Compact Hermitian operators on projective tensor products of Banach algebras (Q1607830)

Summary: Let \(U\) and \(V\) be, respectively, an infinite- and a finite-dimensional complex Banach algebras, and let \(U \otimes_p V\) be their projective tensor product. We prove that (i) every compact Hermitian operator \(T_1\) on \(U\) gives rise to a compact Hermitian operator \(T\) on \(U \otimes_p V\) having the properties that \(\|T_1\|= \|T\|\) and \(\text{s}(T_1) = \text{s}(T)\); (ii) if \(U\) and \(V\) are separable and \(U\) has Hermitian approximation property \((\text{HAP})\), then \(U \otimes_p V\) is also separable and has \((\text{HAP})\); (iii) every compact analytic semigroup \((\text{CAS})\) on \(U\) induces the existence of a \((\text{CAS})\) on \(U \otimes_p V\) having some nice properties. In addition, the converse of the above results are discussed and some open problems are posed.