Compact Hermitian operators on projective tensor products of Banach algebras

Let U and V be, respectively, an infinite- and a finite-dimensional complex Banach algebras, and let U ⊗ 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 ⊗ p V having the properties that ‖ T 1 ‖ = ‖ T ‖ and sp ( T 1 ) = sp ( T ) ; (ii) if U and V are separable and U has Hermitian approximation property ( HAP ) , then U ⊗ p V is also separable and has HAP ; (iii) every compact analytic semigroup ( CAS ) on U induces the existence of a CAS on U ⊗ p V having some nice properties. In addition, the converse of the above results are discussed and some open problems are posed.