Mismatch between equivalent definitions of the Bohr compactification of the reals

Question

I feel I'm overlooking something very silly.

The Bohr compactification of $\mathbb R$ has two equivalent definitions.

1. The set of (possibly discontinuous) homomorphisms $\mathbb R \to \mathbb T$ under the pointwise topology.

2. The maximal ideal space of the set $AP(\mathbb R)$ of almost periodic functions. i.e the set of multiplicative linear functionals under the pointwise topology.

To represent an element of 1 as a functional we use the fact that $AP(G)$ is the closure of functions of the form

$$ \sum c_i e^{i \lambda_i x}$$

Given a homomorphism $\phi: \mathbb R \to \mathbb T$ we can form the corresponding functional by sending each $e^{i \lambda x} \mapsto \phi(\lambda)$. Since exponentials with different $\lambda_i$ are linearly independent, we can extend the above to all linear combinations by linear independence, i.e

$$ \sum c_i e^{i \lambda_i x} \mapsto \sum c_i \phi(\lambda_i).$$

The fact that the above is multiplicative follows from the exponential multiplication rules and how $\phi$ commutes with multiplication.

Finally we extend the functional to all of $AP(\mathbb R)$ by continuity.

What confuses me is that the above makes perfect sense if $\phi$ is instead a homomorphism $\mathbb R \to \mathbb C^ \times$ (complex numbers under multiplication). For simplicity let $\phi(\lambda) = e^\lambda$. Then we can define the functional

$$ \sum c_i e^{i \lambda_i x} \mapsto \sum c_i e^{\lambda_i}.$$

This functional does not correspond to a homomorphism $\phi: \mathbb R \to \mathbb T$.

What am I missing?

Answer 1

This will not be continuous because the sequence of almost periodic functions $e^{-n} e^{i n x}$ converges to zero as an almost periodic function but the value of the functional on each element of the sequence is $e^{-n} e^n=1$.