设M是超有限Ⅱ1型因子．D是M的Cartan子代数,T是对角为D的M 的σ-弱闭的子代数(简称Cartan双模代数)并且生成M．设φ是T到T上的σ-弱连续满线性等距,则Φ可扩张成从M到M上的等距．设φ是T到T上的映射(没假设线性),满足任给a,b∈T,T上存在σ-弱连续满线性等距φa,b(与n,b有关),使得φa,b(a)=φ(a),φa,b(b)=φ(b),则φ是线性等距．

Let M be a hyperfinite factor of type Ⅱ1, D is a Cartan masa of M, T be a Cartan subalgebas of M with diagonal D which generates M. If Φ: T →T be an σ-weakly continuous (Banach) isometry, then Φ can be extended a isometry on M.If a map Φ: T→T satisfies that for every pair a, b ∈ T, there is a oooooooooo-weakly continuous isometry Φa,b on T such that Φa,b(a) = Φ(a), Φ>a,b(b) = Φ(b), then Φ is a linear isometry.