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Characters on the semigroup $(C_{+}^{b}(\mathbb{R}^{d}),+)$, i.e. on bounded positive continuous functions with the ususal pointwise addition, are known to be of the form $C_{+}^{b}(\mathbb{R}^{d})\ni f\mapsto\exp(-\langle\mu,f\rangle)$ with $\mu\in\mathcal{M}^{f}(\mathbb{R}^{d})$, a finite measure.
Are there results already or is there some general way to determine the characters on any of the following semigroups over $\mathbb{R}^{d}$ (each again with pointwise addition, and identical involution)

  1. positive monotone increasing functions
  2. positive monotone increasing, continuous and/or bounded functions?

In other words, what are their dual semigroups? I'm happy about any suggestion, thanks.

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