Characterization of linear mappings on (Banach) ⋆-algebras by similar properties to derivations
-
Behrooz Fadaee
,
Kamal Fallahi
Abstract
Let 𝓐 be a ⋆-algebra, δ : 𝓐 → 𝓐 be a linear map, and z ∈ 𝓐 be fixed. We consider the condition that δ satisfies xδ(y)⋆ + δ(x)y⋆ = δ(z) (x⋆δ(y) + δ(x)⋆y = δ(z)) whenever xy⋆ = z (x⋆y = z), and under several conditions on 𝓐, δ and z we characterize the structure of δ. In particular, we prove that if 𝓐 is a Banach ⋆-algebra, δ is a continuous linear map, and z is a left (right) separating point of 𝓐, then δ is a Jordan derivation. Our proof is based on complex variable techniques. Also, we describe a linear map δ satisfying the above conditions with z = 0 on two classes of ⋆-algebras: zero product determined algebras and standard operator algebras.
(Communicated by Emanuel Chetcuti)
Acknowledgement
The authors would like to express their sincere thanks to the referee(s) of this paper.
References
[1] Alaminos, J.—Brešar, M.—Extremera, J.—Villena, A. R.: Maps preserving zero products, Studia Math. 193 (2009), 131–159.10.4064/sm193-2-3Search in Google Scholar
[2] An, R.—Hou, J.: Characterizations of derivations on triangular rings: additive maps derivable at idempotents, Linear Algebra Appl. 431 (2009), 1070–1080.10.1016/j.laa.2009.04.005Search in Google Scholar
[3] Brešar, M.: Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. Roy. Soc. Edinb. Sect. A. 137 (2007), 9–21.10.1017/S0308210504001088Search in Google Scholar
[4] Brešar, M.—Grašić, M.—Ortega, J. S.: Zero product determined matrix algebras, Linear Algebra Appl. 430 (2009), 1486–1498.10.1016/j.laa.2007.11.018Search in Google Scholar
[5] Brešar, M.: Multiplication algebra and maps determined by zero products, Linear Multilinear Algebra 60 (2012), 763–768.10.1080/03081087.2011.564580Search in Google Scholar
[6] Burgos, M.—Ortega, J. S.: On mappings preserving zero products, Linear Multilinear Algebra 61 (2013), 323–335.10.1080/03081087.2012.678344Search in Google Scholar
[7] Chebotar, M. A.—Ke, W.-F.—Lee, P.-H.: Maps characterized by action on zero products, Pacific. J. Math. 216 (2004), 217–228.10.2140/pjm.2004.216.217Search in Google Scholar
[8] Chuang, C. L.—Lee, T. K.: Derivations modulo elementary operators, J. Algebra 338 (2011), 56–70.10.1016/j.jalgebra.2011.05.009Search in Google Scholar
[9] Ghahramani, H.: Additive mappings derivable at nontrivial idempotents on Banach algebras, Linear Multilinear Algebra 60 (2012), 725–742.10.1080/03081087.2011.628664Search in Google Scholar
[10] Ghahramani, H.: On rings determined by zero products, J. Algebra Appl. 12 (2013), 1–15.10.1142/S0219498813500588Search in Google Scholar
[11] Ghahramani, H.: Additive maps on some operator algebras behaving like (α, β)-derivations or generalized (α, β)-derivations at zero-product elements, Acta Math. Scientia 34B(4) (2014), 1287–1300.10.1016/S0252-9602(14)60085-0Search in Google Scholar
[12] Ghahramani, H.: On derivations and Jordan derivations through zero products, Operator and Matrices 4 (2014), 759–771.10.7153/oam-08-42Search in Google Scholar
[13] Ghahramani, H.: Linear maps on group algebras determined by the action of the derivations or anti-derivations on a set of orthogonal elements, Results Math. 73 (2018), 132–146.10.1007/s00025-018-0898-2Search in Google Scholar
[14] Hou, J. C.—Zhang, X. L.: Ring isomorphisms and linear or additive maps preserving zero products on nest algebras, Linear Algebra Appl. 387 (2004), 343–360.10.1016/j.laa.2004.02.032Search in Google Scholar
[15] Jing, W. S.—Lu, S.—Li, P.: Characterizations of derivations on some operator algebras, Bull. Austr. Math. Soc. 66 (2002), 227–232.10.1017/S0004972700040077Search in Google Scholar
[16] Lu, F.: Characterizations of derivations and Jordan derivations on Banach algebras, Linear Algebra Appl. 430 (2009), 2233–2239.10.1016/j.laa.2008.11.025Search in Google Scholar
[17] Marcoux, L. W.: Projections, commutators and Lie ideals in C⋆-algebras, Math. Proc. R. Ir. Acad. 110A (2010), 31–5510.1353/mpr.2010.0014Search in Google Scholar
[18] Pearcy, C.—Topping, D.: Sum of small numbers of idempotent, Michigan Math. J. 14 (1967), 453–465.10.1307/mmj/1028999848Search in Google Scholar
[19] Ponnusamy, S.—Silverman, H.: Complex Variables with Applications, Birkhäuser, Boston, 2006.Search in Google Scholar
[20] Qi, X.—Hou, J.: Characterizations of derivations of Banach space nest algebras: All-derivable points, Linear Algebra Appl. 432 (2010), 3183–3200.10.1016/j.laa.2010.01.020Search in Google Scholar
[21] Sinclair, A. M.: Jordan homomorphisms and derivations on semisimple Banach algebras, Proc. Amer. Math. Soc. 24 (1970), 209–214.10.2307/2036730Search in Google Scholar
[22] Zhu, J.—Xiong, C. P.: Generalized derivable mappings at zero point on some reflexive operator algebras, Linear Algebra Appl. 397 (2005), 367–379.10.1016/j.laa.2004.11.012Search in Google Scholar
[23] Zhu, J.—Xiong, C. P.: Derivable mappings at unit operator on nest algebras, Linear Algebra Appl. 422 (2007), 721–735.10.1016/j.laa.2006.12.002Search in Google Scholar
[24] Zhu, J.—Zhao, S.: Characterizations all-derivable points in nest algebras, Proc. Amer. Math. Soc. 141(7) (2013), 2343–2350.10.1090/S0002-9939-2013-11511-XSearch in Google Scholar
© 2020 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular papers
- Some relative normality properties in locales
- Upper bounds of some special zeros for the Rankin-Selberg L-function
- Factorization of polynomials over valued fields based on graded polynomials
- Varieties of ∗-regular rings
- On reverse Hölder and Minkowski inequalities
- Coefficient inequalities related with typically real functions
- Existence of wandering and periodic domain in given angular region
- The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*
- Uniqueness problem of meromorphic mappings of a complete Kähler manifold into a projective space
- Long time decay of 3D-NSE in Lei-Lin-Gevrey spaces
- Bn-maximal operator and Bn-singular integral operators on variable exponent Lebesgue spaces
- 𝔻-recurrent ∗-Ricci tensor on three-dimensional real hypersurfaces in nonflat complex space forms
- More on closed non-vanishing ideals in CB(X)
- The Lindley negative-binomial distribution: Properties, estimation and applications to lifetime data
- Multi-opponent James functions
- An alternative distribution to Lindley and Power Lindley distributions with characterizations, different estimation methods and data applications
- A new one-parameter discrete distribution with associated regression and integer-valued autoregressive models
- On the bond pricing partial differential equation in a convergence model of interest rates with stochastic correlation
- Characterization of linear mappings on (Banach) ⋆-algebras by similar properties to derivations
Articles in the same Issue
- Regular papers
- Some relative normality properties in locales
- Upper bounds of some special zeros for the Rankin-Selberg L-function
- Factorization of polynomials over valued fields based on graded polynomials
- Varieties of ∗-regular rings
- On reverse Hölder and Minkowski inequalities
- Coefficient inequalities related with typically real functions
- Existence of wandering and periodic domain in given angular region
- The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*
- Uniqueness problem of meromorphic mappings of a complete Kähler manifold into a projective space
- Long time decay of 3D-NSE in Lei-Lin-Gevrey spaces
- Bn-maximal operator and Bn-singular integral operators on variable exponent Lebesgue spaces
- 𝔻-recurrent ∗-Ricci tensor on three-dimensional real hypersurfaces in nonflat complex space forms
- More on closed non-vanishing ideals in CB(X)
- The Lindley negative-binomial distribution: Properties, estimation and applications to lifetime data
- Multi-opponent James functions
- An alternative distribution to Lindley and Power Lindley distributions with characterizations, different estimation methods and data applications
- A new one-parameter discrete distribution with associated regression and integer-valued autoregressive models
- On the bond pricing partial differential equation in a convergence model of interest rates with stochastic correlation
- Characterization of linear mappings on (Banach) ⋆-algebras by similar properties to derivations
