Surjective norm-additive in modulus maps between real Lipschitz algebras with Lipschitz involution

Authors
Mansureh Mohammadi
Davood Alimohammadi
Arak University
Abstract
Let (X‚ d) and (Y‚ ρ) be compact metric spaces‚ τ be a Lipschitz involution on (X‚ d) and η be Lipschitz involution on (Y‚ ρ). Suppose that for all x∈X‚ for all y∈Y‚ , A is a real subalgebra of C(X‚ τ) which contains Lip(X‚ d‚ τ) and B is a real subalgebra of C(Y‚ η) which contains Lip(Y‚ ρ‚ η). We prove that if T:A→B is a surjetive -homogenous norm-additive in modulus map then there exists a unique bijection such that for all f∈A ‚ y∈Y and . Applying this fact‚ we show that if (X‚d) and (Y‚ρ) are compact metric spaces and is a surjective -homogenous norm-additive in modulus map then there exists a Lipschitz homeomorphism from to such that for all and y∈Ỵ
Keywords
Lipschitz involution‚ norm-additive in modulus‚ real Lipschitz algebra‚ uniform norm
1. Alimohammadi D.‚ Daneshmand S.,̎ Generalized peripherally multiplicative maps between real Lipschitz algebras with involution ̎‚ Khayyam J. Math. 7(1) (2021)‚ 1-31.

2. Alimohammadi D.‚ Ebadian A.,̎ Hedberg's theorem in real Lipschitz algebras ̎‚ Indian J. pure Appl. Math.‚ 32(10) (2001)‚ 1479-1493.

3. Hosseini M.‚ Font J.J.,̎ Norm-additive in modulus maps between function algebras ̎‚ Banach J. Math. Anal.‚ 8(2) (2014)‚ 79-92.

4. Jiménez-Vargas A.‚ Lee K.‚ Luttman A.‚ Villegas-Vallecillos M.,̎ Generalized weak peripheral multiplicativity in algebras of Lipschitz functions ̎‚ Cent. Eur. J. Math.‚ 11(7) (2013). 1197-1211.

5. Kulkarni S.‚ Limaye B.V.,̎ Gleason parts of real function algebras ̎‚ Canadian J. Math.‚ 33(1981)‚ 181-200.

6. Kulkarni S.‚ Limaye B.V.,̎ Real Function Algebras ̎‚ Marcel Dekker‚ 1992.

7. Sherbert D.R.‚ ̎ Banach algebras of Lipschitz functions ̎‚ Pacific J. Math.‚ 13 (1963)‚ 1387-1399.

8. Tonév T.‚ Yates R.,̎ Norm-linear and norm-additive operators between uniform algebras ̎‚ J. Math. Anal. Appl.‚ 357(1) (2009)‚ 45-53.