Mayghani, M., Alimohammadi, D. (2017). Quasicompact and Riesz unital endomorphisms of real Lipschitz algebras of complex-valued functions. Sahand Communications in Mathematical Analysis, (), -.

Maliheh Mayghani; Davood Alimohammadi. "Quasicompact and Riesz unital endomorphisms of real Lipschitz algebras of complex-valued functions". Sahand Communications in Mathematical Analysis, , , 2017, -.

Mayghani, M., Alimohammadi, D. (2017). 'Quasicompact and Riesz unital endomorphisms of real Lipschitz algebras of complex-valued functions', Sahand Communications in Mathematical Analysis, (), pp. -.

Mayghani, M., Alimohammadi, D. Quasicompact and Riesz unital endomorphisms of real Lipschitz algebras of complex-valued functions. Sahand Communications in Mathematical Analysis, 2017; (): -.

Quasicompact and Riesz unital endomorphisms of real Lipschitz algebras of complex-valued functions

Articles in Press, Accepted Manuscript , Available Online from 06 February 2017

^{1}Department of Mathematics, Payame Noor University, P. O. Box: 19359-3697, Tehran, Iran.

^{2}Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran.

Abstract

We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{\mathbb{C}}\longrightarrow E_{\mathbb{C}}$ is quasicompact (Riesz, respectively), where the complex Banach space $E_{\mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{\mathbb{C}}$ associated with $T$. Next, we prove that every unital endomorphism of real Lipschitz algebras of complex-valued functions on compact metric spaces with Lipschitz involutions is a composition operator. Finally, we study some properties of quasicompact and Riesz unital endomorphisms of these algebras.

[1] D. Alimohammadi and A. Ebadian, Hedberg's theorem in real Lipschitz algebras, Indian J. Pure Appl. Math. 32 (10)(2001), 1470-1493.

[2] D. Alimohammadi and S. Sefidgar, Compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions, Journal of Linear and Topological Algebra, 3 (2) (2014), 85-101.

[3] F. Behrouzi, Riesz and quasi-compact endomorphisms of Lipschitz algebras, Houst. J. Math. 36 (2010), 793-802.

[4] F.F. Bonsall and J. Duncan, Complete Normed Algebras, Springer Verlag, New York, 1973.

[5] J.F. Feinstein and H. Kamowitz, Quasicompact and Riesz endomorphisms of Banach algebras, J. Funct. Anal. 225 (2005), 427-438.

[6] J.F. Feinstein and H. Kamowitz, Quasicompact endomorphisms of commutative semiprime Banach algebras, Banach Center Publ. 91 (2010), 159-167.

[7] A. Golbaharan and H. Mahyar, Essential spectral radius of quasicompact endomorphisms of Lipschitz algebras, Rocky Mountain J. Math. 45 (4) (2015), 1149-1164.

[8] A. Jiménez-Vargas, M. Lacruz, and M. Villegas-Vallecillos, Essential norm of composition operators on Banach spaces of Hölder functions, Abstract and Applied Analysis, Hindawi Publishing Corporation, (2011), Article ID 590853, 13 pages.

[9] S.H. Kulkarni and B.V. Limaye, Gleason parts of real function algebras, Canad. J. Math. 33 (1) (1981), 181-200.

[10] S.H. Kulkarni and B.V. Limaye, Real Function Algebras, Marcel Dekker, New York, 1992.

[11] H. Mahyar, Quasicompact and Riesz endomorphisms of infinitely differentiable Lipschitz algebras, Southeast Asian Bull. Math. 35 (2011), 249-259.

[12] H. Mahyar and A.H. Sanatpour, Quasicompact endomorphisms of Lipschitz algebras of analytic functions, Publ. Math. Debr. 76/1-2(2010), 135-145.

[13] H. Mahyar and A.H. Sanatpour, Compact and quasicompact homomorphisms between differentiable Lipschitz algebras, Bull. Belg. Math. Soc. Simon-Stevin 17 (2010), 485-497.

[14] A.H. Sanatpour, Quasicompact composition operators and power-contractive selfmaps, Ann. Funct. Anal. 7 (2) (2016), 281-289.

[15] D.R. Sherbert, Banach algebras of Lipschitz functions, Pacific J. Math. 13 (1963), 1387-1399.

[16] D.R. Sherbert, The structure of ideals and point derivations of Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. 111 (1964), 240-272.