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The primitivity of semigroup algebras of free products. (English) Zbl 0870.16018
Let \(S\) be the free product of semigroups \(A\), \(B\). It is well known that the semigroup algebra \(F[S]\) over any field \(F\) is left and right primitive provided that one of \(A\), \(B\) is nontrivial. In this paper a faithful irreducible representation of \(F[S]\) is constructed. The same is done in the more difficult case of the algebra \(\ell^1(S)\) of infinite sums \(\sum^\infty_{i=1}\alpha_is_i\), \(s_i\in S\), \(\alpha_i\in K\), satisfying \(\sum_i|\alpha_i|<\infty\), over the field \(K\) of reals or complexes. The method is similar to that used by the third author [in Bull. Lond. Math. Soc. 8, 294-298 (1976; Zbl 0344.20002)].

MSC:
16S36 Ordinary and skew polynomial rings and semigroup rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
20M25 Semigroup rings, multiplicative semigroups of rings
20M30 Representation of semigroups; actions of semigroups on sets
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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References:
[1] Barnes, B.A. and J. Duncan,The Banach algebra l 1 (S), J. Funct. Anal.18 (1975), 96–113. · Zbl 0299.46047 · doi:10.1016/0022-1236(75)90032-4
[2] Formanek, E.,Group rings of free products are primitive, J. Algebra26 (1973), 508–511. · Zbl 0266.16008 · doi:10.1016/0021-8693(73)90011-2
[3] Lichtman, A.I.,The primitivity of free products of associative algebras, J. Algebra54 (1978), 153–158. · Zbl 0393.16020 · doi:10.1016/0021-8693(78)90023-6
[4] McGregor, C.M.,A representation for l 1(S), Bull. London Math. Soc.8 (1976), 156–160. · Zbl 0328.43011 · doi:10.1112/blms/8.2.156
[5] ___,On the primitivity of the group ring of a free group, Bull. London Math. Soc.8 (1976), 294–298. · Zbl 0344.20002 · doi:10.1112/blms/8.3.294
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