This book is builded by six papers. First is projection operator and Laplace transforms. This part is begining from Hilbert space for Lapace transforms. This situation is using the orthogonal conditions of projection operators. The Laplace transforms are extended from Now Laplace transforms by Hahn-Banach theory. In this time, I have been adopted to fix the e^as. Second is the negative condition for Laplace transforms. In this time, T(a) operation has a property of ring and field conditions. The Laplace transforms is able to represent by semi-groups. Moreover, this condition is able to obtain special forms by using group-rings. Third is the involution form for Laplace transforms. This involution for Laplace transforms has been payed attention to treat the Hermitian forms.
Fourth is to extend to C* and W*-algebras. In this case, a is extended to infinite number. This extended Laplace transforms is able to treat on C* and W*-algebras. Fifth is that the extended Laplace transforms is able to arrange onto field conditions. This field condition has interesting situation in eigenspaces.
Finally, I concluded it on group conditions as a is finite number. This condition has simple and important properties for Extended Laplace transforms.
Papers
「Projection operators for Laplace transforms」
「The negative condition for Laplace transforms」
「The property of involution for Laplace transforms」
「The ring conditions for Laplace transforms」
「The field conditions for Laplace transforms」
「The group conditions for Laplace transforms」
