Compact Hermitian operators on projective tensor products of Banach algebras

Abstract

Let U and V be, respectively, an infinite- and a finite-dimensional complex Banach algebras, and let U⊗pV be their projective tensor product. We prove that (i) every compact Hermitian operator T1 on U gives rise to a compact Hermitian operator T on U⊗pV having the properties that ‖T1‖=‖T‖ and sp(T1)=sp(T); (ii) if U and V are separable and U has Hermitian approximation property (HAP), then U⊗pV is also separable and has HAP; (iii) every compact analytic semigroup (CAS) on U induces the existence of a CAS on U⊗pV having some nice properties. In addition, the converse of the above results are discussed and some open problems are posed.