A converse to the Grace–Walsh–Szegő theorem

A Converse to Dye’s Theorem

Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of F2 on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel equivalence relation which is not univ...

Ribet’s Converse to Herbrand’s Theorem

On the Converse to a Theorem of Atiyah and Bott

Throughout this paper, C denotes a smooth projective curve of genus at least one, G denotes a reductive linear algebraic group over C, and ξ0 is a C ∞ principal G-bundle over C. The space of all (0, 1)-connections on ξ0 is an affine space A = A(ξ0) associated to the infinite dimensional complex vector space H0,1(C; ad ξ0). Following Shatz [6] for the case G = GL(n), Atiyah and Bott [1] defined ...

Weil Converse Theorem

Hecke generalized this equivalence, showing that an integral form has an associated Dirichlet series which can be analytically continued to C and satisfies a functional equation. Conversely, Weil showed that, if a Dirichlet series satisfies certain functional equations, then it must be associated to some integral form. Our goal in this paper is to describe this work. In the first three sections...

A Converse to a Theorem of Adamyan, Arov and Krein

A classical theorem of Pick gives a criterion for interpolation by analytic functions in the open unit disc D subject to an H∞-norm bound. This result has substantial generalizations in two different directions. On the one hand, one can replace H∞ by the multiplier algebras of certain Hilbert function spaces, some of them having no connection with analyticity [Ag1, Ag2]. On the other hand, one ...