Pontryagin duality for Non-Abelian Locally Compact Groups

概要

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Let $ \chi $ be the set of all functions $ x \colon P_i(G) \rightarrow \mathbf{C} $ satisfying the inequality $ \mid \sum_{i=1}^{n}{{\alpha}_i x(p_i)} - x(p)x(q) \mid \leqq C(x) \sup__{g \in K(x)} \mid \sum_{i=1}^{n} {{\alpha}_i p_i(g)} - p(g)q(g) \mid $, for every $ n, {\alpha}_1, {\alpha}_2, \ldots , {\alpha}_n \in \mathbf{C} $ and $ p_1, p_2, \ldots , p_n, p, q \in P_e(G) $, where $ C(x) > 0 $ and $ K(x) $ is a compact subset of $ G $. Then a topological group structure can be defined on $ \chi $ so that $ \chi $ is isomorphic to $ G $. The inequality can be replaced by $ \mid \sum_{i=1}^{n}{{\alpha}_i x(p_i)} - x(p)x(q) \mid \leqq \sup__{g \in G} \mid f(g) (\sum_{i=1}^{n}{{\alpha}_i p_i(g)} - p(g)q(g)) \mid $, where $ f = f_x \in C_{\infty} (G) $.
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