It is shown that if P is a weak∗-continuous contractive projection on a JBW∗-triple M , then P(M) is of type I or semifinite, respectively, if M is of the corresponding type. We also show that P(M) has no infinite spin part if M is a type I von Neumann algebra.  2002 Elsevier Science (USA). All rights reserved.

Abstract Let $${{\mathcal {M}}}$$ M be a semifinite von Neumann algebra with faithful normal trace $$\tau $$ ? and let {A}}}$$ A an arbitrary subalgebra of . We characterize the class symmetric ideals {E}}}$$ E in such that derivations $$\delta :{{\mathcal {A...

Callias-type (or Dirac-Schrödinger) operators associated to abstract semifinite spectral triples are introduced and their indices computed in terms of an index pairing derived from the triple. The result is then interpreted as theorem for a non-commutative analogue flow. Both even odd considered, both commutative examples given.

A semifinite spectral triple for an algebra canonically associated to canonical quantum gravity is constructed. The algebra is generated by based loops in a triangulation and its barycentric subdivisions. The underlying space can be seen as a gauge fixing of the unconstrained state space of Loop Quantum Gravity. This paper is the second of two papers on the subject. email: johannes.aastrup@uni-...

Example 1 (Spectrum of Multiplication Operators) Let • (X, M, µ) be a σ–finite (actually semifinite will do) measure space, • 1 ≤ p ≤ ∞ and • a : X → C be a bounded measurable function on X. (Aϕ)(x) = a(x)ϕ(x) Point spectrum: Let λ ∈ C. Then λ1l − A is injective ⇐⇒ ϕ ∈ L p (X, M, µ), (λ − a(x))ϕ(x) = 0 a.e. =⇒ ϕ(x) = 0 a.e. ⇐⇒ λ − a(x) = 0 a.e.