## Banach Journal of Mathematical Analysis

### Point multipliers and the Gleason–Kahane–Żelazko theorem

#### Abstract

Let $A$ be a Banach algebra, and let $\mathcal{X}$ be a left Banach $A$-module. In this paper, using the notation of point multipliers on left Banach modules, we introduce a certain type of spectrum for the elements of $\mathcal{X}$ and we also introduce a certain subset of $\mathcal{X}$ which behaves as the set of invertible elements of a commutative unital Banach algebra. Among other things, we use these sets to give some Gleason–Kahane–Żelazko theorems for left Banach $A$-modules.

#### Article information

Source
Banach J. Math. Anal. Volume 11, Number 4 (2017), 864-879.

Dates
Received: 3 August 2016
Accepted: 5 December 2016
First available in Project Euclid: 17 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1502935411

Digital Object Identifier
doi:10.1215/17358787-2017-0021

#### Citation

Ghodrat, Razieh Sadat; Sady, Fereshteh. Point multipliers and the Gleason–Kahane–Żelazko theorem. Banach J. Math. Anal. 11 (2017), no. 4, 864--879. doi:10.1215/17358787-2017-0021. https://projecteuclid.org/euclid.bjma/1502935411

#### References

• [1] B. Aupetit, A primer on spectral theory, Springer, New York, 1991.
• [2] E. Behrends, Multiplier representations and an application to the problem whether $A\otimes_{\varepsilon}\mathit{X}$ determines $A$ and/or $\mathit{X}$, Math. Scand. 52 (1983), no. 1, 117–144.
• [3] D. Blecher and K. Jarosz, Isomorphisms of function modules, and generalized approximation in modulus, Trans. Amer. Math. Soc. 354 (2002), no. 9, 3663–3701.
• [4] D. Blecher and C. Merdy, On function and operator modules, Proc. Amer. Math. Soc. 129 (2001), no. 3, 833–844.
• [5] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, New York, 1973.
• [6] A. Bourhim and J. Mashreghi, “A survey on preservers of and local spectra,” in Invariant Subspaces of the Shift Operator, Contemp. Math., 638 (2015), 45–98.
• [7] J. Bračič, Simple multipliers on Banach modules, Glasg. Math. J. 45 (2003), no. 2, 309–322.
• [8] M. Brešar and P. Šemrl, Linear maps preserving the spectral radius, J. Funct. Anal. 142 (1996), no. 2, 360–368.
• [9] L. Dubarbie, Maps preserving common zeros between subspaces of vector-valued continuous functions, Positivity 14 (2010), no. 4, 695–703.
• [10] D. H. Dunford, Segal algebras and left normed ideals, J. Lond. Math. Soc. (2) 8 (1974), 514–516.
• [11] A. Hausner, Ideals in a certain Banach algebra, Proc. Amer. Math. Soc. 8 (1957), 246–249.
• [12] M. Hosseini and F. Sady, Common zeros preserving maps on vector-valued function spaces and Banach modules, Publ. Mat. 60 (2016), no. 2, 565–582.
• [13] A.A. Jafarian, A survey of invertibility and spectrum preserving linear maps, Bull. Iranian Math. Soc. 35 (2009), no. 2, 1–10.
• [14] A. A. Jafarian and A. R. Sourour, Spectrum-preserving linear maps, J. Funct. Anal. 66 (1986), no. 2, 255–261.
• [15] K. Jarosz, Multipliers in complex Banach spaces and structure of the unit balls, Studia Math. 87 (1987), no. 3, 197–213.
• [16] K. Jarosz, Generalizations of the Gleason-Kahane-Zelazko theorem, Rocky Mountain J. Math. 21 (1991), no. 3, 915–921.
• [17] E. Kaniuth, A course in commutative Banach algebras, Springer, New York, 2009.
• [18] G. Krishna Kumar and S. H. Kulkarni, Linear Maps preserving pseudospectrum and condition spectrum, Banach J. Math. Anal. 6 (2012), no. 1, 45–60.
• [19] D. H. Leung and W. K. Tang, Banach–Stone theorems for maps preserving common zeros, Positivity 14 (2010), no. 1, 17–42.
• [20] L. Li and N. C. Wong, Kaplansky theorem for completely regular spaces, Proc. Amer. Math. Soc. 142 (2014), no. 4, 1381–1389.
• [21] L. Li and N. C. Wong, Banach-Stone theorems for vector valued functions on completely regular spaces, J. Math. Anal. Appl. 395 (2012), no. 1, 265–274.
• [22] J. Mashreghi and T. Ransford, A Gleason–Kahane–Żelazko theorem for modules and applications to holomorphic function spaces, J. Lond. Math. Soc. (2) 47 (2015), no. 6, 1014–1020.
• [23] A. R. Sourour, Invertibility preserving linear maps on L(X), Trans. Amer. Math. Soc. 348 (1996), no. 1, 13–30.