Semigroup crossed products and Hecke algebras arising from number fields
Let \(K\) be a finite extension of the field \(\mathbb{Q}\) of rational numbers and let \({\mathcal O}\) denote the ring of integers in \(K\). Use \({\mathcal O}^\times\) for the multiplicative semigroup of nonzero integers in \(K\). So, any number in \(K\) has the form \(a/b\) where \(a\in{\mathcal O}\) and \(b\in{\mathcal O}^\times\). Consider the discrete Abelian group \(K/{\mathcal O}\) and its group *-algebra \(\mathbb{C}(K/{\mathcal O})\) over the complex numbers. Write \(\delta_x\) for the image of a group element \(x\) in the group algebra. For any \(a\in{\mathcal O}^\times\) and any \(y\in K/{\mathcal O}\), one can look at the set of elements \(x\in K/{\mathcal O}\) such that \(y= ax\). This set is finite and denote by \(N\) the number of elements. Now define \(\alpha_a(\delta_y)= {1\over N} \sum\delta_x\) where the sum runs over all \(x\in K/{\mathcal O}\) satisfying \(y= ax\). It is shown in the paper that \(\alpha\) is an action of the semigroup \({\mathcal O}^\times\) by means of *-endomorphisms of the group *-algebra \(\mathbb{C}(K/{\mathcal O})\). The algebraic crossed product *-algebra, denoted by \(\mathbb{C}(K/{\mathcal O})\times_\alpha{\mathcal O}^\times\) can be characterized as the universal *-algebra generated by elements \(\{u(y)\mid y\in K/{\mathcal O}\}\) and \(\{v_a\mid a\in{\mathcal O}^\times\}\) satisfying (1) \(v^*_av_a= 1\) and \(v_av_b= v_{ab}\) when \(a,b\in{\mathcal O}^\times\), (2) \(u(0)= 1\), \(u(x)^*= u(-x)\) and \(u(x+ y)= u(x)u(y)\) when \(x,y\in K/{\mathcal O}\), (3) \(v_a u(y)v^*_a={1\over N} \sum u(x)\), where \(a\in{\mathcal O}^\times\) and \(y\in K/{\mathcal O}\) and where, as before, the sum is taken over those elements \(x\in K/{\mathcal O}\) satisfying \(y= ax\) and \(N\) is the number of such elements. Similarly, the \(C^*\)-crossed product \(C^*(K/{\mathcal O})\times_\alpha{\mathcal O}^\times\) is characterized as the universal \(C^*\)-algebra generated by these elements, satisfying these relations. A (very) special case is obtained when \(K\) is \(\mathbb{Q}\) itself and hence \({\mathcal O}\) is \(\mathbb{Z}\). This case was studied in an earlier paper by \textit{M. Laca} and \textit{I. Raeburn} [``A semigroup crossed product arising in number theory'', J. Lond. Math. Soc. (to appear)]. The present paper extends results from the previous one to this more general situation. One of the results is that the algebraic crossed product \(\mathbb{C}(K/{\mathcal O})\times_\alpha{\mathcal O}^\times\) is isomorphic with the generalized Hecke algebra as defined by \textit{J.-B. Bost} and \textit{A. Connes} [Sel. Math., New Ser. 1, 412-457 (1995; Zbl 0842.46040)] in the following way. Denote by \(\Gamma_K\) the group of matrices of the form \(\left(\begin{smallmatrix} 1 & y\\ 0 & x\end{smallmatrix}\right)\) where \(x,y\in K\) and \(x\neq 0\). Consider the subgroup \(\Gamma_{\mathcal O}\) given by elements of the form \(\left(\begin{smallmatrix} 1 & a\\ 0 & 1\end{smallmatrix}\right)\) where \(a\in{\mathcal O}\). Now the Hecke algebra \({\mathcal H}(\Gamma_K,\Gamma_{\mathcal O})\) is defined as the *-algebra of functions \(f:\Gamma_K\to \mathbb{C}\) satisfying \(f(\gamma\gamma_1\gamma')= f(\gamma_1)\) whenever \(\gamma_1\in \Gamma_K\) and \(\gamma,\gamma'\in \Gamma_{\mathcal O}\) with convolution product defined by \[ (f* g)(\gamma)= \sum f(\gamma\gamma^{- 1}_1) g(\gamma_1), \] where the sum is taken over the left cosets in \(\Gamma_{\mathcal O}\setminus \Gamma_K\). The involution is given by \(f^*(\gamma)= \overline{f(\gamma^{-1})}\). Another important result obtained in the paper gives the characterization of the faithful representations of the \(C^*\)-crossed product \(C^*(K/{\mathcal O})\times_\alpha {\mathcal O}^\times\) in terms of covariant representations of the system. Such a representation is given by a faithful representation \(\pi\) of the group \(C^*\)-algebra \(C^*(K/{\mathcal O})\) on a Hilbert space and an isometric representation \(v\) of \({\mathcal O}^\times\) on the same Hilbert space such that \(\pi(\alpha_a(\delta_y))= v_a\pi(\delta_y) v^*_a\) whenever \(y\in K/{\mathcal O}\) and \(a\in{\mathcal O}^\times\).
scientific article; zbMATH DE number 1139205