In this paper, we first give a description of a surjective unit-preserving real-linear uniform isometry $ T : A longrightarrow B$,  where $ A $ and $ B $ are complex function spaces on compact Hausdorff spaces $ X $ and $ Y $, respectively, whenever ${rm ER}left (A, Xright ) = {rm Ch}left (A, Xright )$ and ${rm ER}left (B, Yright ) = {rm Ch}left (B, Yright )$. Next, we give a description of $ T...

Let G ⊂ GL(V ) be a complex reductive group where dimV < ∞, and let π : V → V//G be the categorical quotient. Let N := ππ(0) be the null cone of V , and let H be the subgroup of GL(V ) which preserves the ideal I of N . We determine the identity component of H . For adjoint representations we determine H completely. We also investigate the subgroup GF of GL(V ) preserving a fiber F 6= N of π.

Let G ⊂ GL(V ) be a complex reductive group. Let G denote {φ ∈ GL(V ) | p◦φ = p for all p ∈ C[V ]}. We show that, in general, G = G. In case G is the adjoint group of a simple Lie algebra g, we show that G is an order 2 extension of G. We also calculate G for all representations of SL2.