The radial (or Laplacian) masa in a free group factor is the abelian von Neumann algebra generated by the sum of the generators (of the free group) and their inverses. The main result of this paper is that the radial masa is a maximal injective von Neumann subalgebra of a free group factor. We also investigate the tensor products of maximal injective algebras. Given two inclusions B i ⊂ Mi of type I von Neumann algebras in finite von Neumann algebras such that each B i is maximal injective in Mi, we show that the tensor product B1⊗B2 is maximal injective in M 1⊗M2 provided at least one of the inclusions satisfies the asymptotic orthogonality property we establish for the radial masa. In particular, it follows that finite tensor products of generator and radial masas will be maximal injective in the corresponding tensor product of free group factors.
We apply the techniques developed by Marcus, Spielman, and Srivastava, working with principal submatrices in place of rank-$1$ decompositions to give an alternate proof of their results on restricted invertibility. This approach recovers results of theirs’ concerning the existence of well-conditioned column submatrices all the way up to the so-called modified stable rank. All constructions are algorithmic. The main novelty of this approach is that it leads to a new quantitative version of the classical Gauss–Lucas theorem on the critical points of complex polynomials. We show that for any degree $n$ polynomial $p$ and any $c \geq 1/2$, the area of the convex hull of the roots of $p^{(\lfloor cn \rfloor )}$ is at most $4(c-c^2)$ that of the area of the convex hull of the roots of $p$.
We prove a variety of results describing the possible diagonals of tuples of commuting hermitian operators in type II 1 factors. These results are generalisations of the classical Schur-Horn theorem to the infinite dimensional, multivariable setting. Our description of these possible diagonals uses a natural generalisation of the classical notion of majorization to the multivariable setting. In the special case when both the given tuple and the desired diagonal have finite joint spectrum, our results are complete. When the tuples do not have finite joint spectrum, we are able to prove strong approximate results. Unlike the single variable case, the multivariable case presents several surprises and we point out obstructions to extending our complete description in the finite spectrum case to the general case. We also discuss the problem of characterizing diagonals of commuting tuples in B(H) and give approximate characterizations in this case as well.
Let M be a finite von Neumann algebra acting on a Hilbert space H and A be a transitive algebra containing M . In this paper we prove that if A is 2-fold transitive, then A is strongly dense in B(H). This implies that if a transitive algebra containing a standard finite von Neumann algebra (in the sense of [U. Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975) 271-283]) is 2-fold transitive, then A is strongly dense in B(H). Non-selfadjoint algebras related to free products of finite von Neumann algebras, e.g., LF n and (M 2 (C), 1 2 Tr) * (M 2 (C), 1 2 Tr), are studied. Brown measures of certain operators in (M 2 (C), 1 2 Tr) * (M 2 (C), 1 2 Tr) are explicitly computed.
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