Let $R$ be the hyperfinite type $III\_1$ factor,
and let $Aut(R)$ be its group of automorphisms, equipped with the $u$-topology
(topology of pointwise convergence on the predual).
An automorphism $\alpha\in Aut(R)$ is called *inner* if it is of the form $\alpha(x)=uxu^\*$ for some unitary $u\in R$.


I've heard that inner automorhpisms are dense in $Aut(R)$.


Can someone give me an explicit example of a sequence of inner automorphisms that converges to an automorphism that is not inner?


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**PS:** An answer to the same question for the hyperfinite $II\_1$ factor would also be interesting.



Denseness of inner automorphisms inside automorphisms of hyperfinite type III_1 factor
It is relatively easy to give an explicit sequence of inner automorphisms that converges to the flip automorphism $\sigma$. First of all, realize $R$ as an infinite tensor product of matrix algebras $(R,\varphi)=\bigotimes\_n (M\_{k\_n}(\mathbb{C}),\varphi\_n)$. For every $n\in \mathbb{N}$, we find a unitary $u\_n\in M\_{k\_n}(\mathbb{C})\otimes M\_{k\_n}(\mathbb{C})$ that implements the flip automorphism on $M\_{k\_n}(\mathbb{C})\otimes M\_{k\_n}(\mathbb{C})$, i.e. $u\_n(x\otimes y)u\_n^\ast=y\otimes x$. (This follows from the general observation that every automorphism of a type I factor is inner, but it is a good exercise to find the $u\_n$ explicitly).


Now it follows that the sequence of inner automorphisms $\sigma\_n=Ad\_{u\_1\otimes\ldots\otimes u\_n\otimes 1\ldots}$ converges to the flip automorphism $\sigma$: Let $\psi$ be any ultraweakly continuous functional on $R\otimes R$. Consider $R\otimes R$ to be represented on the infinite tensor product space $H=\bigotimes\_n L^2(M\_{k\_n}(\mathbb{C})\otimes M\_{k\_n}(\mathbb{C}), \varphi\_n\otimes\varphi\_n)$. We know that $\psi$ is of the form $\psi(x)=\sum\_k\langle\xi\_k,x\eta\_k\rangle$ for some $\ell^2$-summable sequences $\xi\_k,\eta\_k$ in $H$. Since the finite sequences are dense in the $ell^2$-summable ones, we can assume that $\psi(x)=\langle\xi,x\eta\rangle$. Because the finite tensor products are dense in the infinite ones, we can assume that $\xi,\eta\in H\_N=\bigotimes\_{n=1}^N L^2(M\_{k\_n}(\mathbb{C})\otimes M\_{k\_n}(\mathbb{C}), \varphi\_n\otimes\varphi\_n)$ for some $N\in\mathbb{N}$. Now it follows that $\psi(\sigma\_n(x))=\langle \xi,\sigma\_n(x)\eta\rangle=\langle\sigma\_n^{-1}(\xi),x\sigma\_n^{-1}(\eta)\rangle=\langle\sigma^{-1}(\xi),x\sigma^{-1}(\eta)\rangle=\psi(\sigma(x))$.


