Article
| Title: | A generalized Birkhoff--James orthogonality and norm parallelism in unital $C^*$-algebras and their characterizations (English) |
| Author: | Jalali Ghamsari, Hooriye S. |
| Author: | Dehghani, Mahdi |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 66 |
| Issue: | 1 |
| Year: | 2025 |
| Pages: | 47-69 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $\mathcal{A}$ be a unital $C^*$-algebra and let $a\in\mathcal{A}$ be a positive and invertible element. Suppose that $\mathcal{S}(\mathcal{A})$ is the set of all states on $\mathcal{\mathcal{A}}$ and let $$ \mathcal{S}_a (\mathcal{A})=\Big\{\frac{f}{f(a)} \colon f \in \mathcal{S}(\mathcal{A}), f(a)\neq 0\Big\}. $$ We introduce a family of generalized norms, called $(a,\lambda)$-norms, on $\mathcal{A}$ defined by $$ \|x\|_{a,\lambda}:=\sup\Big\{\sqrt{\lambda\varphi(x^*ax) +(1-\lambda)|\varphi(ax)|^2} \colon \varphi\in\mathcal{S}_a(\mathcal{A})\Big\},\qquad \lambda\in [0,1]. $$ This family of norms generalizes the recently introduced $a$-operator norm, $\|{\cdot}\|_a$ and $a$-numerical radius norm, $v_a({\cdot})$ in unital $C^*$-algebras. The notions of Birkhoff--James orthogonality and norm-parallelism with respect to $\|{\cdot}\|_{a,\lambda}$, which is called, $(a,\lambda)$-Birkhoff--James orthogonality and $(a,\lambda)$-norm parallelism in $\mathcal{A}$, respectively, are introduced and investigated. Characterizations of $(a,\lambda)$-norm parallelism and $(a,\lambda)$-Birkhoff--James orthogonality in terms of the elements of $\mathcal{S}_a(\mathcal{A})$ are obtained. In particular, the relationship between these new concepts are described. Our results extend and cover some known results in this area. (English) |
| Keyword: | $a$-numerical radius |
| Keyword: | $a$-Birkhoff--James orthogonality |
| Keyword: | $a$-norm parallelism |
| Keyword: | $a$-numerical radius parallelism |
| Keyword: | $C^*$-algebra |
| Keyword: | state space |
| Keyword: | $a$-numerical range |
| MSC: | 46B20 |
| MSC: | 46C50 |
| MSC: | 46L05 |
| MSC: | 47A12 |
| DOI: | 10.14712/1213-7243.2026.007 |
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| Date available: | 2026-05-15T05:55:45Z |
| Last updated: | 2026-05-18 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153612 |
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| Reference: | [1] Alahmari A., Mabrouk M., Zamani A.: Further results on the $a$-numerical range in $C^*$-algebras.Banach J. Math. Anal. 16 (2022), no. 2, Paper No. 25, 16 pages. 10.1007/s43037-022-00181-x |
| Reference: | [2] Arambašić L., Guterman A., Kuzma B., Zhilina S.: Birkhoff–James orthogonality: characterizations, preservers, and orthogonality graphs.in: Operator and Norm Inequalities and Related Topics, Trends Math. Birkhäuser/Springer, Cham, 2022, pages 255–302. |
| Reference: | [3] Arambašić L., Rajić R.: The Birkhoff–James orthogonality in Hilbert $C^*$-modules.Linear Algebra Appl. 437 (2012), no. 7, 1913–1929. |
| Reference: | [4] Arias M. L., Corach G., Gonzalez M. C.: Metric properties of projections in semi-Hilbertian spaces.Integral Equations Operator Theory 62 (2008), no. 1, 11–28. 10.1007/s00020-008-1613-6 |
| Reference: | [5] Arias M. L., Corach G., Gonzalez M. C.: Partial isometries in semi-Hilbertian spaces.Linear Algebra Appl. 428 (2008), no. 7, 1460–1475. 10.1016/j.laa.2007.09.031 |
| Reference: | [6] Bhunia P., Feki K., Paul K.: $A$-numerical radius orthogonality and parallelism of semi-Hilbertian space operators and their applications.Bull. Iranian Math. Soc. 47 (2021), no. 2, 435–457. 10.1007/s41980-020-00392-8 |
| Reference: | [7] Bhattacharyya T., Grover P.: Characterization of Birkhoff–James orthogonality.J. Math. Anal. Appl. 407 (2013), no. 2, 350–358. 10.1016/j.jmaa.2013.05.022 |
| Reference: | [8] Bottazzi T., Conde C., Feki K.: On $A$-parallelism and $A$-Birkhoff–James orthogonality of operators.Results Math. 76 (2021), no. 4, Paper No. 209, 27 pages. 10.1007/s00025-021-01515-1 |
| Reference: | [9] Bourhim A., Mabrouk M.: $a$-numerical range on $C^*$-algebras.Positivity 25 (2021), no. 4, 1489–1510. 10.1007/s11117-021-00825-6 |
| Reference: | [10] Enderami S. M., Abtahi M., Zamani A.: An extension of Birkhoff–James orthogonality relations in semi-Hilbertian space operators.Mediterr. J. Math. 19 (2022), no. 5, Paper No. 234, 11 pages. 10.1007/s00009-022-02127-x |
| Reference: | [11] Jalali Ghamsari H. S., Dehghani M.: Characterization of $a$-Birkhoff–James orthogonality in $C^*$-algebras and its applications.Ann. Funct. Anal. 15 (2024), no. 2, Paper No. 36, 20 pages. |
| Reference: | [12] James R. C.: Orthogonality and linear functionals in normed linear spaces.Trans. Amer. Math. Soc. 61 (1947), 265–292. 10.1090/S0002-9947-1947-0021241-4 |
| Reference: | [13] Kittaneh F., Zamani A.: Characterizations of $w_{\rho}$-Birkhoff–James orthogonality and $w_{\rho}$-parallelism.Math. Inequal. Appl. 28 (2025), no. 2, 173–187. |
| Reference: | [14] Mabrouk M., Zamani A.: An extension of the $a$-numerical radius on $C^*$-algebras.Banach J. Math. Anal. 17 (2023), no. 3, Paper No. 42, 23 pages. 10.1007/s43037-023-00265-2 |
| Reference: | [15] Moslehian M. S., Zamani A.: Characterizations of operator Birkhoff–James orthogonality.Canad. Math. Bull. 60 (2017), no. 4, 816–829. 10.4153/CMB-2017-004-5 |
| Reference: | [16] Murphy G. J.: $C^*$-algebras and Operator Theory.Academic Press, Boston, 1990. |
| Reference: | [17] Raeburn I., Williams D. P.: Morita Equivalence and Continuous-trace $C^*$-algebras.Mathematical Surveys and Monographs, 60, American Mathematical Society, Providence, 1998. 10.1090/surv/060/05 |
| Reference: | [18] Serb I.: Rectangular modulus, Birkhoff orthogonality and characterizations of inner product spaces.Comment. Math. Univ. Carolin. 40 (1999), no. 1, 107–119. |
| Reference: | [19] Williams J. P.: Finite operators.Proc. Amer. Math. Soc. 26 (1970), 129–136. 10.1090/S0002-9939-1970-0264445-6 |
| Reference: | [20] Wójcik P.: The Birkhoff orthogonality in pre-Hilbert $C^*$-modules.Oper. Matrices 10 (2016), no. 3, 713–729. 10.7153/oam-10-44 |
| Reference: | [21] Wójcik P.: Norm-parallelism in classical M-ideals.Indag. Math. (N.S.) 28 (2017), no. 2, 287–293. 10.1016/j.indag.2016.07.001 |
| Reference: | [22] Wójcik P., Zamani A.: From norm derivatives to orthogonalities in Hilbert $C^*$-modules.Linear Multilinear Algebra 71 (2023), no. 6, 875–888. 10.1080/03081087.2022.2046688 |
| Reference: | [23] Zamani A.: Birkhoff–James orthogonality of operators in semi-Hilbertian spaces and its applications.Ann. Funct. Anal. 10 (2019), no. 3, 433–445. 10.1215/20088752-2019-0001 |
| Reference: | [24] Zamani A.: Characterization of numerical radius parallelism in $C^*$-algebras.Positivity 23 (2019), no. 2, 397–411. 10.1007/s11117-018-0613-2 |
| Reference: | [25] Zamani A., Moslehian M. S.: Exact and approximate operator parallelism.Canad. Math. Bull. 58 (2015), no. 1, 207–224. 10.4153/CMB-2014-029-4 |
| Reference: | [26] Zamani A., Moslehian M. S.: Norm-parallelism in the geometry of Hilbert $C^*$-modules.Indag. Math. (N.S.) 27 (2016), no. 1, 266–281. 10.1016/j.indag.2015.10.008 |
| Reference: | [27] Zamani A., Wójcik P.: Numerical radius orthogonality in $C^*$-algebras.Ann. Funct. Anal. 11 (2020), no. 4, 1081–1092. 10.1007/s43034-020-00071-z |
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