Spectral analysis for automorphisms of UHF \(C^*\)-algebras
Let \(M_ n\) be the \(n\times n\) complex matrices with normalized trace \(\tau\). Let \(u\in M_ n\) be a unitary operator and let \(u=\sum \exp (it_ k)e_ k\) be the spectral decomposition of u. Let \(\nu_ u\) be the measure defined on the torus \({\mathbb{T}}\) by \(\nu_ u=\sum \tau (e_ k)\delta_{t_ k}\) where \(\delta_ t\) denotes the Dirac measure. The authors show that, given two unitaries u and v in \(M_ n\), there is an automorphism \(\theta\) of \(M_ n\) such that \(\| u-\theta (v)\| \leq \epsilon\) if and only if \(\nu_ u(I)\leq \nu_ v(I_{\epsilon})\) for every open interval of \({\mathbb{T}}\). Here \(I_{\epsilon}=\{z\in {\mathbb{T}}\| z-z'| <\epsilon\) for some z'\(\in I\}.\) The authors use this result to study automorphisms of UHF algebras. Let \(\{n(k)\}\) be an unbounded sequence of positive integers and let A be the UHF algebra \(A=\otimes^{\infty}_{1}M_{n(k)}\). Then the cyclic product automorphism \(\alpha_ 0\) defined by \(\alpha_ 0=\otimes^{\infty}_{1}ad s(n(k))\) is outer conjugate to a product automorphism \(\beta =\otimes^{\infty}_{1}ad u_ k\) if and only if \(\sum_{k}(1-| {\mathcal F}\nu_{u(k)}(m)|)=+\infty\) for all nonzero integers m. Here \(s(n(k))\) is the unitary giving the left regular representation of \({\mathbb{Z}}_{n(k)}\) and \({\mathcal F}\) denotes the Fourier transform. As a corollary, the authors show that \(\beta \otimes \alpha_ 0\) is outer conjugate to \(\alpha_ 0\) on \(A\otimes A\) if \(\beta\) is a product action \(\beta =\otimes ad v(k).\) The authors also show that an automorphism of the universal UHF algebra is centrally stable if and only if it satisfies a Rohlin property.
scientific article; zbMATH DE number 4019789