×
El Azhari, M.

Formes positives dans les \(BP^*\)-algèbres. (French) Zbl 0857.46033

Math. Slovaca 45, No. 3, 281-285 (1995).
Summary: Let \(A\) be a \(BP^*\)-algebra with a unit \(e\), \(P_1(A)\) be the set of all positive linear functionals \(f\) on \(A\) such that \(f(e)= 1\), and let \(M_s(A)\) be the set of all non-zero Hermitian multiplicative linear functionals on \(A\). We prove that \(M_s(A)\) is the set of extremal points of \(P_1(A)\). We also prove that, if \(M_s(A)\) is equicontinuous, then every positive linear functional on \(A\) is continuous. Finally, we give an example of a \(BP^*\)-algebra whose topological dual is not included in the vector space generated by \(P_1(A)\), which gives a negative answer to a question posed by M. A. Hennings [J. Math. Anal. Appl. 140, No. 2, 289-300 (1989; Zbl 0705.46032); question E].

MSC:

46J20 Ideals, maximal ideals, boundaries
46H99 Topological algebras, normed rings and algebras, Banach algebras

Keywords:

*-representation; \(BP^*\)-algebra; positive linear functionals; non-zero Hermitian multiplicative linear functionals; set of extremal points; equicontinuous; topological dual

Citations:

Zbl 0705.46032
Cite Review PDF
Full Text: EuDML

References:

[1] AKKAR M., EL AZHARI M., OUDADESS M.: On an expression of the spectrum in BP*-algebras. Period. Math. Hungar. 19 (1988), 65-67. · Zbl 0649.46052 · doi:10.1007/BF01848009
[2] ALLAN G. R.: On a class of locally convex algebra. Proc. London Math. Soc. (3) 17 (1967), 91-114. · Zbl 0147.33503 · doi:10.1112/plms/s3-17.1.91
[3] HENNINGS M. A.: Topologies on BP*-algebras. J. Math. Anal. Appl. 140 (1989), 289-300. · Zbl 0705.46032 · doi:10.1016/0022-247X(89)90064-4
[4] HUSAIN T., WARSI S. A.: Positive functionals on BP*-algebras. Period. Math. Hungar. 8 (1977), 15-28. · Zbl 0354.46034 · doi:10.1007/BF02018042
[5] HUSAIN T., WARSI S. A.: Representations of BP*-algebras. Math. Japon. 21 (1976), 237-247. · Zbl 0344.46104
[6] RICHART C. E. : General Theory of Banach Algebras. Van Nostrand, Princeton N. J., 1960.
[7] ROBERTSON A. P., ROBERTSON W.: Topological Vector Spaces. Cambridge Tracts in Math. and Math. Phys. 53, Cambridge Univ. Press, London, 1973. · Zbl 0251.46002
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.