| Summary: | 博士 === 國立中山大學 === 應用數學系研究所 === 100 === Let A and B be real or complex C*-algebras. Let V and W be real or complex (right) full
Hilbert C*-modules over A and B, respectively. Let T be a linear bijective map from V onto
W. We show the following four statements are equivalent.
(a) T is a unitary operator, i.e., there is a ∗-isomorphism α : A → B such that
<Tx,Ty> = α(<x,y>), ∀ x,y∈ V ;
(b) T preserves TRO products, i.e., T(x<y,z>) =Tx<Ty,Tz>, ∀ x,y,z in V ;
(c) T is a 2-isometry;
(d) T is a complete isometry.
Moreover, if A and B are commutative, the four statements are also equivalent to
(e) T is a isometry.
On the other hand, if V and W are complex Hilbert C*-modules over complex C*-algebras,
then T is unitary if and only if it is a module map, i.e.,
T(xa) = (Tx)α(a), ∀ x ∈ V,a ∈ A.
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