Countable groups of isometries on Banach spaces
Countable groups of isometries on Banach spaces
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Banach 空间上的可数等距群
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发表时间：
2007
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通讯作者：
E. Galego
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作者：
V. Ferenczi;E. Galego
如果G同构于某等价范数中X上的等距群，则群G在巴拿赫空间X中是可表示的。在几种一般情况下证明了可分实Banach空间X中可数群G是可表示的，包括当G≃{-1,1}× H， H有限且暗淡X > |H|，或当G含有两个元素的正规子群且X的形式为co(Y)或l p (Y), 1≤p < +∞。这是由S. Bellenot（1986）的方法所启发的结果的结果，该结果表明，在相当一般的条件下，在可分离实Banach空间X和X上包含- Id的同构的可数有界群G上，我们还扩展了K. Jarosz（1988）的方法，证明了任何至少2维的复巴拿赫空间都可以用等价的复范数改造成只允许平凡实等距，并且任何巴拿赫空间的复化都可以用等价的复范数改造成只允许平凡实等距和共轭实等距。由此可见，每一个至少为4维且具有复杂结构的实巴拿赫空间都可以被改造为允许两个复杂结构达到等距，每一个实笛卡尔方形都可以被改造为允许一个唯一的复杂结构达到等距。
A group G is representable in a Banach space X if G is isomorphic to the group of isometries on X in some equivalent norm. We prove that a countable group G is representable in a separable real Banach space X in several general cases, including when G ≃ {-1,1} × H, H finite and dim X > |H|, or when G contains a normal subgroup with two elements and X is of the form co(Y) or l p (Y), 1 ≤ p < +∞. This is a consequence of a result inspired by methods of S. Bellenot (1986) and stating that under rather general conditions on a separable real Banach space X and a countable bounded group G of isomorphisms on X containing - Id, there exists an equivalent norm on X for which G is equal to the group of isometries on X. We also extend methods of K. Jarosz (1988) to prove that any complex Banach space of dimension at least 2 may be renormed with an equivalent complex norm to admit only trivial real isometries, and that any complexification of a Banach space may be renormed with an equivalent complex norm to admit only trivial and conjugation real isometries. It follows that every real Banach space of dimension at least 4 and with a complex structure may be renormed to admit exactly two complex structures up to isometry, and that every real Cartesian square may be renormed to admit a unique complex structure up to isometry.