## Semigroup crossed products and Hecke algebras arising from number fields.(English)Zbl 0940.47062

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 M. Laca and 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 J.-B. Bost and 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 $$a\in{\mathcal O}^\times$$.

### MSC:

 47L65 Crossed product algebras (analytic crossed products) 46L55 Noncommutative dynamical systems 11R04 Algebraic numbers; rings of algebraic integers 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations

Zbl 0842.46040
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