Extensions of positive definite functions on amenable groups

Let $S$ be a subset of a amenable group $G$ such that $e\in S$ and $S^{-1}=S$. The main result of the paper states that if the Cayley graph of $G$ with respect to $S$ has a certain combinatorial property, then every positive definite operator-valued function on $S$ can be extended to a positive definite function on $G$. Several known extension results are obtained as a corollary. New applications are also presented

Redheffer Products and Characteristic Functions

AbstractFuhrmann [Israel J. Math.16 (1973), 162-176], and subsequently Ball and Lubin [Pacific J. Math.63 (1976), 309-324] have studied a class of perturbations of completely nonunitary contractions. We extend their results concerning the computation of the characteristic function by using the "Redheffer product" machinery [J. Math. Phys.39 (1960), 269-286]. This has been familiar to system theory experts for many years and has been recently revived by Foias and Frazho ["The Commutant Lifting Approach to Interpolation Problems," Birkhäuser, Basel, 1990] to obtain alternate proofs in the theory of intertwining dilations of contractions on a Hilbert space. The proof obtained is conceptually surprisingly simple. An application is the recapture, from a point of view different from the original one, of a result concerning de Branges′ spaces, which have received renewed attention in recent years

Two remarks about nilpotent operators of order two

We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator.Comment: 7 pages. To appear in Proceedings of the AM

Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor

It is known that the eigenvalues of selfadjoint elements a,b,c with a+b+c=0 in the factor R^omega (ultrapower of the hyperfinite II1 factor) are characterized by a system of inequalities analogous to the classical Horn inequalities of linear algebra. We prove that these inequalities are in fact true for elements of an arbitrary finite factor. A matricial (`complete') form of this result is equivalent to an embedding question formulated by Connes.Comment: 41 pages, many figure