*-regularity and uniqueness of \(C^*\)-norm for tensor products of *- algebras (Q753095)

The authors study *-algebras that are not necessarily Banach algebras, but admit \(C^*\)-norms. One *-algebra A is called a \(U^*\)-algebra, if every element in \(A\oplus 1\) (the algebra with adjoint unit) is a finite linear combination of unitary elements. Such an algebra is said to be \(C^*\)-unique, if there exists only one \(C^*\)-norm on A, and *- regular, if there exists only one \(C^*\)-norm on A/I\(\cap A\) for every closed ideal I in \(C^*(A)\)- its enveloping \(C^*\)-algebra. The purpose of the paper is to give criteria for \(A\otimes B\) to be *- regular or \(C^*\)-unique. Here is the main result: Let A and B be two reduced \(U^*\)-algebras. Then \(A\otimes B\) is *- regular \((C^*\)-unique), if and only if A, B and \(C^*(A)\otimes C^*(B)\) are *-regular \((C^*\)-unique). Under more restrictive conditions, when A and B are GCR-algebras, this result was first obtained by \textit{B. A. Barnes} [Trans. Am. Math. Soc. 279, 841-859 (1983; Zbl 0532.46031)]. Corollary: Let G, H be two locally compact groups. Then \(L^ 1(G\times H)\) is *-regular \((C^*\)-unique) if and only if \(L^ 1(G)\) and \(L^ 1(H)\) are *-regular \((C^*\)-unique).