International Journal of Mathematics and Mathematical Sciences
Volume 29 (2002), Issue 3, Pages 167-178
doi:10.1155/S0161171202004659
Compact Hermitian operators on projective tensor
products of Banach algebras
Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India
Received 24 February 2000
Copyright © 2002 T. K. Dutta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.