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In mathematics, a locally compact topological group ''G'' has property (T) if the trivial representation is an isolated point in its unitary dual equipped with the Fell topology. Informally, this means that if ''G'' acts unitarily on a Hilbert space and has "almost invariant vectors", then it has a nonzero invariant vector. The formal definition, introduced by David Kazhdan (1967), gives this a precise, quantitative meaning. Although originally defined in terms of irreducible representations, property (T) can often be checked even when there is little or no explicit knowledge of the unitary dual. Property (T) has important applications to group representation theory, lattices in algebraic groups over local fields, ergodic theory, geometric group theory, expanders, operator algebras and the theory of networks. ==Definitions== Let ''G'' be a σ-compact, locally compact topological group and π : ''G'' → ''U''(''H'') a unitary representation of ''G'' on a (complex) Hilbert space ''H''. If ε > 0 and ''K'' is a compact subset of ''G'', then a unit vector ξ in ''H'' is called an (ε, ''K'')-invariant vector if : The following conditions on ''G'' are all equivalent to ''G'' having property (T) of Kazhdan, and any of them can be used as the definition of property (T). (1) The trivial representation is an isolated point of the unitary dual of ''G'' with Fell topology. (2) Any sequence of continuous positive definite functions on ''G'' converging to 1 uniformly on compact subsets, converges to 1 uniformly on ''G''. (3) Every unitary representation of ''G'' that has an (ε, ''K'')-invariant unit vector for any ε > 0 and any compact subset ''K'', has a non-zero invariant vector. (4) There exists an ε > 0 and a compact subset ''K'' of ''G'' such that every unitary representation of ''G'' that has an (ε, ''K'')-invariant unit vector, has a nonzero invariant vector. (5) Every continuous affine isometric action of ''G'' on a ''real'' Hilbert space has a fixed point (property (FH)). If ''H'' is a closed subgroup of ''G'', the pair (''G'',''H'') is said to have relative property (T) of Margulis if there exists an ε > 0 and a compact subset ''K'' of ''G'' such that whenever a unitary representation of ''G'' has an (ε, ''K'')-invariant unit vector, then it has a non-zero vector fixed by ''H''. 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「Kazhdan's property (T)」の詳細全文を読む スポンサード リンク
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