On a property ofL p spaces on semifinite von Neumann algebras

Abstract

ABSTRACT. A characterization of the traces in a broad class of weights on yon Neumann algebras is obtained. A new property of the "domain ideals ~ of these traces is proved. In the semifinite case, a relation for a faithful normal trace is established. This result is new even for the algebra of all bounded operators on a Hilbert space. Applications of the main result to the structure theory of yon Neumann algebras and to the K5the duality theory for ideal spares of Segal measurable operators are given.KEY woRDS: von Neumann algebra, affiliated operator, trace, weight, Jordan algebra, logic of orthoprojections, measure on projections, noncommutative integration, KSthe duality.w Characterization of traces on yon Neumann algebras Let M be a yon Neumann algebra acting on a Hilbert space H. We denote by M erm , M uni , M + , and M'P r~ its Hermitian, unitary, and positive part, and the logic of orthoprojections, respectively. Next, let e be the identity element of M and II 9 II the C'-no~m on M.

General Mazur–Ulam Type Theorems and Some Applications

“…We assert that this implies that l is tracial, i.e.,l (X Y ) = l (Y X ), X , Y ∈ A . To verify this, we apply an idea from the proof of Lemma 1 in [4]. Namely, select an arbitrary pair P,Q of projections in A , define S = I − 2P and compute…”

Commutativity of projections and characterization of traces on Von Neumann algebras

“…In [15] (also see [16]) the author obtained the affirmative answer and constructed an example of p, q ∈ B(H ) pr with pqp ∈ S 1 but pq / ∈ S 1 , and in [16] he showed that for all ideals J of B(H ) distinct from the ideals of finite-dimensional and compact operators (for separable H with dim H = ∞) there exist p, q ∈ B(H ) pr with pqp ∈ J but pq / ∈ J. Thus, the operators pq and pqp are not always similar, and a cyclic permutation of projections in products under a trace in the general case of dim H = ∞ is inadmissible (also see [4]).…”

Chern Numbers, Localisation and the Bulk-edge Correspondence for Continuous Models of Topological Phases

“…Note that |a|(1 + D 2 ) −s/2 ∈ L 1 (N , τ ) by the polar decomposition a = v|a|, which does not require |a| to be in A. For the definition of spectral dimension to have meaning, we require that τ (a(1 + D 2 ) −s/2 ) ≥ 0 for a ≥ 0, a fact that follows from [12,Theorem 3]. For a semifinite spectral triple (A, H, D) to be finitely summable with spectral dimension p, it is a necessary condition that A ⊂ B 1 (D, p) [ We see that to define δ k (T ), we require that T : H k → H k for H k = k l=0 Dom(D l ).…”