We consider the Aluthge transform T̃ = |T |1/2U |T |1/2 of a Hilbert space operator T , where T = U |T | is the polar decomposition of T . We prove that the map T 7→ T̃ is continuous with respect to the norm topology and with respect to the ∗–SOT topology on bounded sets. We consider the special case in a tracial von Neumann algebra when U implements an automorphism of the von Neumann algebra gen...

Let Cm×n, Cm×n r , C m ≥ , C m > , and In denote the set of m × n complex matrices, subset of Cm×n consisting of matrices with rank r, set of the Hermitian nonnegative definite matrices of order m, subset of C ≥ consisting of positive-definite matrices and n × n unit matrix, respectively. Without specification, we always assume that m > n >max{r, s} and the given weight matrices M ∈ C > ,N ∈ C ...

The non self-adjointness of the radial momentum operator has been noted before by several authors, but the various proofs are incorrect. We give a rigorous proof that the n-dimensional radial momentum operator is not self-adjoint and has no selfadjoint extensions. The main idea of the proof is to show that this operator is unitarily equivalent to the momentum operator on L[(0,∞), dr] which is n...

A fundamental result of von Neumann’s identifies unitarily invariant matrix norms as symmetric gauge functions of the singular values. Identifying the subdifferential of such a norm is important in matrix approximation algorithms, and in studying the geometry of the corresponding unit ball. We show how to reduce many convex-analytic questions of this kind to questions about the underlying gauge...

In 2005, Shen introduced a new invariant, G(N), of a di use von Neumann algebra N with a xed faithful trace, and he used this invariant to give a uni ed approach to showing that large classes of II1 factors M are singly generated. This paper focuses on properties of this invariant. We relate G(M) to the number of self-adjoint generators of a II1 factor M : if G(M) < n/2, then M is generated by ...