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Suppose, $ O_{\infty} $ is the cuntz algebra generated by the orthogonal isometries $ \{S_i\}_{i\in \mathbb{N}} $,i.e. $ S_i^*S_j=\delta_{ij}$ and $ O_{\infty}=C^*(\{S_i\}_{i\in \mathbb{N}}) $.

Then the span of $ \{S_iS_j\}_{i,j\in \mathbb{N}} $ forms a Hilbert space $ H $ with the inner product being $ \langle u,v\rangle 1=v^*u$.

Given a unitary operator $ U:H\to H $ so that $ U $ commutes with the flip operator $ \sigma: H\to H $ (defined by $ \sigma(S_iS_j)=S_jS_i \forall i,j$),i.e. $ U\sigma=\sigma U $, does there exist an automorphism $ \phi $ of $ O_{\infty} $ so that $ \phi(x)=U(x) \forall x\in H $.

Note that, if we consider the Hilbert space $ H' $ to be the span of $ \{S_i\}_{i\in \mathbb{N}} $ with inner product defined similar as before, then $ H=H'\otimes H' $ and if we take $ U $ to be $ V\otimes V $ where $ V $ is a unitary on $ H' $, then clearly the above statement is true for $ U $.

I am trying to see if its true when we take the strong closure of the span of elements of the form $ V\otimes V $($ V $ is in $ U(H') $) in $ B(H) $ which is basically the commutant of $ \sigma $.

It seemed to me that this is not always possible. Actually, I couldn't find $ \phi $ when $ U=\sigma $. So in that case, is there any nice characterization of the unitaries on $ H $ for which this is possible?

Also, can anyone give a reference to any paper where automorphisms of $ O_{\infty} $ has been studied? I only found papers on automorphisms of $ O_n $ but not of $ O_{\infty} $.

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