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The socle and finite-dimensionality of a semiprime Banach algebra. (English) Zbl 0691.46036
Let A be a Banach algebra, presumably complex, which is semiprime, i.e., if J is a two-sided ideal such that \(J^ 2=\{0\}\), then \(J=\{0\}\). It is known that there is a two-sided ideal which is the smallest left ideal containing all minimal left ideals and also the smallest right ideal containing all minimal right ideals; this two-sided ideal is called the socle; if A has no minimal ideals the socle is defined to be \(\{0\}\). The authors show that the intersection of the socle and the radical is 0, and also that an element t is in the socle if and only if tAt is finite- dimensional; hence if A is finite-dimensional then A equals its socle and A is semisimple. A. W. Tullo [Proc. Edinburgh Math. Soc., II. Ser. 20, 1-5 (1976; Zbl 0328.46051)] has shown that if A equals its socle then A is semisimple and finite-dimensional, so the three conditions, A semisimple, A equals its socle and A finite-dimensional, are equivalent.
Reviewer: A.Wulfsohn

46H10 Ideals and subalgebras
46L05 General theory of \(C^*\)-algebras
Zbl 0328.46051
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