Bipositive isomorphisms on semigroup algebras

Document Type : Original Article

Authors

Faculty of Basic Sciences, Babol University of Technology, Babol, Iran

Abstract

Let $S$ be a locally compact foundation semigroup with identity and $ M_{a}(S)$ be its semigroup algebra.  In the present article, we show that if $T$ is a bipositive isomorphism from $ M_{a}(S_{1})$ onto $ M_{a}(S_{2})$, then $T$ is an isometry  and $S_{1}$ and $S_{2}$  are isomorphic locally compact semigroups. Indeed, we have obtained a generalization of a well-known result of Wendel  [9] and Kawada  [6] for locally compact groups to a more general setting of locally compact foundation semigroups. Also we show that if $T$ is a bipositive isomorphism from $M_{a}(S_{1})^{**}$ onto $M_{a}(S_{2})^{**}$,  then  $S_{1}$ and $S_{2}$  are isomorphic locally compact semigroups.

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References

[1] Aliprantis, C.D., & Burkinshaw, O. (1985). Positive operators. Volume 119 of Pure and Applied Mathematics, Academic Press, New York.
[2] Bonsall, F.F., & Duncan, J. (1973). Complete normed algebras. Springer-Verlag, Berlin.
[3] Conway, J.B. (1990). A course in functional analysis. Springer-Verlag, New York.
[4] Farhadi, H. (1998). Bipositive isomorphisms between the second duals of group algebras of locally compact groups. Math. Proc. Cambridge Philos. Soc, 123, 95–99.
[5] Hewitt, E., & Ross, K.A. (1963). Abstract Harmonic Analysis. Springer-Verlag.
[6] Kawada, Y. (1948). On the group ring of a topological group. Math. Japon, 1, 1–5.
[7] Lashkarizadeh Bami, M. (1998). Ideal of M(S) as ideals of LUC(S)∗ of a compactly cancellative semigroup S. Math. Japon, 48, 363–366. 
[8] Lashkarizadeh Bami, M., Mohammadzadeh, B., & Samea, H. (2007). Derivations on certain semigroup algebras. Journal of Sciences. Islamic Republic of Iran, 18(4), 339–345.
[9] Wendel, J.G. (1951). On isometric isomorphism of group algebras. Pacific J. Math, 1, 305–311. 

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