Vol. 103, No. 1, 1982

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Quasi-isometric dilations of operator-valued measures and Grothendieck’s inequality

Milton Rosenberg

Vol. 103 (1982), No. 1, 135–161

Let M() be a strongly countably-additive (s.c.a.) (continuous linear) operator-valued measure on an arbitrary σ-algebra of subsets of an arbitrary set Ω from a Hilbert space W to a Hilbert space . Is there a Hilbert space 𝒦and a s.c.a. quasi-isometric measure M() (cf. Masani, BAMS 76 (1970), 427–528) on from W to 𝒦 such that M() = P M() where P is the projection on 𝒦 onto ? In other words, has such an M() a “quasi-isometric dilation M()”? We show that when W or 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.

Mathematical Subject Classification 2000
Primary: 46G10
Secondary: 28B05
Received: 9 March 1981
Revised: 1 July 1981
Published: 1 November 1982
Milton Rosenberg