Let M(⋅) be a strongly
countably-additive (s.c.a.) (continuous linear) operator-valued measure on an
arbitrary σ-algebra ℬ of subsets of an arbitrary set Ω from a Hilbert space W to a
Hilbert space ℋ. Is there a Hilbert space 𝒦⊇ℋ and a s.c.a. quasi-isometric measure
M(⋅) (cf. Masani, BAMS 76 (1970), 427–528) on ℬ from W to 𝒦 such that
M(⋅) = P ∘M(⋅) where P is the projection on 𝒦 onto ℋ? In other words,
has such an M(⋅) a “quasi-isometric dilation M(⋅)”? We show that when W
or ℋ is finite-dimensional the answer is affirmative, and that when W is
finite-dimensional there is a unique (up to isomorphism) quasi-isometric dilation
M(⋅) of M(⋅) such that trace (M(Ω)∗M(Ω)) is a minimum. This generalizes results
of Miamee and Salehi, and Niemi. Our results depend on Grothendieck’s
inequality.
