Palle Jorgensen, Feng Tian (2014), arXiv preprint (a 442-page lecture-notes monograph, v8 2015-08-23). A book by the same authors titled “Non-commutative Analysis” was published by World Scientific (Crossref: DOI 10.1142/10317, dated 2016-09-27); that it is this text is likely but unverified. — ARXIV-1408.1164
Key takeaways
From the reading, NOTE-829, which covers the cited passages only.
- The kernel construction, named GNS. Remark 1.34 builds a Hilbert space from a positive definite ϕ on a set: form Σc̄_xd_yϕ(x,y) on finite sums, quotient the null vectors, complete. It calls this “the GNS construction”. Chapter 4’s GNS applies the same recipe to ϕ(a*b) on a *-algebra. The book does not state a reduction of one to the other.
- Uniqueness, stated not proved. Corollary 1.35: ϕ is positive definite iff ϕ(x,y) = ⟨Φ(x), Φ(y)⟩ for some Φ: X → H, and two minimal realizations are related by a unitary. The proof is “left as an exercise to the student”. Equation (1.24) is printed “UΦ₁(x) = Φ₂(x)U”, a typo for UΦ₁(x) = Φ₂(x).
- Uniqueness, proved, for GNS. Theorem 4.35: for a state, the cyclic representation is unique up to unitary equivalence, by the isometry π₁(A)u₁ ↦ π₂(A)u₂ extended by density. The same argument proves Corollary 1.35’s uniqueness with Φ_i(x) in place of π_i(A)u_i.
- Commutative GNS is L²(μ). Example 4.28: in the abelian case states are Borel probability measures. Example 4.86: the GNS space of f ↦ ∫f dμ on C(X) is L²(μ), with multiplication operators and cyclic vector 1. Lemma 3.28 does the same inside the proof of the spectral theorem. Both are for compact X.
- RKHS by Riesz. Def. 11.1: bounded evaluation gives K_s “by Riesz”; Claim 11.2, proved, K_s = E_s*(1).
Standing in the record
Filed on 2026-09-26 while pursuing, at the owner’s request, the connection GNS, unitary equivalence and the convergence of representations (see the curation entry of that date). The cited passages were read closely on 2026-10-14 in NOTE-829, a Skimmed note: the rest of the book was not read.
It is a source of THEORY-004, whose promote_when names this close reading as the outstanding half. The reading is done. It finds that Corollary 1.35 is stated without proof, so the book cannot be said to prove it, as THEORY-004 now says. The uniqueness follows by the book’s own argument for Theorem 4.35. It is also cited by THEORY-005 for L²(p) as the commutative GNS space; Example 4.86 is the explicit statement. No theory was filed from the reading.
The open check of the seed, an explicit L²(μ) statement of the commutative GNS space, is answered by Example 4.86 and Lemma 3.28. Theorem 11.10 (Kolmogorov) and Chapter 5 (Stinespring) were looked at only in their statements.