Strong Singularity of Singular Masas in Ii1 Factors

A singular masa A in a II1 factor N is defined by the property that any unitary w ∈ N for which A = wAw∗ must lie in A. A strongly singular masa A is one that satisfies the inequality ‖EA − EwAw∗‖∞,2 ≥ ‖w − EA(w)‖2 for all unitaries w ∈ N , where EA is the conditional expectation of N onto A, and ‖ · ‖∞,2 is defined for bounded maps φ : N → N by sup{‖φ(x)‖2 : x ∈ N, ‖x‖ ≤ 1}. Strong singularity easily implies singularity, and the main result of this paper shows the reverse implication. ∗Partially supported by a grant from the National Science Foundation.