ar X iv : 0 90 6 . 54 08 v 1 [ m at h . O A ] 3 0 Ju n 20 09 Murphy ’ s Positive definite kernels and Hilbert C ∗ – modules

The paper the title refers to is that in Proceedings of the Edinburgh Mathematical Society, 40 (1997), 367-374. Taking it as an excuse we intend to realize a twofold purpose: 1o to atomize that important result showing by the way connections which are out of favour, 2o to rectify a tiny piece of history. The objective 1o is going to be achieved by adopting means adequate to goals; it is of great gravity and this is just Mathematics. The other, 2o, comes from author’s internal need of showing how ethical values in Mathematics are getting depreciated. The latter have nothing to do with the previous issue; the coincidence is totally accidental. Reproducing kernel Hilbert C–modules Rudiments of the theory of Hilbert C–modules. Let A be a C–algebra with its norm denoted 1 by ‖ · ‖A. An inner product A–module is a right A–module Ξ (with scalar multiplication compatible with this in Ξ as well as that in A) with a mapping Ξ ×Ξ ∋ (ξ, η) 7−→ 〈ξ, η〉 ∈ A such that (a) it is C–linear in the second variable; (b) 〈ξ, ηa〉 = 〈ξ, η〉a, ξ, η ∈ Ξ, a ∈ A; (c) 〈η, ξ〉 = 〈ξ, η〉, ξ, η ∈ Ξ; (d) 〈ξ, ξ〉 > 0; if 〈ξ, ξ〉 = 0 then ξ = 0. If the second condition in (d) is dropped we call Ξ semi–inner product A–module. 1991 Mathematics Subject Classification. Primary 46L08; Secondary 46E22.