| Title:
|
The kh-socle of a commutative semisimple Banach algebra (English) |
| Author:
|
Hadder, Youness |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
145 |
| Issue:
|
4 |
| Year:
|
2020 |
| Pages:
|
387-399 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\mathcal {A}$ be a commutative complex semisimple Banach algebra. Denote by ${\rm kh}({\rm soc}(\mathcal {A}))$ the kernel of the hull of the socle of $\mathcal {A}$. In this work we give some new characterizations of this ideal in terms of minimal idempotents in $\mathcal {A}$. This allows us to show that a ``result'' from Riesz theory in commutative Banach algebras is not true. (English) |
| Keyword:
|
commutative Banach algebra |
| Keyword:
|
socle |
| Keyword:
|
kh-socle |
| Keyword:
|
inessential element |
| MSC:
|
46J05 |
| MSC:
|
46J20 |
| MSC:
|
47A10 |
| idZBL:
|
07286020 |
| idMR:
|
MR4221841 |
| DOI:
|
10.21136/MB.2019.0106-18 |
| . |
| Date available:
|
2020-11-18T09:56:43Z |
| Last updated:
|
2021-04-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148431 |
| . |
| Reference:
|
[1] Aiena, P.: Fredholm and Local Spectral Theory, with Applications to Multipliers.Kluwer Academic Publishers, Dordrecht (2004). Zbl 1077.47001, MR 2070395, 10.1007/1-4020-2525-4 |
| Reference:
|
[2] Alexander, J. C.: Compact Banach algebras.Proc. Lond. Math. Soc., III. Ser. 18 (1968), 1-18. Zbl 0184.16502, MR 0229040, 10.1112/plms/s3-18.1.1 |
| Reference:
|
[3] Al-Moajil, A. H.: The compactum of a semi-simple commutative Banach algebra.Int. J. Math. Math. Sci. 7 (1984), 821-822. Zbl 0599.46072, MR 0780844, 10.1155/s0161171284000855 |
| Reference:
|
[4] Androulakis, G., Schlumprecht, T.: Strictly singular, non-compact operators exist on the space of Gowers and Maurey.J. Lond. Math. Soc., II. Ser. 64 (2001), 655-674. Zbl 1015.46007, MR 1843416, 10.1112/s0024610701002769 |
| Reference:
|
[5] Aupetit, B.: A Primer on Spectral Theory.Universitext. Springer, New York (1991). Zbl 0715.46023, MR 1083349, 10.1007/978-1-4612-3048-9 |
| Reference:
|
[6] Aupetit, B., Mouton, H. du T.: Spectrum preserving linear mappings in Banach algebras.Studia Math. 109 (1994), 91-100. Zbl 0829.46039, MR 1267714, 10.4064/sm-109-1-91-100 |
| Reference:
|
[7] Barnes, B. A.: A generalized Fredholm theory for certain maps in the regular representations of an algebra.Can. J. Math. 20 (1968), 495-504. Zbl 0159.18502, MR 0232208, 10.4153/cjm-1968-048-2 |
| Reference:
|
[8] Barnes, B. A.: The Fredholm elements of a ring.Can. J. Math. 21 (1969), 84-95. Zbl 0175.44101, MR 0237542, 10.4153/cjm-1969-009-1 |
| Reference:
|
[9] Barnes, B. A., Murphy, G. J., Smyth, M. R. F., West, T. T.: Riesz and Fredholm Theory in Banach Algebras.Research Notes in Mathematics 67. Pitman Advanced Publishing Program, Boston (1982). Zbl 0534.46034, MR 0668516 |
| Reference:
|
[10] Boudi, N., Hadder, Y.: On linear maps preserving generalized invertibility and related properties.J. Math. Anal. Appl. 345 (2008), 20-25. Zbl 1178.47023, MR 2422630, 10.1016/j.jmaa.2008.03.066 |
| Reference:
|
[11] Puhl, J.: The trace of finite and nuclear elements in Banach algebras.Czech. Math. J. 28 (1978), 656-676. Zbl 0394.46041, MR 0506439, 10.21136/CMJ.1978.101567 |
| Reference:
|
[12] Rickart, C. E.: General Theory of Banach Algebras.The University Series in Higher Mathematics. D. Van Nostrand, Princeton (1960). Zbl 0095.09702, MR 0115101 |
| Reference:
|
[13] Smyth, M. R. F.: Riesz theory in Banach algebras.Math. Z. 145 (1975), 145-155. Zbl 0298.46049, MR 0394210, 10.1007/bf01214779 |
| Reference:
|
[14] Wang, X., Cao, P.: Spectral characterization of the kh-socle in Banach Jordan algebras.J. Math. Anal. Appl. 466 (2018), 567-572. Zbl 06897081, MR 3818131, 10.1016/j.jmaa.2018.06.003 |
| . |
