Closed and Σ-finite Measures on the Orthogonal Projections

Let A be a von Neumann algebra acting in a separable complex Hilbert space H and let A be the set of all orthogonal projections (=idempotents) from A. A subset M ⊆ A is said to be ideal of projections if: a) p ≤ q where p ∈ A, q ∈ M ⇒ p ∈ M; b) p, q ∈ M and ‖pq‖ < 1 ⇒ p ∨ q ∈ M; c) sup{p : p ∈ M} = I. Put Mp := {q : q ∈ M, q ≤ p}, ∀p ∈ A . Note that A is the ideal of projections, 0 ∈ Mp, ∀p, and the conditions 1), 2) are fulfilled on Mp. A function μ : M → [0,+∞] is said to be a measure if μ(e) = ∑ μ(ei) for any representation e = ∑ ei. Let μ1 : M1 → [0,+∞] and μ2 : M2 → [0,+∞] are measures. The measure μ2 is said to be the continuation of μ1 if M1 ⊂ M2 and μ1(p) = μ2(p), ∀p ∈ M1. A projection p ∈ A pr is called to be: projection of finite μ-measure if sup{q ∈ Mp} = p and sup{μ(q) : q ∈ Mp} < +∞; hereditary projection of finite μ-measure if q is the projection of finite μmeasure for any q ∈ A, q ≤ p. The measure μ is called to be: finite if μ(p) < ∞, ∀p; infinite if there exists p ∈ M such that μ(p) = +∞; fully finite if sup{μ(p) : p ∈ M} < +∞; closed if μ is finite and p ∈ M if p is the hereditary projection of finite μmeasure; σ-finite if M = A and there exists a sequence {pn} ⊂ A pr such that pn ր I and μ(pn) < +∞, ∀n. The following Proposition will be needed in Theorem 3. Proposition 1. Let A be a finite von Neumann algebra acting in the separable Hilbert space H and let M ⊆ A be the ideal of projections. Then