An Ando–Choi–Effros lifting theorem respecting subspaces

A ZPP Lifting Theorem

The complexity class ZPP (corresponding to zero-error randomized algorithms with access to one NP oracle query) is known to have a number of curious properties. We further explore this class in the settings of time complexity, query complexity, and communication complexity. r For starters, we provide a new characterization: ZPP equals the restriction of BPP where the algorithm is only allowed t...

Notes on the Lifting Theorem

We have seen that the proof of existence of inverses for elements of Ext(X) can be based on a lifting theorem for (completely) positive maps of C(X) into a quotient C∗-algebra of the form E/K, where E ⊆ B(H) is a C∗-algebra containing the compact operators K. That argument works equally well for arbitrary C∗-algebras in place of C(X) whenever a completely positive lifting exists. Thus we are le...

A Bollobás-type theorem for affine subspaces

ROSENTHAL’S THEOREM FOR SUBSPACES OF NONCOMMUTATIVE Lp

We show that a reflexive subspace of the predual of a von Neumann algebra embeds into a noncommutative Lp space for some p > 1. This is a noncommutative version of Rosenthal’s result for commutative Lp spaces. Similarly for 1 ≤ q < 2, an infinite dimensional subspace X of a noncommutative Lq space either contains lq or embeds in Lp for some q < p < 2. The novelty in the noncommutative setting i...

Lifting map automorphisms and MacBeath's theorem

The purpose of the present note is essentially twofold . First we wish to indicate how coverings of maps can be obtained easily and in a direct way by using a cohomological construction. Secondly, we wish to apply this construction to obtain in an explicit way infinitely many examples of finite maps which are extremal in the sense of having 84(g− 1) automorphisms, where g is the genus of the ma...