By F. Hirsch, G. Mokobodzki
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Example text
Px E [XQ] for x E [XQ] because xo = x\e0 = Px\e0 + {x - Px)\e0 = Px\e0- Moreover, Px = Py for x,y £ [x0] because Px = Py -^ Px\e0 = Py\e0can define x'0 = Px for x E [xo] independent of x. We also note that [x'0,x'0] < [x,x], Hence we x £ [x0] because [x,x] = [x'0,x'0] + [x - x'0,x - x'Q] > [x'0,x'0]. H")-modules. k. for (X 0 , [-, -]o)- Take any XQ E X 0 and t 6 0 o . Then it holds that x0(t) = (Ux0){t) = [Cteo, i f («,-)] = [ i o , r 0 ( t , - ) ] 0 since r f (t, •) £ X* and UT0{t, ■) = r f (*, ■) for t € 0 .
F : G ->• i,k T(H) is said to be weakly continuous if tr(aF(-)) is continuous for a € B(H). ) of G on X is a mapping [/(■) from G into A(X) for which U(s) is gramian unitary for every s e G and satisfies that U(e) = I and U(st) = U{s)U(t) for s,t 6 G, where e is the identity of G. r. [/(•) of G on X is said to be weakly continuous if (f/(-)ar,y) is continuous for x,y E X. r. , the closed submodule generated by the set {U(s)xQ : s € G) coincides with the whole space X. 5. HARMONIC ANALYSIS FOR NORMAL HILBERT B(H)-MODULES If r : G -> T{H), we put F(s,t) = Fist'1) for s,t € G.
For more infor mation relevant to this chapter we refer to Ambrose [1](1945), Giellis [l](1972), Kakihara [4](1983), Saworotnow [5](1976) and Smith [1](1974). 1. Normal Hilbert B(H)-modules. A (normal) Hilbert 5(if)-module was intro duced by Kakihara and Terasaki [l](1979) to treat Hilbert space valued stochastic processes. ff)-module is a natural abstraction of Lg(fi; H). 2 is esssentially due to Kaplansky [1] and Pashke [1]. 5 is due to Ozawa [1](1980). 2. Submodules, operators and functionals.