Annihilator Banach ∗-algebras and the Peter-Weyl theorem

The Peter-Weyl Theorem and Generalizations

A Local Peter-weyl Theorem

An Ad K invariant inner product on the Lie algebra of a compact connected Lie group K extends to a Hermitian inner product on the Lie algebra of the complexified Lie group Kc. The Laplace-Beltrami operator, ∆, on Kc induced by the Hermitian inner product determines, for each number a > 0, a Green’s function ra by means of the identity (a2−∆/4)−1 = ra∗. The Hilbert space of holomorphic functions...

THE PETER–WEYL THEOREM FOR CLASSICAL AND QUANTUM slN

as (G,G)-bimodules, and where the sum is over all irreducible representations of G. Let H be a Hopf algebra. If H is finite dimensional, then H∗ is also a Hopf algebra by dualizing all operations from H. We run into issues if H is infinite-dimensional, but we can find a fix. For a finite-dimensional H-module V , an element v ∈ V , and f ∈ V ∗, define a linear functional on H by cf,v(u) = f(uv)....

A constructive proof of the Peter-Weyl theorem

We present a new and constructive proof of the Peter-Weyl theorem on the representations of compact groups. We use the Gelfand representation theorem for commutative C*-algebras to give a proof which may be seen as a direct generalization of Burnside’s algorithm [3]. This algorithm computes the characters of a finite group. We use this proof as a basis for a constructive proof in the style of B...

Zel’manov’s Theorem for Primitive Jordan–banach Algebras

In fact, if X is any vector space on which the primitive Banach algebra A acts faithfully and irreducibly, then X can be converted in a Banach space in such a way that the requirements in Theorem 0 are satisfied and even the inclusion A9BL(X ) is contractive. Roughly speaking, the aim of this paper is to prove the appropriate Jordan variant of Theorem 0. The notion of primitiveness for Jordan a...