×

zbMATH — the first resource for mathematics

The socle and finite dimensionality of some Banach algebras. (English) Zbl 1098.46034
Summary: The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. One of the main results in Theorem 2 which states that for a locally compact group \(G\), \(G\) is compact if there exists a measure \(\mu\) in \(\text{Soc}(L^1(G))\) such that \(\mu(G)\neq 0\). We also prove that \(G\) is finite if \(\text{Soc} (M(G))\) is closed and every nonzero left ideal in \(M(G)\) contains a minimal left ideal.

MSC:
46H20 Structure, classification of topological algebras
46H10 Ideals and subalgebras
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Al-Moajil A H, The compactum and finite dimensionality in Banach algebras,Int. J. Math. & Math. Sci. 5 (1982) 275–280 · Zbl 0481.46024 · doi:10.1155/S0161171282000246
[2] Baker J W and Filali M, On minimal ideals in some Banach algebras associated with a locally compact group,J. London Math. Soc. 63 (2001) 83–98 · Zbl 1010.43002 · doi:10.1112/S0024610700001733
[3] Bresar M and Semrl P, Finite rank elements in semisimple Banach algebras,Studia Math. 128 (1998) 287–298
[4] Dales H G, Banach algebras and automatic continuity (New York, Oxford: Oxford Uni-versity Press Inc.) (2000)
[5] Dalla L, Giotopoulos S and Katseli N, The Socle and finite dimensionality of a semiprime Banach algebra,Studia Math. 92 (1989) 201–204 · Zbl 0691.46036
[6] Filali M, Finite dimensional left ideals in some Banach algebras associated with a locally compact group,Proc. Am. Math. Soc. 127 (1999) 2325–2333 · Zbl 0918.43003 · doi:10.1090/S0002-9939-99-04793-0
[7] Filali M, Finite dimension right ideals in some Banach algebras associated with a locally compact group,Proc. Am. Math. Soc. 127 (1999) 1729–1734 · Zbl 0918.43002 · doi:10.1090/S0002-9939-99-04631-6
[8] Filali M, The ideal structure of some Banach algebras,Math. Proc. Camb. Philos. Soc. 111 (1992) 567–576 · Zbl 0765.46032 · doi:10.1017/S0305004100075642
[9] Ghaffari A, Convolution operators on semigroup algebras,Southeast Asian Bull. Math. 27 (2004) 1025–1036 · Zbl 1052.22001
[10] Hewitt E and Ross K A, Abstract harmonic analysis (Heidelberg and New York: Springer-Verlag, Berlin) (1963) vol.1 · Zbl 0115.10603
[11] Hewitt E and Ross K A, Abstract harmonic analysis (Heidelberg and New York: Springer-Verlag, Berlin) (1970) vol.II · Zbl 0213.40103
[12] Takahasi S E, Finite dimensionality in socle of Banach algebras,Int. J. Math & Math. Sci. 7 (1984) 519–522 · Zbl 0581.46043 · doi:10.1155/S0161171284000570
[13] Tullo A W, Condition on Banach algebras which imply finite dimensionality,Proc. Edin-burg Math. Soc. 20 (1976) 1–5 · Zbl 0328.46051 · doi:10.1017/S0013091500015698
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.