Noncommutative Lp structure encodes exactly Jordan structure

We prove some structure results for isometries between noncommutative Lp spaces associated to von Neumann algebras. We find that an isometry T: Lp(M_1) to Lp(M_2) (1 le p < infty, p not 2) can be canonically expressed in a certain simple form whenever M_1 has variants of Watanabe's extension property. Conversely, this form always defines an isometry provided that M_1 is "approximately semifinite" (defined below). Although neither of these properties is fully understood, we sh...

We prove some noncommutative analogues of a theorem by Plotkin and Rudin about isometries between subspaces of Lp-spaces. Let 0<p<\infty, p not an even integer. The main result of this paper states that in the category of unital subspaces of noncommutative probability Lp-spaces, under some boundedness condition, the unital completely isometric maps come from *-isomorphisms of the underlying von Neumann algebras. Some applications are given, including to non commutative H^p ...

Surjective isometries between unital C*-algebras were classified in 1951 by Kadison. In 1972 Paterson and Sinclair handled the nonunital case by assuming Kadison's theorem and supplying some supplementary lemmas. Here we combine an observation of Paterson and Sinclair with variations on the methods of Yeadon and the author, producing a fundamentally new proof of the structure of surjective isometries between (nonunital) C*-algebras. In the final section we indicate how our te...

We characterize completely bounded normal Jordan $*$-homomorphisms acting on von Neumann algebras. We also characterize completely positive isometries acting on noncommutative $\mathrm{L}^p$-spaces.

In this paper we extend previous results of Banach, Lamperti and Yeadon on isometries of Lp-spaces to the non-tracial case first introduced by Haagerup. Specifically, we use operator space techniques and an extrapolation argument to prove that every 2-isometry T : Lp(M) to Lp(N) between arbitrary noncommutative Lp-spaces can always be written in the form T(phi^{1/p}) = w (phi circ pi^{-1} circ E)^{1/p}, for phi in M_*^+. Here pi is a normal *-isomorphism from M onto the von N...

Let $1\leq p \leq +\infty$. We show that the positive part of the closed unit ball of a non-commmutative $L^p$-space, as a metric space, is a complete Jordan $^*$-invariant for the underlying von Neumann algebra.

We prove that unital surjective spectral isometries on certain non-simple unital C*-algebras are Jordan isomorphisms. Along the way, we establish several general facts in the setting of semisimple Banach algebras.

Let N and M be von Neumann algebras. It is proved that L^p(N) does not Banach embed in L^p(M) for N infinite, M finite, 1 < or = p < 2. The following considerably stronger result is obtained (which implies this, since the Schatten p-class C_p embeds in L^p(N) for N infinite). Theorem: Let 1 < or = p < 2 and let X be a Banach space with a spanning set (x_{ij}) so that for some C < or = 1: (i) any row or column is C-equivalent to the usual ell^2-basis; (ii) (x_{i_k,j_k}) is C-e...

We investigate surjective isometries between projection lattices of two von Neumann algebras. We show that such a mapping is characterized by means of Jordan $^*$-isomorphisms. In particular, we prove that two von Neumann algebras without type I$_1$ direct summands are Jordan $^*$-isomorphic if and only if their projection lattices are isometric. Our theorem extends the recent result for type I factors by G.P. Geh\'er and P. \v{S}emrl, which is a generalization of Wigner's th...

We prove that the multiplication map L^a(M)\otimes_M L^b(M)\to L^{a+b}(M) is an isometric isomorphism of (quasi)Banach M-M-bimodules. Here L^a(M)=L_{1/a}(M) is the noncommutative L_p-space of an arbitrary von Neumann algebra M and \otimes_M denotes the algebraic tensor product over M equipped with the (quasi)projective tensor norm, but without any kind of completion. Similarly, the left multiplication map L^a(M)\to Hom_M(L^b(M),L^{a+b}(M)) is an isometric isomorphism of (quas...