×

Additive Hermitian idempotent preservers between operator algebras. (English) Zbl 1512.47068

Let \(M_{n}\) be the set of \(n\times n\) matrices and let \(H_n\) be the subset of Hermitian matrices. In the first half of the paper, the authors give a detailed description of the additive maps \(L : H_n \to H_m\) that preserve Hermitian idempotents. In particular, they show that they behave like a Jordan \(*\)-homomorphism on \(\operatorname{span}_{\mathbb{Q}} P(M_n)\). In the second half, they extend this result to von Neumann algebras in the spirit of the Bunce-Wright-Dye theorem [L. J. Bunce and J. D. M. Wright, Expo. Math. 11, No. 1, 91–95 (1993; Zbl 0772.46037); H. A. Dye, Ann. Math. (2) 61, 73–89 (1955; Zbl 0064.11002)].

MSC:

47B49 Transformers, preservers (linear operators on spaces of linear operators)
15A86 Linear preserver problems
46L10 General theory of von Neumann algebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brown, L. G.; Pedersen, G. K., \( C^\ast \)-algebras of real rank zero, J. Funct. Anal., 99, 131-149 (1991) · Zbl 0776.46026
[2] Bunce, L. J.; Wright, J. D.M., On Dye’s theorem for Jordan operator algebras, Expo. Math., 11, 91-95 (1993) · Zbl 0772.46037
[3] Cho, C.; Zhang, X., Additive operators preserving idempotent matrices over fields and applications, Linear Algebra Appl., 248, 327-338 (1996) · Zbl 0861.15017
[4] Dolinar, G., Maps on matrix algebras preserving idempotents, Linear Algebra Appl., 371, 287-300 (2003) · Zbl 1031.15003
[5] Dye, H. A., On the geometry of projections in certain operator algebras, Ann. Math., 61, 73-89 (1955) · Zbl 0064.11002
[6] Fillmore, P. A., On sums of projections, J. Funct. Anal., 4, 146-152 (1969) · Zbl 0176.43404
[7] Goldstein, S.; Paszkiewicz, A., Linear combinations of projections in von Neumann algebras, Proc. Am. Math. Soc., 116, 175-183 (1992) · Zbl 0768.47017
[8] Hamhalter, J., Quantum Measure Theory (2003), Springer Science & Business Media · Zbl 1038.81003
[9] Ke, W.-F.; Lai, K.-F.; Lee, T.-L.; Wong, N.-C., Preconditioning random Toeplitz systems, J. Nonlinear Convex Anal., 17, 4, 757-770 (2016) · Zbl 1349.47063
[10] C.-K. Li, M.-C. Tsai, Y.-S. Wang, N.-C. Wong, Nonsurjective zero product preservers between matrices over an arbitrary field, preprint.
[11] Li, C.-K.; Tsing, N.-K., Linear preserver problems: a brief introduction and some special techniques, Linear Algebra Appl., 162-164, 217-235 (1992) · Zbl 0762.15016
[12] Liu, J.-H.; Chou, C.-Y.; Liao, C.-J.; Wong, N.-C., Linear disjointness preservers of operator algebras and related structures, Acta Sci. Math. (Szeged), 84, 277-307 (2018) · Zbl 1413.47063
[13] Yao, H. M.; Cao, C. G.; Zhang, X., Additive preservers of idempotence and Jordan homomorphisms between rings of square matrices, Acta Math. Sin. Engl. Ser., 25, 4, 639-648 (2009) · Zbl 1187.15031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.