Banach algebras with unique uniform norm. II (Q2759176)

A uniform norm on a normed algebra \(({\mathcal A}, \|\cdot\|)\) is a submultiplicative norm \(|\cdot|\) satisfying the square property \(|x^2|=|x|^2\). If \({\mathcal A}\) admits a uniform norm then \({\mathcal A}\) is semisimple and commutative; further, if \(({\mathcal A}, \|\cdot\|)\) is complete, then the spectral radius \(r(\cdot)=r_A(\cdot)\) is (in fact, the greatest) uniform norm. A normed algebra \({\mathcal A}\) has unique norm property \((UUNP)\) if it admits exactly one uniform norm. The authors continue studies of commutative semisimple \(UUNP\) Banach algebras undertaken in their papers Proc. Indian. Acad. Sci. Math. Sci. 105, No. 4, 405-409 (1995; Zbl 0866.46029) and part I, Proc. Am. Math. Soc. 124, No. 2, 579-584 (1996; Zbl 0840.46032). The property \(UUNP\) is characterized in terms of the Hausdorff completion \(U({\mathcal A})\) of \({\mathcal A}\) in the spectral radius. Among other results are proved: NEWLINENEWLINENEWLINE\({\mathcal A}\) has \(UUNP\) iff \(U({\mathcal A})\) has \(UUNP\) and for any non-zero spectral synthesis ideal \({\mathcal I}\) of \(U({\mathcal A})\), \({\mathcal I}\cap{\mathcal A}\) is non-zero; NEWLINENEWLINENEWLINE\({\mathcal A}\) is regular iff \(U({\mathcal A})\) is regular, and the quotient algebra \({\mathcal A}/{\mathcal I}\) has \(UUNP\) for every spectral synthesis ideal \({\mathcal I}\) of \({\mathcal A}\); NEWLINENEWLINENEWLINE\({\mathcal A}\) is regular iff \(U({\mathcal A})\) is regular, and for any spectral synthesis ideal \({\mathcal I}\) of \(U({\mathcal A})\), \({\mathcal I}=k(h({\mathcal A}\cap{\mathcal I}))\) (hulls and kernels in \(U({\mathcal A})\)). NEWLINENEWLINENEWLINESeveral classes of Banach algebras that are weakly regular but not regular, as well as those that are not weakly regular but have \(UUNP\) are exhibited. The results are applied to multivariate holomorphic function algebras as well as to the measure algebra of a locally compact abelian group \(G\). For a continuous weight \(\omega\) on \(G\), the Beurling algebra \(L^1(G, \omega)\) (assumed semisimple) has \(UUNP\) iff it is regular.