Abstract
We show that conditional expectations, optimal hypotheses, disintegrations and adjoints of unital completely positive maps are all instances of Bayesian inverses. We study the existence of the latter by means of the Tomita-Takesaki modular group and we provide extensions of a theorem of Takesaki as well as a theorem of Accardi and Cecchini to the setting of not necessarily faithful states on finite-dimensional C∗-algebras.
| Original language | English |
|---|---|
| Pages (from-to) | 975-1014 |
| Number of pages | 40 |
| Journal | Quarterly Journal of Mathematics |
| Volume | 74 |
| Issue number | 3 |
| DOIs | |
| State | Published - Sep 1 2023 |
Funding
This work was supported by the European Union's Horizon 2020 research and innovation programme H2020-MSCA-IF-2017 under Grant Agreement n. 795151 ``Beyond Rationality in Algebraic CFT: mathematical structures and models”; by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program under Grant Agreement n. 677368 ``Quantum Algebraic Structures In Field Theories”; by the Ministero dell'Istruzione, dell'Universit\`a e della Ricerca “MIUR Excellence Department Project” awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006; by the “MIUR Excellence Department Project MatMod@TOV” awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C23000330006; by the University of Rome Tor Vergata funding OAQM, CUP E83C22001800005; and by MEXT-JSPS Grant-in-Aid for Transformative Research Areas (A) “Extreme Universe'', No. 21H05183. We also acknowledge support for collaboration in the occasion of the conferences ``AMS-MAA Joint Mathematics Meeting 2019” in Baltimore, Maryland, and “Operator Algebras and Quantum Physics” at the Simons Center for Geometry and Physics at Stony Brook, New York, in June 2019, provided by the National Science Foundation (NSF) Division of Mathematical Sciences (DMS) Grant n. 1641020; by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program under Grant Agreement n. 669240 “Quantum Algebraic Structures and Models”; and by the Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni of the Istituto Nazionale di Alta Matematica “Francesco Severi” (GNAMPA-INdAM). Notice: This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a non-exclusive, paid up, irrevocable, world-wide license to publish or reproduce the published form of the manuscript, or allow others to do so, for U.S. Government purposes. The DOE will provide public access to these results in accordance with the DOE Public Access Plan ( http://energy.gov/downloads/doe-public-access-plan ). This work was supported by the European Union's Horizon 2020 research and innovation programme H2020-MSCA-IF-2017 under Grant Agreement n. 795151 "Beyond Rationality in Algebraic CFT: mathematical structures and models"; by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program under Grant Agreement n. 677368 "Quantum Algebraic Structures In Field Theories"; by the Ministero dell'Istruzione, dell'Università e della Ricerca "MIUR Excellence Department Project" awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006; by the "MIUR Excellence Department Project MatMod@TOV" awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C23000330006; by the University of Rome Tor Vergata funding OAQM, CUP E83C22001800005; and by MEXT-JSPS Grant-in-Aid for Transformative Research Areas (A) "Extreme Universe", No. 21H05183. We also acknowledge support for collaboration in the occasion of the conferences "AMS-MAA Joint Mathematics Meeting 2019" in Baltimore, Maryland, and "Operator Algebras and Quantum Physics" at the Simons Center for Geometry and Physics at Stony Brook, New York, in June 2019, provided by the National Science Foundation (NSF) Division of Mathematical Sciences (DMS) Grant n. 1641020; by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program under Grant Agreement n. 669240 "Quantum Algebraic Structures and Models"; and by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni of the Istituto Nazionale di Alta Matematica "Francesco Severi" (GNAMPA-INdAM).
| Funders | Funder number |
|---|---|
| European Union's Horizon 2020 research and innovation programme H2020-MSCA-IF-2017 | 795151 |
| GNAMPA-INdAM | |
| Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni of the Istituto Nazionale di Alta Matematica | |
| Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni of the Istituto Nazionale di Alta Matematica | |
| MEXT-JSPS | 21H05183 |
| Ministero dell'Istruzione, dell'Universit | |
| National Science Foundation | |
| U.S. Department of Energy | |
| Division of Mathematical Sciences | 1641020, 669240 |
| European Research Council | |
| Ministero dell’Istruzione, dell’Università e della Ricerca | CUP E83C18000100006, CUP E83C23000330006, CUP E83C22001800005 |
| Horizon 2020 | 677368 |