On the Structure of Isometries between Noncommutative L Spaces

We prove some structure results for isometries between noncommutative L spaces associated to von Neumann algebras. We find that an isometry T : L(M1) → L(M2) (1 ≤ p < ∞, p 6= 2) can be canonically expressed in a certain simple form whenever M1 has variants of Watanabe’s extension property [W2]. Conversely, this form always defines an isometry provided that M1 is “approximately semifinite” (defined below). Although neither of these properties is fully understood, we show that they are enjoyed by all semifinite algebras and hyperfinite algebras (with no summand of type I2), plus others. Thus the classification is stronger than Yeadon’s theorem [Y1] for semifinite algebras (and its recent improvement in [JRS]), and the proof uses independent techniques. Related to this, we examine the modular theory for positive projections from a von Neumann algebra onto a Jordan image of another von Neumann algebra, and use such projections to construct new L isometries by interpolation. Some complementary results and questions are also presented.