By Hyman Kestelman
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For a unique b ∈ M∞ (M ) which satisfies b = qbq). (xiv) A vector ξ in an M -module X is said to be a bounded vector if the following equivalent conditions are satisfied: (a) there exists a constant K > 0 such that ||x · ξ||2 ≤ K trM (x∗ x) ∀ x ∈ M ; (b) there exists a (unique) M -linear operator Rξ : M L2 (M ) → M X such that Rξ (Ω) = ξ. (xv) The set of bounded vectors in an M -module X is denoted by the symbol X 0 . Then, we have: (a) the subspace X 0 is M -stable and dense; further T (X 0 ) ⊆ Y 0 ∀ T ∈ Hom(M X, M Y ) ; (b) conversely, any ‘M -linear’ map from X 0 to Y 0 extends to a unique M -linear bounded operator in Hom(M X, M Y ).
2001 CRC Press LLC Appeal again to the unitarity condition to note that d(Z1 )d(A) Z(ξ14 , ξ24 , ξ4 , ξ34 ) Z(ξ14 , ξ24 , µ, ξ34 ) ξ24 ,Z1 ,ξ34 = δ(ξ14 ,A,ξ4 ),(ξ14 ,A,µ) , and hence that d(Z1 ) Z(ξ14 , ξ24 , ξ4 , ξ34 ) Z(ξ14 , ξ24 , µ, ξ34 ) = ξ24 ,Z1 ,ξ34 1 δξ ,µ . 21) in the last step; the proof of the verification of invariance under moves of type (1-4), and hence under choice of the underlying triangulation, is complete. Hence the quantity < (M, ∆, ≤, B) > is actually an invariant of the oriented manifold M .
M, δ, ≤); φ > extends uniquely to a conjugate-linear functional on W (Σ, δ, ≤). Proof: (a) Note that the case when Σ is empty has already been established. , one requires that the isotopy f satisfies f (x, t) = x ∀x ∈ ∂M, t ∈ [0, 1]. (b) Since ξφ = ⊗f ∈F ace(δ) ξ˜f , where we write ξf = φ2 (f ), and since the tensor product is ‘multilinear in its factors’, it is enough to verify the following: if φ, ψ ∈ S(Σ, δ) satisfy φ1 = ψ 1 , φ2 (f ) = ψ 2 (f ) ∀f = f0 , φ2 (f0 ) = ξ, ψ 2 (f0 ) = η and if we define χ ∈ S(δ, ≤) by χ1 = φ1 , χ2 (f ) = φ2 (f ) ∀f = f0 and χ2 (f ) = ξ + αη, where α ∈ C, then < (M, δ, ≤); χ >=< (M, δ, ≤); φ > +¯ α < (M, δ, ≤); ψ > .