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In mathematics, a uniformly bounded representation of a locally compact group on a Hilbert space is a homomorphism into the bounded invertible operators which is continuous for the strong operator topology. In 1947 Béla Szőkefalvi-Nagy established that any uniformly bounded representation of the integers or the real numbers is unitarizable, i.e. conjugate by an invertible operator to a unitary representation. For the integers this gives a criterion for an invertible operator to be similar to a unitary operator: the operator norms of all the positive and negative powers must be uniformly bounded. The result on unitarizability of uniformly bounded representations was extended in 1950 by Dixmier, Day and Nakamura-Takeda to all locally compact amenable groups, following essentially the method of proof of Sz-Nagy. The result is known to fail for non-amenable groups such as SL(2,R) and the free group on two generators. conjectured that a locally compact group is amenable if and only if every uniformly bounded representation is unitarizable. ==Statement== Let ''G'' be a locally compact amenable group and let ''T''''g'' be a homomorphism of ''G'' into ''GL''(''H''), the group of an invertible operators on a Hilbert space such that *for every ''x'' in ''H'' the vector-valued ''gx'' on ''G'' is continuous; *the operator norms of the operators ''T''''g'' are uniformly bounded. Then there is a positive invertible operator ''S'' on ''H'' such that ''S'' ''T''''g'' ''S''−1 is unitary for every ''g'' in ''G''. As a consequence, if ''T'' is an invertible operator with all its positive and negative powers unformly bounded in operator norm, then ''T'' is conjugate by a positive invertible operator to a unitary. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Uniformly bounded representation」の詳細全文を読む スポンサード リンク
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