RENORMALIZATIONS OF MEASURABLE OPERATOR IDEAL SPACES AFFILIATED TO SEMI-FINITE VON NEUMANN ALGEBRA

This work is devoted to non-commutative analogues of classical methods of constructing functional spaces. Let a von Neumann algebra M of operators act in a Hilbert space H, tau be a faithful normal semi-finite trace M. Let (M) over tilde be an *-algebra of tau-measurable operators, vertical bar X vertical bar = root X*X for X is an element of (M) over tilde. A lineal epsilon in (M) over tilde is called ideal space on (M, tau) if 1) X is an element of epsilon implies X* is an element of epsilon; 2) X is an element of epsilon, Y is an element of (M) over tilde and vertical bar Y vertical bar <= vertical bar X vertical bar imply Y is an element of epsilon. Let epsilon, F be ideal spaces on (M, tau). We propose a method of constructing a mapping (p) over tilde: epsilon -> [0, +infinity] with nice properties by employing a mapping p on a positive cone epsilon(+). At that, if epsilon = M and rho = tau, then (rho) over tilde (X) = tau(vertical bar X vertical bar) and if the trace tau is finite, then (rho) over tilde (X) = parallel to X parallel to(1) for all X is an element of M. We study the case as (rho) over tilde (X) is equivalent to the original mapping rho(vertical bar X vertical bar). Employing mappings on epsilon and F, we construct a new mapping with nice properties on the sum epsilon + F. We provide examples of such mappings. The results are new also for *-algebra M = B(H) of all bounded linear operators in H equipped with a canonical trace tau = tr.