A Note on Faithful Traces on a Von Neumann Algebra
نویسنده
چکیده
It is known that a semifinite von Neumann algebra always has a faithful semifinite trace. This trace can be used, for instance, to build up a non-commutative integration and, consequently, to define non commutative L-spaces. In this note we give some techniques for constructing, starting from a family F of semifinite traces, a faithful one which is closely related to the family F. Let F = {ηα; α ∈ I} be a family of normal, semifinite traces on M. We say that the family F is sufficient if for X ∈ M, X ≥ 0 and ηα(X) = 0 for every α ∈ I, then X = 0 (clearly, if F = {η}, then F is sufficient if, and only if, η is faithful). In this case, M is a semifinite von Neumann algebra [3, ch.5]. The analysis would really be simplified if, from a given family F of normal semifinite traces, one could extract a sufficient subfamily G of traces with mutually orthogonal supports. Apart from quite simple situations (for instance when F is finite), we do not know if this is possible or not. There are however at least two relevant cases where this can be done without many difficulties. The first case occurs when F is countable and the second when F is a convex and w∗-compact family of finite traces on M. These two situations will be discussed here. In the sequel we will need the following Lemmas.
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