The extension of positive definite operator-valued functions defined on a symmetric interval of an ordered group

Abstract

Abstract. Let G 1 be an ordered abelian group and a ∈ G 1 . Let G 2 be an abelian group and f an operator-valued positive definite function on (−a, a) × G 2 . We prove that f admits a positive definite extension to G 1 ×G 2 , generalizing in this way existing results for the case when G 1 = R and f is continuous.

Extending positive definiteness

“…For more discussion of the role played by symmetry we refer to Section 5.2. The results contained in [4,5] and [15] also corroborate the importance of symmetry, the latter concerns extensions to indefinite forms with finite number of negative squares.…”

“…. By the Radon-Nikodym theorem, there exists a system {h i,j } m i,j=1 of M-measurable complex functions on X S such that µ(σ)(e i , e j ) = σ h i,j dν for all σ ∈ M. Since [µ(σ)(e i , e j )] m i,j=1 0 for every σ ∈ M, one can show (see the proof of [39,Theorem 6.4]) that there exists Z ∈ M such that ν(X S \ Z) = 0 and [h k,l (χ)] m k,l=1 0 for 4 With the natural identification of bounded linear operators with sesquilinear forms, our definition subsumes the classical semispectral operator-valued measures (cf. [39]).…”

“…Indeed, take a nonzero system {c m,n } (m,n)∈T ⊂ C satisfying (iv) and suppose that, contrary to our claim, the system satisfies (iv) Z for all C[z, z]-determining sets Z ⊂ C. In particular, this is the case for Z 1 = {z ∈ C : |z| 1} and Z 2 = {z ∈ C : 2 |z| 3} (for their determining property see Lemma 5). By Lemma 25, {c m,m } ∞ m=0 is a Stieltjes moment sequence having two representing measures supported in [0, 1] and [4,9], respectively. Since each Hamburger moment sequence with a compactly supported representing measure is determinate (cf.…”

Extensions of Positive Definite Functions on Amenable Groups

“…In this paper, we extend the result to a class of subsets of amenable groups that satisfy a certain combinatorial condition. The result turns out to be more general than the main result in [1], and it is obtained by much simpler means. Our main result was also influenced by [5], where a version of Nehari's Problem was solved for operator functions on totally ordered amenable groups.…”

“…The above proposition has the following consequence that represents the main result of [1]. The proof derived here is much simpler.…”