Positive definite functions of infinitely many variables in a layer

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

“…M. G. Krein proved [10] that every positive definite continuous scalar function on a real interval (−a, a) admits a continuous positive definite extension to R. A. P. Artjomenko [2] (see also Theorem 4.2.3 in [15]) provided a new proof for Krein's Extension Theorem without the continuity requirement. Y. M. Berezansky and I. M. Gali ( [6], see also Theorem 5.4.4.2 in [5]) proved the following extension of Krein's Theorem: "Given a Hilbert space H, and a positive definite function k on a layer in H that is J continuous at 0, then k can be extended to a positive definite function on H with the same property of continuity." By a similar proof it follows that every continuous positive definite function on (−a, a) × G, where G is a topological group, can be extended to a continuous positive definite function on…”