Formes positives dans les \(BP^*\)-algèbres. (French) Zbl 0857.46033
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 dualCitations:
Zbl 0705.46032References:
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