The operator system of Toeplitz matrices
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- by Douglas Farenick HTML | PDF
- Trans. Amer. Math. Soc. Ser. B 8 (2021), 999-1023
Abstract:
A recent paper of A. Connes and W.D. van Suijlekom [Comm. Math. Phys. 383 (2021), pp. 2021–2067] identifies the operator system of $n\times n$ Toeplitz matrices with the dual of the space of all trigonometric polynomials of degree less than $n$. The present paper examines this identification in somewhat more detail by showing explicitly that the Connes–van Suijlekom isomorphism is a unital complete order isomorphism of operator systems. Applications include two special results in matrix analysis: (i) that every positive linear map of the $n\times n$ complex matrices is completely positive when restricted to the operator subsystem of Toeplitz matrices and (ii) that every linear unital isometry of the $n\times n$ Toeplitz matrices into the algebra of all $n\times n$ complex matrices is a unitary similarity transformation.
An operator systems approach to Toeplitz matrices yields new insights into the positivity of block Toeplitz matrices, which are viewed herein as elements of tensor product spaces of an arbitrary operator system with the operator system of $n\times n$ complex Toeplitz matrices. In particular, it is shown that min and max positivity are distinct if the blocks themselves are Toeplitz matrices, and that the maximally entangled Toeplitz matrix $\xi _n$ generates an extremal ray in the cone of all continuous $n\times n$ Toeplitz-matrix valued functions $f$ on the unit circle $S^1$ whose Fourier coefficients $\hat f(k)$ vanish for $|k|\geq n$. Lastly, it is noted that all positive Toeplitz matrices over nuclear C$^*$-algebras are approximately separable.
References
- T. Andô, Truncated moment problems for operators, Acta Sci. Math. (Szeged) 31 (1970), 319–334. MR 290157
- T. Ando, Structure of operators with numerical radius one, Acta Sci. Math. (Szeged) 34 (1973), 11–15. MR 318920
- T. Ando, Cones and norms in the tensor product of matrix spaces, Linear Algebra Appl. 379 (2004), 3–41. Tenth Conference of the International Linear Algebra Society. MR 2039295, DOI 10.1016/j.laa.2003.06.005
- Martín Argerami, The matricial range of $E_{21}$, Expo. Math. 37 (2019), no. 1, 48–83. MR 3964216, DOI 10.1016/j.exmath.2017.12.002
- William Arveson, Subalgebras of $C^{\ast }$-algebras. II, Acta Math. 128 (1972), no. 3-4, 271–308. MR 394232, DOI 10.1007/BF02392166
- William Arveson, A short course on spectral theory, Graduate Texts in Mathematics, vol. 209, Springer-Verlag, New York, 2002. MR 1865513, DOI 10.1007/b97227
- William Arveson, The noncommutative Choquet boundary, J. Amer. Math. Soc. 21 (2008), no. 4, 1065–1084. MR 2425180, DOI 10.1090/S0894-0347-07-00570-X
- Man Duen Choi and Edward G. Effros, Injectivity and operator spaces, J. Functional Analysis 24 (1977), no. 2, 156–209. MR 0430809, DOI 10.1016/0022-1236(77)90052-0
- Alain Connes and Walter D. van Suijlekom, Spectral truncations in noncommutative geometry and operator systems, Comm. Math. Phys. 383 (2021), no. 3, 2021–2067. MR 4244265, DOI 10.1007/s00220-020-03825-x
- Michael A. Dritschel, On factorization of trigonometric polynomials, Integral Equations Operator Theory 49 (2004), no. 1, 11–42. MR 2057766, DOI 10.1007/s00020-002-1198-4
- Douglas Farenick, Arveson’s criterion for unitary similarity, Linear Algebra Appl. 435 (2011), no. 4, 769–777. MR 2807232, DOI 10.1016/j.laa.2011.01.039
- Douglas Farenick, Ali S. Kavruk, Vern I. Paulsen, and Ivan G. Todorov, Operator systems from discrete groups, Comm. Math. Phys. 329 (2014), no. 1, 207–238. MR 3207002, DOI 10.1007/s00220-014-2037-6
- Douglas Farenick, Mitja Mastnak, and Alexey I. Popov, Isometries of the Toeplitz matrix algebra, J. Math. Anal. Appl. 434 (2016), no. 2, 1612–1632. MR 3415742, DOI 10.1016/j.jmaa.2015.09.057
- Douglas Farenick and Vern I. Paulsen, Operator system quotients of matrix algebras and their tensor products, Math. Scand. 111 (2012), no. 2, 210–243. MR 3023524, DOI 10.7146/math.scand.a-15225
- Leonid Gurvits and Howard Burnam, Largest separable balls around the maximally mixed bipartite quantum states, Phys. Rev. A 66 (2002), 062311.
- Masamichi Hamana, Injective envelopes of operator systems, Publ. Res. Inst. Math. Sci. 15 (1979), no. 3, 773–785. MR 566081, DOI 10.2977/prims/1195187876
- Samuel J. Harris and Se-Jin Kim, Crossed products of operator systems, J. Funct. Anal. 276 (2019), no. 7, 2156–2193. MR 3912803, DOI 10.1016/j.jfa.2018.11.017
- Henry Helson, Lectures on invariant subspaces, Academic Press, New York-London, 1964. MR 0171178
- B. Jeuris and R. Vandebril, The Kähler mean of block-Toeplitz matrices with Toeplitz structured blocks, SIAM J. Matrix Anal. Appl. 37 (2016), no. 3, 1151–1175. MR 3543154, DOI 10.1137/15M102112X
- Ali Ş. Kavruk, Nuclearity related properties in operator systems, J. Operator Theory 71 (2014), no. 1, 95–156. MR 3173055, DOI 10.7900/jot.2011nov16.1977
- A. Samil Kavruk, On a non-commutative analogue of a classical result of Namioka and Phelps, J. Funct. Anal. 269 (2015), no. 10, 3282–3303. MR 3401618, DOI 10.1016/j.jfa.2015.09.002
- Ali Kavruk, Vern I. Paulsen, Ivan G. Todorov, and Mark Tomforde, Tensor products of operator systems, J. Funct. Anal. 261 (2011), no. 2, 267–299. MR 2793115, DOI 10.1016/j.jfa.2011.03.014
- Ali S. Kavruk, Vern I. Paulsen, Ivan G. Todorov, and Mark Tomforde, Quotients, exactness, and nuclearity in the operator system category, Adv. Math. 235 (2013), 321–360. MR 3010061, DOI 10.1016/j.aim.2012.05.025
- Eberhard Kirchberg and Simon Wassermann, $C^\ast$-algebras generated by operator systems, J. Funct. Anal. 155 (1998), no. 2, 324–351. MR 1624549, DOI 10.1006/jfan.1997.3226
- Vern Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR 1976867
- Kôtarô Tanahashi and Jun Tomiyama, Indecomposable positive maps in matrix algebras, Canad. Math. Bull. 31 (1988), no. 3, 308–317. MR 956361, DOI 10.4153/CMB-1988-044-4
- Pei Yuan Wu, A numerical range characterization of Jordan blocks, Linear and Multilinear Algebra 43 (1998), no. 4, 351–361. MR 1616464, DOI 10.1080/03081089808818536
Additional Information
- Douglas Farenick
- Affiliation: Department of Mathematics & Statistics, University of Regina, Regina, Saskatchewan S4S 0A2, Canada
- MR Author ID: 243969
- ORCID: 0000-0002-7151-213X
- Email: douglas.farenick@uregina.ca
- Received by editor(s): April 3, 2021
- Received by editor(s) in revised form: May 22, 2021, and June 7, 2021
- Published electronically: November 17, 2021
- Additional Notes: This work was supported in part by the NSERC Discovery Grant program
- © Copyright 2021 by the author under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 999-1023
- MSC (2020): Primary 46L07, 47L05
- DOI: https://doi.org/10.1090/btran/83
- MathSciNet review: 4340832