| Title:
|
The operation $ABA$ in operator algebras (English) |
| Author:
|
Gaál, Marcell |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
61 |
| Issue:
|
4 |
| Year:
|
2020 |
| Pages:
|
513-521 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The binary operation $aba$, called Jordan triple product, and its variants (such as e.g. the sequential product $\sqrt{a} b \sqrt{a}$ or the inverted Jordan triple product $a b^{-1} a$) appear in several branches of operator theory and matrix analysis. In this paper we briefly survey some analytic and algebraic properties of these operations, and investigate their intimate connection to Thompson type isometries in different operator algebras. (English) |
| Keyword:
|
loop |
| Keyword:
|
gyrogroup |
| Keyword:
|
Jordan triple product |
| Keyword:
|
Thompson metric |
| Keyword:
|
JB-algebra |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 07332725 |
| idMR:
|
MR4230956 |
| DOI:
|
10.14712/1213-7243.2020.041 |
| . |
| Date available:
|
2021-02-25T12:42:03Z |
| Last updated:
|
2023-01-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148661 |
| . |
| Reference:
|
[1] Abe T., Akiyama S., Hatori O.: Isometries of the special orthogonal group.Linear Algebra Appl. 439 (2013), no. 1, 174–188. MR 3045229 |
| Reference:
|
[2] Beneduci R., Molnár L.: On the standard K-loop structure of positive invertible elements in a $C^{*}$-algebra.J. Math. Anal. Appl. 420 (2014) no. 1, 551–562. MR 3229839, 10.1016/j.jmaa.2014.05.009 |
| Reference:
|
[3] Gaál M.: On certain generalized isometries of the special orthogonal group.Arch. Math. (Basel) 110 (2018), no. 1, 61–70. MR 3742293, 10.1007/s00013-017-1122-4 |
| Reference:
|
[4] Hatori O.: Isometries on the special unitary group.in Function Spaces in Analysis, Contemp. Math., 645, Amer. Math. Soc., Providence, 2015, pages 119–134. MR 3382409, 10.1090/conm/645/12925 |
| Reference:
|
[5] Hatori O., Molnár L.: Isometries of the unitary groups and Thompson isometries of the spaces of invertible positive elements in $C^{*}$-algebras.J. Math. Anal. Appl. 409 (2014), no. 1, 158–167. MR 3095026, 10.1016/j.jmaa.2013.06.065 |
| Reference:
|
[6] Hatori O., Molnár L.: Generalized isometries of the special unitary group.Arch. Math. (Basel) 106 (2016), no. 2, 155–163. MR 3453990, 10.1007/s00013-015-0856-0 |
| Reference:
|
[7] Hatori O., Molnár L.: Spectral conditions for Jordan $^*$-isomorphisms on operator algebras.Studia Math. 236 (2017), no. 2, 101–126. MR 3610684, 10.4064/sm8393-8-2016 |
| Reference:
|
[8] Isidro J. M., Rodríguez-Palacios Á.: Isometries of JB-algebras.Manuscripta Math. 86 (1995), no. 3, 337–348. MR 1323796, 10.1007/BF02567998 |
| Reference:
|
[9] Lemmens B., Roelands M., Wortel M.: Hilbert and Thompson isometries on cones in JB-algebras.Math. Z. 292 (2019), no. 3–4, 1511–1547. MR 3980302, 10.1007/s00209-018-2144-8 |
| Reference:
|
[10] Molnár L.: General Mazur–Ulam type theorems and some applications.Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, Oper. Theory Adv. Appl., 250, Birkhäuser, 2015, pages 311–342. MR 3468225 |
| Reference:
|
[11] Sabinin L. V., Sabinina L. L., Sbitneva L. V.: On the notion of gyrogroup.Aequationes Math. 56 (1998), no. 1–2, 11–17. MR 1628291, 10.1007/s000100050039 |
| Reference:
|
[12] Ungar A. A.: Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity.World Scientific Publishing, Hackensack, 2008. Zbl 1147.83004, MR 2396580 |
| . |
