[{"quality_controlled":"1","ec_funded":1,"article_type":"original","publisher":"Springer Nature","author":[{"id":"D8F41E38-9E66-11E9-A9E2-65C2E5697425","first_name":"Haonan","last_name":"Zhang","full_name":"Zhang, Haonan"}],"scopus_import":"1","_id":"13271","title":"Some convexity and monotonicity results of trace functionals","date_created":"2023-07-23T22:01:15Z","article_processing_charge":"No","department":[{"_id":"JaMa"}],"publication_status":"epub_ahead","acknowledgement":"I am grateful to Boguslaw Zegarliński for asking me the questions in [3] and for helpful communication. I also want to thank Paata Ivanisvili for drawing [25] to my attention and for useful correspondence. Many thanks to the anonymous referee for the valuable comments and for pointing out some errors in an earlier version of the paper. This work is partially supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411 and the Lise Meitner fellowship, Austrian Science Fund (FWF) M3337.","external_id":{"isi":["001025709100001"],"arxiv":["2108.05785"]},"isi":1,"year":"2023","citation":{"apa":"Zhang, H. (2023). Some convexity and monotonicity results of trace functionals. <i>Annales Henri Poincare</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00023-023-01345-7\">https://doi.org/10.1007/s00023-023-01345-7</a>","ama":"Zhang H. Some convexity and monotonicity results of trace functionals. <i>Annales Henri Poincare</i>. 2023. doi:<a href=\"https://doi.org/10.1007/s00023-023-01345-7\">10.1007/s00023-023-01345-7</a>","chicago":"Zhang, Haonan. “Some Convexity and Monotonicity Results of Trace Functionals.” <i>Annales Henri Poincare</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00023-023-01345-7\">https://doi.org/10.1007/s00023-023-01345-7</a>.","ieee":"H. Zhang, “Some convexity and monotonicity results of trace functionals,” <i>Annales Henri Poincare</i>. Springer Nature, 2023.","short":"H. Zhang, Annales Henri Poincare (2023).","mla":"Zhang, Haonan. “Some Convexity and Monotonicity Results of Trace Functionals.” <i>Annales Henri Poincare</i>, Springer Nature, 2023, doi:<a href=\"https://doi.org/10.1007/s00023-023-01345-7\">10.1007/s00023-023-01345-7</a>.","ista":"Zhang H. 2023. Some convexity and monotonicity results of trace functionals. Annales Henri Poincare."},"date_updated":"2023-12-13T11:33:46Z","abstract":[{"lang":"eng","text":"In this paper, we prove the convexity of trace functionals (A,B,C)↦Tr|BpACq|s,\r\nfor parameters (p, q, s) that are best possible, where B and C are any n-by-n positive-definite matrices, and A is any n-by-n matrix. We also obtain the monotonicity versions of trace functionals of this type. As applications, we extend some results in Carlen et al. (Linear Algebra Appl 490:174–185, 2016), Hiai and Petz (Publ Res Inst Math Sci 48(3):525-542, 2012) and resolve a conjecture in Al-Rashed and Zegarliński (Infin Dimens Anal Quantum Probab Relat Top 17(4):1450029, 2014) in the matrix setting. Other conjectures in Al-Rashed and Zegarliński (Infin Dimens Anal Quantum Probab Relat Top 17(4):1450029, 2014) will also be discussed. We also show that some related trace functionals are not concave in general. Such concavity results were expected to hold in different problems."}],"day":"08","arxiv":1,"doi":"10.1007/s00023-023-01345-7","language":[{"iso":"eng"}],"publication":"Annales Henri Poincare","month":"07","project":[{"grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"_id":"eb958bca-77a9-11ec-83b8-c565cb50d8d6","grant_number":"M03337","name":"Curvature-dimension in noncommutative analysis"}],"oa_version":"Preprint","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2108.05785"}],"type":"journal_article","date_published":"2023-07-08T00:00:00Z","oa":1,"publication_identifier":{"issn":["1424-0637"]}},{"publication":"Mathematische Annalen","project":[{"name":"Curvature-dimension in noncommutative analysis","grant_number":"M03337","_id":"eb958bca-77a9-11ec-83b8-c565cb50d8d6"}],"oa_version":"Published Version","month":"07","language":[{"iso":"eng"}],"type":"journal_article","date_published":"2023-07-24T00:00:00Z","publication_identifier":{"issn":["0025-5831"],"eissn":["1432-1807"]},"oa":1,"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s00208-023-02680-0"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","scopus_import":"1","_id":"13318","author":[{"full_name":"Volberg, Alexander","last_name":"Volberg","first_name":"Alexander"},{"full_name":"Zhang, Haonan","last_name":"Zhang","first_name":"Haonan","id":"D8F41E38-9E66-11E9-A9E2-65C2E5697425"}],"department":[{"_id":"JaMa"}],"date_created":"2023-07-30T22:01:03Z","article_processing_charge":"No","publication_status":"epub_ahead","title":"Noncommutative Bohnenblust–Hille inequalities","quality_controlled":"1","publisher":"Springer Nature","article_type":"original","citation":{"apa":"Volberg, A., &#38; Zhang, H. (2023). Noncommutative Bohnenblust–Hille inequalities. <i>Mathematische Annalen</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00208-023-02680-0\">https://doi.org/10.1007/s00208-023-02680-0</a>","ama":"Volberg A, Zhang H. Noncommutative Bohnenblust–Hille inequalities. <i>Mathematische Annalen</i>. 2023. doi:<a href=\"https://doi.org/10.1007/s00208-023-02680-0\">10.1007/s00208-023-02680-0</a>","ieee":"A. Volberg and H. Zhang, “Noncommutative Bohnenblust–Hille inequalities,” <i>Mathematische Annalen</i>. Springer Nature, 2023.","chicago":"Volberg, Alexander, and Haonan Zhang. “Noncommutative Bohnenblust–Hille Inequalities.” <i>Mathematische Annalen</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00208-023-02680-0\">https://doi.org/10.1007/s00208-023-02680-0</a>.","short":"A. Volberg, H. Zhang, Mathematische Annalen (2023).","mla":"Volberg, Alexander, and Haonan Zhang. “Noncommutative Bohnenblust–Hille Inequalities.” <i>Mathematische Annalen</i>, Springer Nature, 2023, doi:<a href=\"https://doi.org/10.1007/s00208-023-02680-0\">10.1007/s00208-023-02680-0</a>.","ista":"Volberg A, Zhang H. 2023. Noncommutative Bohnenblust–Hille inequalities. Mathematische Annalen."},"year":"2023","date_updated":"2023-12-13T11:36:20Z","external_id":{"arxiv":["2210.14468"],"isi":["001035665500001"]},"isi":1,"day":"24","doi":"10.1007/s00208-023-02680-0","arxiv":1,"abstract":[{"lang":"eng","text":"Bohnenblust–Hille inequalities for Boolean cubes have been proven with dimension-free constants that grow subexponentially in the degree (Defant et al. in Math Ann 374(1):653–680, 2019). Such inequalities have found great applications in learning low-degree Boolean functions (Eskenazis and Ivanisvili in Proceedings of the 54th annual ACM SIGACT symposium on theory of computing, pp 203–207, 2022). Motivated by learning quantum observables, a qubit analogue of Bohnenblust–Hille inequality for Boolean cubes was recently conjectured in Rouzé et al. (Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum Boolean functions, 2022. arXiv preprint arXiv:2209.07279). The conjecture was resolved in Huang et al. (Learning to predict arbitrary quantum processes, 2022. arXiv preprint arXiv:2210.14894). In this paper, we give a new proof of these Bohnenblust–Hille inequalities for qubit system with constants that are dimension-free and of exponential growth in the degree. As a consequence, we obtain a junta theorem for low-degree polynomials. Using similar ideas, we also study learning problems of low degree quantum observables and Bohr’s radius phenomenon on quantum Boolean cubes."}],"acknowledgement":"The research of A.V. is supported by NSF DMS-1900286, DMS-2154402 and by Hausdorff Center for Mathematics. H.Z. is supported by the Lise Meitner fellowship, Austrian Science Fund (FWF) M3337. This work is partially supported by NSF DMS-1929284 while both authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Harmonic Analysis and Convexity program."},{"language":[{"iso":"eng"}],"month":"03","oa_version":"Published Version","project":[{"name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"_id":"eb958bca-77a9-11ec-83b8-c565cb50d8d6","grant_number":"M03337","name":"Curvature-dimension in noncommutative analysis"},{"grant_number":"716117","name":"Optimal Transport and Stochastic Dynamics","_id":"256E75B8-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2"}],"publication":"Annales Henri Poincare","has_accepted_license":"1","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file":[{"success":1,"access_level":"open_access","relation":"main_file","file_id":"14051","creator":"dernst","date_created":"2023-08-14T11:38:28Z","checksum":"8c7b185eba5ccd92ef55c120f654222c","file_size":554871,"date_updated":"2023-08-14T11:38:28Z","content_type":"application/pdf","file_name":"2023_AnnalesHenriPoincare_Wirth.pdf"}],"oa":1,"publication_identifier":{"issn":["1424-0637"]},"date_published":"2023-03-01T00:00:00Z","type":"journal_article","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"article_type":"original","publisher":"Springer Nature","file_date_updated":"2023-08-14T11:38:28Z","page":"717-750","quality_controlled":"1","ec_funded":1,"title":"Curvature-dimension conditions for symmetric quantum Markov semigroups","intvolume":"        24","publication_status":"published","date_created":"2022-09-11T22:01:57Z","department":[{"_id":"JaMa"}],"article_processing_charge":"Yes (via OA deal)","author":[{"last_name":"Wirth","first_name":"Melchior","full_name":"Wirth, Melchior","orcid":"0000-0002-0519-4241","id":"88644358-0A0E-11EA-8FA5-49A33DDC885E"},{"id":"D8F41E38-9E66-11E9-A9E2-65C2E5697425","full_name":"Zhang, Haonan","last_name":"Zhang","first_name":"Haonan"}],"_id":"12087","scopus_import":"1","ddc":["510"],"volume":24,"acknowledgement":"H.Z. is supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411 and the Lise Meitner fellowship, Austrian Science Fund (FWF) M3337. M.W. acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 716117) and from the Austrian Science Fund (FWF) through grant number F65. Both authors would like to thank Jan Maas for fruitful discussions and helpful comments. Open access funding provided by Austrian Science Fund (FWF).","abstract":[{"text":"Following up on the recent work on lower Ricci curvature bounds for quantum systems, we introduce two noncommutative versions of curvature-dimension bounds for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet–Myers theorem and concavity of entropy power in the noncommutative setting. We also provide examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz–Schur multipliers over group algebras and generalized depolarizing semigroups.","lang":"eng"}],"arxiv":1,"doi":"10.1007/s00023-022-01220-x","day":"01","isi":1,"external_id":{"isi":["000837499800002"],"arxiv":["2105.08303"]},"date_updated":"2023-08-14T11:39:28Z","citation":{"chicago":"Wirth, Melchior, and Haonan Zhang. “Curvature-Dimension Conditions for Symmetric Quantum Markov Semigroups.” <i>Annales Henri Poincare</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00023-022-01220-x\">https://doi.org/10.1007/s00023-022-01220-x</a>.","ieee":"M. Wirth and H. Zhang, “Curvature-dimension conditions for symmetric quantum Markov semigroups,” <i>Annales Henri Poincare</i>, vol. 24. Springer Nature, pp. 717–750, 2023.","ama":"Wirth M, Zhang H. Curvature-dimension conditions for symmetric quantum Markov semigroups. <i>Annales Henri Poincare</i>. 2023;24:717-750. doi:<a href=\"https://doi.org/10.1007/s00023-022-01220-x\">10.1007/s00023-022-01220-x</a>","apa":"Wirth, M., &#38; Zhang, H. (2023). Curvature-dimension conditions for symmetric quantum Markov semigroups. <i>Annales Henri Poincare</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00023-022-01220-x\">https://doi.org/10.1007/s00023-022-01220-x</a>","ista":"Wirth M, Zhang H. 2023. Curvature-dimension conditions for symmetric quantum Markov semigroups. Annales Henri Poincare. 24, 717–750.","mla":"Wirth, Melchior, and Haonan Zhang. “Curvature-Dimension Conditions for Symmetric Quantum Markov Semigroups.” <i>Annales Henri Poincare</i>, vol. 24, Springer Nature, 2023, pp. 717–50, doi:<a href=\"https://doi.org/10.1007/s00023-022-01220-x\">10.1007/s00023-022-01220-x</a>.","short":"M. Wirth, H. Zhang, Annales Henri Poincare 24 (2023) 717–750."},"year":"2023"},{"volume":302,"acknowledgement":"Yu. K. thanks Professor Waldemar Hebisch for valuable discussions on the general context of multipliers on Lie groups. This work was started during an ICL-CNRS fellowship of the second\r\nnamed author at the Imperial College London. Yu. K. is supported by the ANR-19-CE40-0002 grant of the French National Research Agency (ANR). H. Z. is supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411 and the Lise Meitner fellowship, Austrian Science Fund (FWF) M3337. R. A. was supported by the EPSRC grant EP/R003025. M. R. is supported by the EPSRC grant EP/R003025/2 and by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations.","arxiv":1,"doi":"10.1007/s00209-022-03143-z","day":"01","abstract":[{"lang":"eng","text":"The aim of this paper is to find new estimates for the norms of functions of a (minus) distinguished Laplace operator L on the ‘ax+b’ groups. The central part is devoted to spectrally localized wave propagators, that is, functions of the type ψ(L−−√)exp(itL−−√), with ψ∈C0(R). We show that for t→+∞, the convolution kernel kt of this operator satisfies\r\n∥kt∥1≍t,∥kt∥∞≍1,\r\nso that the upper estimates of D. Müller and C. Thiele (Studia Math., 2007) are sharp. As a necessary component, we recall the Plancherel density of L and spend certain time presenting and comparing different approaches to its calculation. Using its explicit form, we estimate uniform norms of several functions of the shifted Laplace-Beltrami operator Δ~, closely related to L. The functions include in particular exp(−tΔ~γ), t>0,γ>0, and (Δ~−z)s, with complex z, s."}],"date_updated":"2023-08-04T09:22:14Z","year":"2022","citation":{"chicago":"Akylzhanov, Rauan, Yulia Kuznetsova, Michael Ruzhansky, and Haonan Zhang. “Norms of Certain Functions of a Distinguished Laplacian on the Ax + b Groups.” <i>Mathematische Zeitschrift</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s00209-022-03143-z\">https://doi.org/10.1007/s00209-022-03143-z</a>.","ieee":"R. Akylzhanov, Y. Kuznetsova, M. Ruzhansky, and H. Zhang, “Norms of certain functions of a distinguished Laplacian on the ax + b groups,” <i>Mathematische Zeitschrift</i>, vol. 302, no. 4. Springer Nature, pp. 2327–2352, 2022.","ama":"Akylzhanov R, Kuznetsova Y, Ruzhansky M, Zhang H. Norms of certain functions of a distinguished Laplacian on the ax + b groups. <i>Mathematische Zeitschrift</i>. 2022;302(4):2327-2352. doi:<a href=\"https://doi.org/10.1007/s00209-022-03143-z\">10.1007/s00209-022-03143-z</a>","apa":"Akylzhanov, R., Kuznetsova, Y., Ruzhansky, M., &#38; Zhang, H. (2022). Norms of certain functions of a distinguished Laplacian on the ax + b groups. <i>Mathematische Zeitschrift</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00209-022-03143-z\">https://doi.org/10.1007/s00209-022-03143-z</a>","ista":"Akylzhanov R, Kuznetsova Y, Ruzhansky M, Zhang H. 2022. Norms of certain functions of a distinguished Laplacian on the ax + b groups. Mathematische Zeitschrift. 302(4), 2327–2352.","short":"R. Akylzhanov, Y. Kuznetsova, M. Ruzhansky, H. Zhang, Mathematische Zeitschrift 302 (2022) 2327–2352.","mla":"Akylzhanov, Rauan, et al. “Norms of Certain Functions of a Distinguished Laplacian on the Ax + b Groups.” <i>Mathematische Zeitschrift</i>, vol. 302, no. 4, Springer Nature, 2022, pp. 2327–52, doi:<a href=\"https://doi.org/10.1007/s00209-022-03143-z\">10.1007/s00209-022-03143-z</a>."},"isi":1,"external_id":{"arxiv":["2101.00584"],"isi":["000859680700001"]},"publisher":"Springer Nature","article_type":"original","page":"2327-2352","quality_controlled":"1","ec_funded":1,"publication_status":"published","date_created":"2023-01-16T09:45:31Z","department":[{"_id":"JaMa"}],"article_processing_charge":"No","title":"Norms of certain functions of a distinguished Laplacian on the ax + b groups","intvolume":"       302","_id":"12210","scopus_import":"1","author":[{"last_name":"Akylzhanov","first_name":"Rauan","full_name":"Akylzhanov, Rauan"},{"full_name":"Kuznetsova, Yulia","first_name":"Yulia","last_name":"Kuznetsova"},{"full_name":"Ruzhansky, Michael","first_name":"Michael","last_name":"Ruzhansky"},{"first_name":"Haonan","last_name":"Zhang","full_name":"Zhang, Haonan","id":"D8F41E38-9E66-11E9-A9E2-65C2E5697425"}],"issue":"4","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2101.00584"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","status":"public","publication_identifier":{"eissn":["1432-1823"],"issn":["0025-5874"]},"oa":1,"date_published":"2022-12-01T00:00:00Z","type":"journal_article","language":[{"iso":"eng"}],"keyword":["General Mathematics"],"oa_version":"Preprint","project":[{"call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411"},{"name":"Curvature-dimension in noncommutative analysis","grant_number":"M03337","_id":"eb958bca-77a9-11ec-83b8-c565cb50d8d6"}],"month":"12","publication":"Mathematische Zeitschrift"},{"publisher":"Elsevier","article_type":"original","page":"289-310","quality_controlled":"1","file_date_updated":"2023-01-27T08:08:39Z","publication_status":"published","article_processing_charge":"Yes (via OA deal)","date_created":"2023-01-16T09:46:38Z","department":[{"_id":"JaMa"}],"title":"Monotonicity versions of Epstein's concavity theorem and related inequalities","intvolume":"       654","_id":"12216","scopus_import":"1","author":[{"full_name":"Carlen, Eric A.","first_name":"Eric A.","last_name":"Carlen"},{"id":"D8F41E38-9E66-11E9-A9E2-65C2E5697425","first_name":"Haonan","last_name":"Zhang","full_name":"Zhang, Haonan"}],"volume":654,"acknowledgement":"Work partially supported by the Lise Meitner fellowship, Austrian Science Fund (FWF) M3337.","ddc":["510"],"doi":"10.1016/j.laa.2022.09.001","day":"01","abstract":[{"text":"Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this paper we prove several new monotonicity results, the first of which is a monotonicity theorem that has as a simple corollary a celebrated concavity theorem of Epstein. Our starting points are the monotonicity versions of the Lieb Concavity and the Lieb Convexity Theorems. We also give two new proofs of these in their general forms using interpolation. We then prove our new monotonicity theorems by several duality arguments.","lang":"eng"}],"date_updated":"2023-08-04T09:24:51Z","year":"2022","citation":{"ista":"Carlen EA, Zhang H. 2022. Monotonicity versions of Epstein’s concavity theorem and related inequalities. Linear Algebra and its Applications. 654, 289–310.","short":"E.A. Carlen, H. Zhang, Linear Algebra and Its Applications 654 (2022) 289–310.","mla":"Carlen, Eric A., and Haonan Zhang. “Monotonicity Versions of Epstein’s Concavity Theorem and Related Inequalities.” <i>Linear Algebra and Its Applications</i>, vol. 654, Elsevier, 2022, pp. 289–310, doi:<a href=\"https://doi.org/10.1016/j.laa.2022.09.001\">10.1016/j.laa.2022.09.001</a>.","chicago":"Carlen, Eric A., and Haonan Zhang. “Monotonicity Versions of Epstein’s Concavity Theorem and Related Inequalities.” <i>Linear Algebra and Its Applications</i>. Elsevier, 2022. <a href=\"https://doi.org/10.1016/j.laa.2022.09.001\">https://doi.org/10.1016/j.laa.2022.09.001</a>.","ieee":"E. A. Carlen and H. Zhang, “Monotonicity versions of Epstein’s concavity theorem and related inequalities,” <i>Linear Algebra and its Applications</i>, vol. 654. Elsevier, pp. 289–310, 2022.","ama":"Carlen EA, Zhang H. Monotonicity versions of Epstein’s concavity theorem and related inequalities. <i>Linear Algebra and its Applications</i>. 2022;654:289-310. doi:<a href=\"https://doi.org/10.1016/j.laa.2022.09.001\">10.1016/j.laa.2022.09.001</a>","apa":"Carlen, E. A., &#38; Zhang, H. (2022). Monotonicity versions of Epstein’s concavity theorem and related inequalities. <i>Linear Algebra and Its Applications</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.laa.2022.09.001\">https://doi.org/10.1016/j.laa.2022.09.001</a>"},"isi":1,"external_id":{"isi":["000860689600014"]},"language":[{"iso":"eng"}],"keyword":["Discrete Mathematics and Combinatorics","Geometry and Topology","Numerical Analysis","Algebra and Number Theory"],"oa_version":"Published Version","project":[{"_id":"eb958bca-77a9-11ec-83b8-c565cb50d8d6","grant_number":"M03337","name":"Curvature-dimension in noncommutative analysis"}],"month":"12","publication":"Linear Algebra and its Applications","has_accepted_license":"1","file":[{"checksum":"cf3cb7e7e34baa967849f01d8f0c1ae4","file_size":441184,"date_created":"2023-01-27T08:08:39Z","content_type":"application/pdf","file_name":"2022_LinearAlgebra_Carlen.pdf","date_updated":"2023-01-27T08:08:39Z","relation":"main_file","success":1,"access_level":"open_access","creator":"dernst","file_id":"12415"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","status":"public","publication_identifier":{"issn":["0024-3795"]},"oa":1,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"date_published":"2022-12-01T00:00:00Z","type":"journal_article"}]