Invertible preservers and algebraic groups

An Algebraic Property of an Isometry between the Groups of Invertible Elements in Banach Algebras

We show that if T is an isometry (as metric spaces) between the invertible groups of unital Banach algebras, then T is extended to a surjective real-linear isometry up to translation between the two Banach algebras. Furthermore if the underling algebras are closed unital standard operator algebras, (T (eA)) −1 T is extended to a surjective real algebra isomorphism; if T is a surjective isometry...

Algebraic Groups and Arithmetic Groups

Embedding Differential Algebraic Groups in Algebraic Groups

Quantifier Elimination We work in K a large rich differentially closed field. All other differential fields are assumed to be small subfields of K. Let L = {+,−, ·, δ, 0, 1} be the language of differential rings. We let L− = {+,−, ·, 0, 1}, the language of rings. If k is a differential field, we can view k either as an L-structure or an L−-structure. Theorem 1.1 (Quantifier Elimination) For any...

Algebraic Groups

APPROACH We sketch a more abstract version of the proof of the smoothness of CG.H/. LEMMA 16.24. Let G and H be algebraic groups over k. Let R be a k-algebra, let R0 D R=I with I 2 D 0, and let 0 denote base change R! R0. The obstruction to lifting a homomorphism u0WH0!G0 to R is a class in H .H0;Lie.G0/ ̋I ); if the class is zero, then the set of lifts modulo the action of Ker.G.R/!G.R0// by co...

Compact groups as algebraic groups

Aclassic result of [Tannaka:1939], in the formulation of [Chevalley:1946], asserts that every compact subgroup of GLn(C) is the group of real points on an algebraic group defined over R. Chevalley introduced a more algebraic approach to the topic, but his underlying argument is not so different from Tannaka’s. The literature since then has followed one of two threads. One is algebraic (as in Ch...