Bost, Jean Benoît; Connes, Alain Euler products and type III factors. (Produits eulériens et facteurs de type III.) (French. English summary) Zbl 0781.46045 C. R. Acad. Sci., Paris, Sér. I 315, No. 3, 279-284 (1992). There is a standard way to construct a partition function from a Hecke \(C^*\)-algebra with a one-parameter group of automorphisms. The authors make such an explicit construction and get this way the Riemann \(\zeta\)- function. Phase transition with spontaneous symmetry breaking for this dynamical system turns out to correspond to a certain action of the Galois group \(\overline {\mathbb{Q}}_{ab}/\mathbb{Q}\).If \(T\) is a contractive operator in a Hilbert space, then for the Bose- quantization \(\Gamma_ s(T)\) we have \[ \text{Trace } \Gamma_ s(T)=\text{det}(I-T)^{-1}. \tag{*} \] It follows that \(T^{-s}\), \(\text{re}(s)>1\), yields this way the Riemann \(\zeta\)-function by applying Euler product to (*). They consider diagonal operators on \(\ell^ 2(\mathbb{N}^*)\) defined from a fixed orthonormal basis \(\{\varepsilon_ n\}^ \infty_{n=0}\) by \(H_ \Gamma \varepsilon_ n=(\log n) \varepsilon_ n\), and \(\mu_ n \varepsilon_ k= \varepsilon_{kn}\), \(k\in\mathbb{N}^*\). For each prime \(p\) let \({\mathcal T}_ p\) be the Toeplitz algebra of \(\mu_ p\), and set \(C^*(\mathbb{N}^*):=\oplus {\mathcal T}_ p\) where the tensor product is over all primes \(p\) in some prescribed set \({\mathcal P}\) of primes. Then the one- parameter group \(\{\sigma_ t\}_{t\in\mathbb{R}}\) is defined by \[ \sigma_ t(x):= e^{itH_ \Gamma} xe^{-itH_ \Gamma}. \] They show that for \(\beta>0\), the \((\beta,\sigma_ t)\)-KMS state \(\varphi_ \beta\) is of type \(I_ \infty\) and given by \(\varphi_ \beta(x)= \zeta(\beta)^{-1} \text{ Trace}(e^{-\beta H_ \Gamma} x)\) for \(x\in C^*(\mathbb{N}^*)\), while for \(\beta=1\), \(\varphi_ \beta\) is a type \(\text{III}_ 1\)- factor state given by \(\varphi_ 1(x)=\text{Trace}_ \omega (e^{-H_ \Gamma} x)\), \(\forall x\in C^*(\mathbb{N}^*)\) where \(\text{Trace}_ \omega(\cdot)\) denotes the Dixmier trace, i.e., \[ \underset {N\to\infty} {\text{Lim}_ \omega} \left( {1 \over {\log N}} \sum^ N_{n=0} \nu_ n (e^{-H_ \Gamma})x\right) \] where \(\nu_ n\) denotes the proper value (counting index) of the operator. For \(0<\beta\leq 1\), they show that \(\varphi_ \beta\) is a factor state of type \(\text{III}_ 1\) and the factor is Araki-Woods \(R_ \infty\). They use that the Dixmier trace is also the residue at \(s=1\) of the complex function, \(s\mapsto \text{Trace}(e^{-sH} x)\). If \(A_ f\) denotes the finite adèles on \(\mathbb{Q}\), they give a similar construction for \(C^*(P_{A_ f})\) where \(P_{A_ f}\) denotes the group \({{1\;n} \choose {0\;h}}\) defined requiring \(h\) invertible. Then \(\varphi_ \beta\) will be a dual weight for a measure \(\mu_ \beta\) on \(A^*_ f\) given by \[ \mu_ \beta(f)= \zeta(\beta)^{-1} \int_{A^*_ f} | t|^ \beta f(t)d^*t \] where \(d^*t\) denotes Haar measure on \(A^*_ f\). Moreover, \(\varphi_ \beta\) is factorial of type \(I_ \infty\). The \(\beta\)-KMS states for \(C^*(\mathbb{Q}/\mathbb{Z})\) are also found, along with a character formula, \(\chi: \mathbb{Q}/\mathbb{Z}\to \mathbb{C}^*\), (\(e_ \gamma\) = projection), \[ \varphi_{\beta,\chi} (e_ \gamma)= \zeta(\beta)^{-1} \sum^ \infty_{n=1} n^{-\beta} \chi(\gamma)^ n. \] Reviewer: P.E.T.Jorgensen (Iowa City) Cited in 3 ReviewsCited in 3 Documents MSC: 46L10 General theory of von Neumann algebras 46L35 Classifications of \(C^*\)-algebras 11M06 \(\zeta (s)\) and \(L(s, \chi)\) Keywords:phase transition with spontaneous symmetry breaking; partition function; Hecke \(C^*\)-algebra with a one-parameter group of automorphisms; Riemann \(\zeta\)-function; Galois group; Bose-quantization; diagonal operators; Toeplitz algebra; tensor product; one-parameter group; Dixmier trace; factor state; finite adèles; Haar measure PDFBibTeX XMLCite \textit{J. B. Bost} and \textit{A. Connes}, C. R. Acad. Sci., Paris, Sér. I 315, No. 3, 279--284 (1992; Zbl 0781.46045)