A construction of a BSE-algebra of type I which is isomorphic to no \( C^\ast\)-algebras (Q1697460)

It is shown that, for a locally compact abelian group \(G\), \(C_0(G)\cap L^p(G)\), with norm \(\|\cdot\|_{\infty}+\|\cdot\|_p\), \(1\leq p<\infty\), is a BSE algebra of type~I which is not isomorphic to any \(C^*\)-algebra. BSE stands for Bochner-Schönberg-Eberlein and BSE algebras, introduced in [\textit{S.-E. Takahasi} and \textit{O. Hatori}, Proc. Am. Math. Soc. 110, No.~1, 149--158 (1990; Zbl 0722.46025)], are those commutative Banach algebras for which an abstract analogue of the classical BSE theorem holds. Type~I here means that all bounded continuous functions on the Gelfand space of \(A\) are the Gelfand transforms of elements of the multiplier algebra of \(A\). The authors use and cite an earlier paper [Rocky Mt. J. Math. 44, No.~2, 539--589 (2014; Zbl 1319.46040)] on Segal algebras over commutative Banach algebras. Such algebras have been studied by many authors since their introduction by \textit{J. T. Burnham} [Segal algebras and generalizations. University of Iowa (Ph.D. Thesis) (1972; MR2622880)], see also [\textit{H. Reiter}, \(L^1\)-algebras and Segal algebras. Lecture Notes in Mathematics 231. Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0219.43003)], and their properties are well known. Segal algebras of the authors, in this paper and in [Inoue and Takahashi, loc. cit.], are those of Burnham with an extra assumption on approximate units.