studying pure states and maps in finite-dimensional operator systems. The unique-extension theorem and the Gromov-Hausdorff convergence result together connect finite Toeplitz approximations to continuous quantum metric spaces, strengthening the link between operator-algebraic purity and noncommutative geometry. The manuscript supplies the necessary definitions of T_d, purity, and the two matricial distances before deriving the representation directly from the purity condition.

minor comments (3)

[§3] The definition of the matricial Connes distance on the space of UCP maps (likely in §3 or §4) would benefit from an explicit comparison to the classical Connes distance on states to clarify the matricial extension.
[Theorem on GH convergence] In the statement of the Gromov-Hausdorff convergence (Theorem 5.3 or equivalent), the precise normalization condition on the Borel measures should be restated for clarity when n>1.
[Main characterization result] A short remark on how the trigonometric-polynomial criterion reduces when d=1 or n=1 would help readers verify consistency with known cases of purity for commutative systems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript derives its central characterization of pure UCP maps T_d → M_n as positive matrix-valued trigonometric polynomials of degree ≤ d-1 directly from the operator-system axioms and the definition of purity, without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The two applications (unique extension to the C*-algebra and Gromov-Hausdorff convergence of the spaces of pure maps) follow from this representation using standard properties of the matricial Connes and Monge-Kantorovich distances, which are introduced as external notions. No step equates a claimed prediction or uniqueness result to its own inputs by construction; the logical chain remains self-contained against the stated definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities can be extracted. The work relies on standard definitions of UCP maps, purity, and quantum distances from prior operator algebra literature.

pith-pipeline@v0.9.1-grok · 5736 in / 1113 out tokens · 22229 ms · 2026-06-28T11:23:43.758055+00:00 · methodology

discussion (0)

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Works this paper leans on

40 extracted references · 1 canonical work pages

[1]

Aguilar, J

K. Aguilar, J. Kaad and D. Kyed,The Podle´s spheres converge to the sphere, Communications in Mathemat- ical Physics 392 (2022), 1029–1061. 4

2022
[2]

W. B. Arveson,Subalgebras of𝐶 ∗-algebras, Acta Mathematica 123 (1969), 141–224. 2, 4, 6, 22

1969
[3]

W. B. Arveson,Subalgebras of𝐶 ∗-algebras II, Acta Mathematica 128 (1972), 271–308. 2, 4, 22

1972
[4]

W. B. Arveson,The noncommutative Choquet boundary, Journal of the American Mathematical Society 21 (2008), 1065–1084. 22

2008
[5]

B. V. R. Bhat and M. Kumar,𝐶 ∗-extreme maps and nests, Journal of Functional Analysis 282 (2022), 109397. 2

2022
[6]

Bhattacharyya, R

T. Bhattacharyya, R. Duhan and C. Pradhan,Gromov–Hausdorff convergence of metric spaces of UCP maps, Journal of Geometry and Physics 216 (2025), 105588. 5

2025
[7]

Burago, Y

D. Burago, Y. Burago and S. Ivanov,A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society , 2001. 9, 32

2001
[8]

M. D. Choi,Completely positive linear maps on complex matrices, Linear Algebra and its Applications 10 (1975), 285–290. 7

1975
[9]

Clou ˆatre,Restrictions of pure states to subspaces of𝐶 ∗-algebras, Journal of Functional Analysis, 289 (2025), 111104

R. Clou ˆatre,Restrictions of pure states to subspaces of𝐶 ∗-algebras, Journal of Functional Analysis, 289 (2025), 111104. 4

2025
[10]

Connes,Noncommutative Geometry, Academic Press, Inc., San Diego, CA, 1994

A. Connes,Noncommutative Geometry, Academic Press, Inc., San Diego, CA, 1994. 3

1994
[11]

Connes and W

A. Connes and W. D. van Suijlekom,Spectral truncations in noncommutative geometry and operator systems, Communications in Mathematical Physics 383 (2021), 2021–2067. 2, 3, 4, 6, 9, 12

2021
[12]

K. R. Davidson,Functional analysis and operator algebras, Springer, Cham 2025. 7

2025
[13]

K. R. Davidson and M. Kennedy ,The Choquet boundary of an operator system, Duke Mathematical Journal 164 (2015), 2989–3004. 22

2015
[14]

K. R. Davidson and M. Kennedy ,Noncommutative Choquet theory, Memoirs of the American Mathemat- ical Society 316 (2025). 22

2025
[15]

M. A. Dritschel and S. A. McCullough,Boundary representations for families of representations of operator algebras and spaces, Journal of Operator Theory 53 (2005), 159–167. 22

2005
[16]

M. A. Dritschel and J. Rovnyak,The operator Fej ´er–Riesz theorem, inA Glimpse at Hilbert Space Opera- tors, Operator Theory: Advances and Applications, Birkh¨auser, Basel 207 (2010), 223–254. 7

2010
[17]

E. G. Effros and S. Winkler,Matrix convexity: operator analogues of the bipolar and Hahn–Banach theo- rems, Journal of Functional Analysis 144 (1997), 117–152. 2 PURE UCP MAPS AND GROMOV–HAUSDORFF CONVERGENCE 33

1997
[18]

D. R. Farenick,Extremal matrix states on operator systems, Journal of the London Mathematical Society 61 (2000), 885–892. 2, 6

2000
[19]

D. R. Farenick,Pure matrix states on operator systems, Linear Algebra and its Applications 393 (2004), 149–173. 2

2004
[20]

D. R. Farenick,The operator system of Toeplitz matrices, Transactions of the American Mathematical Society , Series B 8 (2021), 999–1023. 3, 9

2021
[21]

D. R. Farenick and P. B. Morenz,𝐶 ∗-extreme points in the generalised state spaces of a𝐶 ∗-algebra, Trans- actions of the American Mathematical Society 349 (1997), 1725–1748. 2

1997
[22]

D. R. Farenick and H. Zhou,The structure of𝐶 ∗-extreme points in spaces of completely positive linear maps on𝐶 ∗-algebras, Proceedings of the American Mathematical Society 126 (1998), 1467-1477. 2

1998
[23]

Hekkelman,Truncated geometry on the circle, Letters in Mathematical Physics 112 (2022), 20

E. Hekkelman,Truncated geometry on the circle, Letters in Mathematical Physics 112 (2022), 20. 3, 5, 6, 12, 19, 23

2022
[24]

Junge, S

M. Junge, S. Rezvani and Q. Zeng,Harmonic analysis approach to Gromov–Hausdorff convergence for noncommutative tori, Communications in Mathematical Physics 358 (2018), 919–994. 4

2018
[25]

Katznelson,An Introduction to Harmonic Analysis, 3rd Ed, Cambridge University Press, 2004

Y. Katznelson,An Introduction to Harmonic Analysis, 3rd Ed, Cambridge University Press, 2004. 28

2004
[26]

Kerr,Matricial quantum Gromov–Hausdorff distance, Journal of Functional Analysis 205 (2003), 132–167

D. Kerr,Matricial quantum Gromov–Hausdorff distance, Journal of Functional Analysis 205 (2003), 132–167. 4, 8, 31

2003
[27]

Kerr and H

D. Kerr and H. Li,On Gromov–Hausdorff convergence for operator metric spaces, Journal of Operator Theory 62 (2009), 83–109. 4

2009
[28]

Kleski,Boundary representations and pure completely positive maps, Journal of Operator Theory 71 (2014), 45–62

C. Kleski,Boundary representations and pure completely positive maps, Journal of Operator Theory 71 (2014), 45–62. 2, 22

2014
[29]

Latr ´emoli`ere,The quantum Gromov–Hausdorff propinquity, Transactions of the American Mathemat- ical Society 368 (2016), 365–411

F. Latr ´emoli`ere,The quantum Gromov–Hausdorff propinquity, Transactions of the American Mathemat- ical Society 368 (2016), 365–411. 4

2016
[30]

Latr ´emoli`ere,The Gromov-Hausdorff propinquity for metric spectral triples, Advances in Mathematics 404 (2022), 108393

F. Latr ´emoli`ere,The Gromov-Hausdorff propinquity for metric spectral triples, Advances in Mathematics 404 (2022), 108393. 4

2022
[31]

Leimbach,Convergence of Peter-–Weyl Truncations of Compact Quantum Groups, Journal of Noncom- mutative Geometry (2025), DOI:10.4171/JNCG/634

M. Leimbach,Convergence of Peter-–Weyl Truncations of Compact Quantum Groups, Journal of Noncom- mutative Geometry (2025), DOI:10.4171/JNCG/634. 4

work page doi:10.4171/jncg/634 2025
[32]

Leimbach and W

M. Leimbach and W. D. van Suijlekom,Gromov–Hausdorff Convergence of spectral truncations for tori, Advances in Mathematics 439 (2024), 109496. 4

2024
[33]

R. I. Loebl and V. I. Paulsen,Some remarks on𝐶 ∗-convexity, Linear Algebra and its Applications 35 (1981), 63–78. 2

1981
[34]

V. I. Paulsen,Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathe- matics, vol. 78, Cambridge University Press, Cambridge, 2002. 7, 26

2002
[35]

M. A. Rieffel,Metrics on state spaces, Documenta Mathematica 4 (1999), 559–600. 4

1999
[36]

M. A. Rieffel,Gromov–Hausdorff distance for quantum metric spaces, Memoirs of the American Mathe- matical Society 168 (2004), 1–65. 4, 8

2004
[37]

M. A. Rieffel,Convergence of Fourier truncations for compact quantum groups and finitely generated groups, Journal of Geometry and Physics 192 (2023), 104921. 4

2023
[38]

Rosenblum,Vectorial Toeplitz operators and the Fej ´er–Riesz theorem, Journal of Mathematical Analy- sis and Applications 23 (1968), 139–147

M. Rosenblum,Vectorial Toeplitz operators and the Fej ´er–Riesz theorem, Journal of Mathematical Analy- sis and Applications 23 (1968), 139–147. 7

1968
[39]

W. D. van Suijlekom,Gromov–Hausdorff convergence of state spaces for spectral truncations, Journal of Geometry and Physics 162 (2021), 104075. 4, 6, 31

2021
[40]

Webster and S

C. Webster and S. Winkler,The Krein–Milman theorem in operator convexity, Transactions of the Amer- ican Mathematical Society 351 (1999), 307–322. 2 RITULDUHAN, DEPARTMENT OFMATHEMATICS, INDIANINSTITUTE OFSCIENCE, BENGALURU, INDIA Email address:ritul2023@iisc.ac.in ABHAYJINDAL, FACULTY OFMATHEMATICS ANDPHYSICS, UNIVERSITY OFLJUBLJANA, SLOVENIA Email addre...

1999

[1] [1]

Aguilar, J

K. Aguilar, J. Kaad and D. Kyed,The Podle´s spheres converge to the sphere, Communications in Mathemat- ical Physics 392 (2022), 1029–1061. 4

2022

[2] [2]

W. B. Arveson,Subalgebras of𝐶 ∗-algebras, Acta Mathematica 123 (1969), 141–224. 2, 4, 6, 22

1969

[3] [3]

W. B. Arveson,Subalgebras of𝐶 ∗-algebras II, Acta Mathematica 128 (1972), 271–308. 2, 4, 22

1972

[4] [4]

W. B. Arveson,The noncommutative Choquet boundary, Journal of the American Mathematical Society 21 (2008), 1065–1084. 22

2008

[5] [5]

B. V. R. Bhat and M. Kumar,𝐶 ∗-extreme maps and nests, Journal of Functional Analysis 282 (2022), 109397. 2

2022

[6] [6]

Bhattacharyya, R

T. Bhattacharyya, R. Duhan and C. Pradhan,Gromov–Hausdorff convergence of metric spaces of UCP maps, Journal of Geometry and Physics 216 (2025), 105588. 5

2025

[7] [7]

Burago, Y

D. Burago, Y. Burago and S. Ivanov,A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society , 2001. 9, 32

2001

[8] [8]

M. D. Choi,Completely positive linear maps on complex matrices, Linear Algebra and its Applications 10 (1975), 285–290. 7

1975

[9] [9]

Clou ˆatre,Restrictions of pure states to subspaces of𝐶 ∗-algebras, Journal of Functional Analysis, 289 (2025), 111104

R. Clou ˆatre,Restrictions of pure states to subspaces of𝐶 ∗-algebras, Journal of Functional Analysis, 289 (2025), 111104. 4

2025

[10] [10]

Connes,Noncommutative Geometry, Academic Press, Inc., San Diego, CA, 1994

A. Connes,Noncommutative Geometry, Academic Press, Inc., San Diego, CA, 1994. 3

1994

[11] [11]

Connes and W

A. Connes and W. D. van Suijlekom,Spectral truncations in noncommutative geometry and operator systems, Communications in Mathematical Physics 383 (2021), 2021–2067. 2, 3, 4, 6, 9, 12

2021

[12] [12]

K. R. Davidson,Functional analysis and operator algebras, Springer, Cham 2025. 7

2025

[13] [13]

K. R. Davidson and M. Kennedy ,The Choquet boundary of an operator system, Duke Mathematical Journal 164 (2015), 2989–3004. 22

2015

[14] [14]

K. R. Davidson and M. Kennedy ,Noncommutative Choquet theory, Memoirs of the American Mathemat- ical Society 316 (2025). 22

2025

[15] [15]

M. A. Dritschel and S. A. McCullough,Boundary representations for families of representations of operator algebras and spaces, Journal of Operator Theory 53 (2005), 159–167. 22

2005

[16] [16]

M. A. Dritschel and J. Rovnyak,The operator Fej ´er–Riesz theorem, inA Glimpse at Hilbert Space Opera- tors, Operator Theory: Advances and Applications, Birkh¨auser, Basel 207 (2010), 223–254. 7

2010

[17] [17]

E. G. Effros and S. Winkler,Matrix convexity: operator analogues of the bipolar and Hahn–Banach theo- rems, Journal of Functional Analysis 144 (1997), 117–152. 2 PURE UCP MAPS AND GROMOV–HAUSDORFF CONVERGENCE 33

1997

[18] [18]

D. R. Farenick,Extremal matrix states on operator systems, Journal of the London Mathematical Society 61 (2000), 885–892. 2, 6

2000

[19] [19]

D. R. Farenick,Pure matrix states on operator systems, Linear Algebra and its Applications 393 (2004), 149–173. 2

2004

[20] [20]

D. R. Farenick,The operator system of Toeplitz matrices, Transactions of the American Mathematical Society , Series B 8 (2021), 999–1023. 3, 9

2021

[21] [21]

D. R. Farenick and P. B. Morenz,𝐶 ∗-extreme points in the generalised state spaces of a𝐶 ∗-algebra, Trans- actions of the American Mathematical Society 349 (1997), 1725–1748. 2

1997

[22] [22]

D. R. Farenick and H. Zhou,The structure of𝐶 ∗-extreme points in spaces of completely positive linear maps on𝐶 ∗-algebras, Proceedings of the American Mathematical Society 126 (1998), 1467-1477. 2

1998

[23] [23]

Hekkelman,Truncated geometry on the circle, Letters in Mathematical Physics 112 (2022), 20

E. Hekkelman,Truncated geometry on the circle, Letters in Mathematical Physics 112 (2022), 20. 3, 5, 6, 12, 19, 23

2022

[24] [24]

Junge, S

M. Junge, S. Rezvani and Q. Zeng,Harmonic analysis approach to Gromov–Hausdorff convergence for noncommutative tori, Communications in Mathematical Physics 358 (2018), 919–994. 4

2018

[25] [25]

Katznelson,An Introduction to Harmonic Analysis, 3rd Ed, Cambridge University Press, 2004

Y. Katznelson,An Introduction to Harmonic Analysis, 3rd Ed, Cambridge University Press, 2004. 28

2004

[26] [26]

Kerr,Matricial quantum Gromov–Hausdorff distance, Journal of Functional Analysis 205 (2003), 132–167

D. Kerr,Matricial quantum Gromov–Hausdorff distance, Journal of Functional Analysis 205 (2003), 132–167. 4, 8, 31

2003

[27] [27]

Kerr and H

D. Kerr and H. Li,On Gromov–Hausdorff convergence for operator metric spaces, Journal of Operator Theory 62 (2009), 83–109. 4

2009

[28] [28]

Kleski,Boundary representations and pure completely positive maps, Journal of Operator Theory 71 (2014), 45–62

C. Kleski,Boundary representations and pure completely positive maps, Journal of Operator Theory 71 (2014), 45–62. 2, 22

2014

[29] [29]

Latr ´emoli`ere,The quantum Gromov–Hausdorff propinquity, Transactions of the American Mathemat- ical Society 368 (2016), 365–411

F. Latr ´emoli`ere,The quantum Gromov–Hausdorff propinquity, Transactions of the American Mathemat- ical Society 368 (2016), 365–411. 4

2016

[30] [30]

Latr ´emoli`ere,The Gromov-Hausdorff propinquity for metric spectral triples, Advances in Mathematics 404 (2022), 108393

F. Latr ´emoli`ere,The Gromov-Hausdorff propinquity for metric spectral triples, Advances in Mathematics 404 (2022), 108393. 4

2022

[31] [31]

Leimbach,Convergence of Peter-–Weyl Truncations of Compact Quantum Groups, Journal of Noncom- mutative Geometry (2025), DOI:10.4171/JNCG/634

M. Leimbach,Convergence of Peter-–Weyl Truncations of Compact Quantum Groups, Journal of Noncom- mutative Geometry (2025), DOI:10.4171/JNCG/634. 4

work page doi:10.4171/jncg/634 2025

[32] [32]

Leimbach and W

M. Leimbach and W. D. van Suijlekom,Gromov–Hausdorff Convergence of spectral truncations for tori, Advances in Mathematics 439 (2024), 109496. 4

2024

[33] [33]

R. I. Loebl and V. I. Paulsen,Some remarks on𝐶 ∗-convexity, Linear Algebra and its Applications 35 (1981), 63–78. 2

1981

[34] [34]

V. I. Paulsen,Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathe- matics, vol. 78, Cambridge University Press, Cambridge, 2002. 7, 26

2002

[35] [35]

M. A. Rieffel,Metrics on state spaces, Documenta Mathematica 4 (1999), 559–600. 4

1999

[36] [36]

M. A. Rieffel,Gromov–Hausdorff distance for quantum metric spaces, Memoirs of the American Mathe- matical Society 168 (2004), 1–65. 4, 8

2004

[37] [37]

M. A. Rieffel,Convergence of Fourier truncations for compact quantum groups and finitely generated groups, Journal of Geometry and Physics 192 (2023), 104921. 4

2023

[38] [38]

Rosenblum,Vectorial Toeplitz operators and the Fej ´er–Riesz theorem, Journal of Mathematical Analy- sis and Applications 23 (1968), 139–147

M. Rosenblum,Vectorial Toeplitz operators and the Fej ´er–Riesz theorem, Journal of Mathematical Analy- sis and Applications 23 (1968), 139–147. 7

1968

[39] [39]

W. D. van Suijlekom,Gromov–Hausdorff convergence of state spaces for spectral truncations, Journal of Geometry and Physics 162 (2021), 104075. 4, 6, 31

2021

[40] [40]

Webster and S

C. Webster and S. Winkler,The Krein–Milman theorem in operator convexity, Transactions of the Amer- ican Mathematical Society 351 (1999), 307–322. 2 RITULDUHAN, DEPARTMENT OFMATHEMATICS, INDIANINSTITUTE OFSCIENCE, BENGALURU, INDIA Email address:ritul2023@iisc.ac.in ABHAYJINDAL, FACULTY OFMATHEMATICS ANDPHYSICS, UNIVERSITY OFLJUBLJANA, SLOVENIA Email addre...

1999