Locally Convex Quasi *-Algebras and their Representations by Maria Fragoulopoulou & Camillo Trapani

Author:Maria Fragoulopoulou & Camillo Trapani
Language: eng
Format: epub
ISBN: 9783030377052
Publisher: Springer International Publishing

(ii)there exists a positive (resp. faithful) sesquilinear form φ on satisfying the following properties: (ii)1 ;

(ii)2 ;

(ii)3 ;

further, is standard, if and only if, (ii)4 .

Proof

(i) ⇒ (ii) We put

where 〈⋅|⋅〉 is the inner product defined by the Hilbertian norm ∥⋅∥′ on . Then, it is easily shown that φ is a positive sesquilinear form on satisfying the conditions (ii)1–(ii)3. If is standard, then (ii)4 follows from Lemma 5.5.3.

(ii) ⇒ (i) We put . Then, is a subspace of and due to the positivity of φ, the quotient space is a pre-Hilbert space with inner product given by

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