Semigroup crossed products and Hecke algebras arising from number fields
type: Item
instance of: scholarly article
title: Semigroup crossed products and Hecke algebras arising from number fields
zbMATH Open document ID: 0940.47062
author: Jane Arledge
author: Marcelo Laca
author: Iain Raeburn
published in: Documenta Mathematica
publication date: 1998-04-14T00:00:00Z
full work available at URL: https://eudml.org/doc/224704
full work available at URL: vol-02.html
review text: 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 c…
reviewed by: Alfons Van Daele
Mathematics Subject Classification ID: 47L65
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 22D25
zbMATH DE Number: 1139205
zbMATH Keywords: covariant representation
zbMATH Keywords: algebraic crossed product *-algebra
zbMATH Keywords: universal \(C^*\)-algebra
zbMATH Keywords: Hecke algebra
zbMATH Keywords: convolution product
zbMATH Keywords: faithful representations
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1139205