Constructing Extensions of Ultraweakly Continuous Linear Functionals

✍ Scribed by Douglas Bridges; Luminiţa Vı̂ţă (Dediu)

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 133 KB
• Volume: 178
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2000.3657

✦ Synopsis

Let R be a linear subset of the space B(H) of bounded operators on a Hilbert space H with an orthonormal basis. It is shown constructively that if the unit ball of R is weak-operator totally bounded, then an ultraweakly continuous linear functional on R extends to one on B(H), and the extended functional has the form

Let H be a Hilbert space over C, and R a linear subset of the space B(H) of all bounded linear operators on H. In this paper we consider, constructively, the extension and characterisation of linear functionals on R that are continuous with respect to the following topologies on B(H) : v the weak-operator topology { w the weakest topology such that the mapping T [ (Tx, y) is continuous on B(H) for all x, y in H; v the ultraweak-operator topology { _w the weakest topology such that the mapping T [ n=1 (Tx n , y n ) is continuous on B(H) for all elements (x n ) n=1 and ( y n ) n=1 of the direct sum H = n=1 H of a sequence of copies of H.

Note that these two topologies coincide on the unit ball B 1 (H)=[T # B(H) : \x # H (&Tx& &x&)] of B(H), which is { w -, and hence { _w -, totally bounded [6].

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