Isometries of real and complex Hilbert C*-modules

• Main Authors: Ming-Hsiu Hsu, 許銘修
• Other Authors: Ngai-Ching Wong
• Format: Others
• Language: en_US
• Published: 2012
• Online Access: http://ndltd.ncl.edu.tw/handle/65738727022766899829

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.