Complex group rings and group C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^*$$\end{document}-algebras of group extensions

Let N and H be groups, and let G be an extension of H by N. In this article, we describe the structure of the complex group ring of G in terms of data associated with N and H. In particular, we present conditions on the building blocks N and H guaranteeing that G satisfies the zero-divisor and idempotent conjectures. Moreover, for central extensions involving amenable groups we present conditions on the building blocks guaranteeing that the Kadison–Kaplansky conjecture holds for the group C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^*$$\end{document}-algebra of G.

This problem is known as the idempotent problem, and the corresponding conjecture is called the idempotent conjecture. Using algebraic methods as well as analytical methods, a lot of progress (see e.g. [2,10] and also [13,20,21,27]) has been made on Problem 3. Nevertheless, for a general group G, the answer to Problem 3 remains unknown. In the last two decades, however, Problem 3 has regained interest, mainly due to its intimate connection with the Baum-Connes conjecture in operator algebras (see e.g. [29]) via the so-called Kadison-Kaplansky conjecture for reduced group C *algebras. Recall that the Kadison-Kaplansky conjecture asserts that the reduced group C * -algebra of a (discrete) torsion-free group has no non-trivial idempotents.
There is a mutual hierarchy between Problems 1, 2 and 3. Indeed, for fixed K and G, it is easy to see that an affirmative answer to Problem 1 yields that Problem 3 has an affirmative answer. Furthermore, using a result of D. S. Passman's (see [26, Chap. 13, Lem. 1.2]), we conclude that an affirmative answer to Problem 2 yields an affirmative answer to Problem 1. For a thorough account of the development on the above problems (mainly) during the 1970s, we refer the reader to [26].
In this article we shall restrict our attention to complex group rings, i.e. the case where K = C. Our aim is to contribute to a better understanding of Kaplansky's conjectures by studying complex group rings of group extensions. More concretely, let N and H be two groups. Furthermore, let G be an extension of H by N . Our main objective is to investigate the structure of C[G] in terms of data associated with the building blocks N and H . This article is organized as follows.
In Sect. 2 we record the most important preliminaries and notation. In particular, we discuss crossed products and crossed systems.
In Sect. 3 we represent C[G] as a crossed product of the complex group ring C[N ] and H , where the respective crossed system is associated with the factor system of the underlying group extension (see Theorem 3.2). Although this might be well known to experts (cf. [24, p. 4]), we have not found such a statement explicitly discussed in the literature. Moreover, as an application, we show that if C[N ] is a domain and H is a unique product group, then G satisfies Kaplansky's complex zero-divisor conjecture (see Theorem 3.4 and Corollary 3.6). We conclude the section with several examples and remarks.
In Sect. 4 we consider central extensions, i.e., N is central in G, from an alternative C * -algebraic perspective. To this end, we employ a group-adapted version of the Dauns-Hofmann Theorem (cf. [6,14]) to represent the group C * -algebra C * (G) as a C * -algebra of global continuous sections of a C * -algebraic bundle over the dual group N . In this way we are able to show, under some technical assumptions, that if C * (H ) contains no non-trivial idempotent, then the same assertion holds for C * (G) (see Lemma 4.1 and Corollary 4.2).

Preliminaries and notation
Our study revolves around the structure of complex group rings of group extensions. Consequently, we blend tools from algebraic representation theory and the theory of group extensions. In this preliminary section, we provide the most important definitions and notation which are repeatedly used in this article. In general, given a group G, we shall always write e G , or simply 1 or e, for its identity element.

Group extensions and factor systems
Let 1 → N → G q → H → 1 be a short exact sequence of groups. We first recall a description of the extension G in terms of data associated with N and H . For this purpose, let σ : H → G be a section of q, which is normalized in the sense that σ (e H ) = e G . Then the map N × H → G, (n, h) → nσ (h) is a bijection and may be turned into an isomorphism of groups by endowing N × H with the multiplication where S := C N • σ : H → Aut(N ) with C N : G → Aut(N ), C N (g)(n) := gng −1 and The pair (S, ω) is called a factor system for N and H and we write N × (S,ω) H for the set N × H endowed with the group multiplication defined in (1). We also recall that the maps S and ω satisfy the relations for all h, h , h ∈ H . For a detailed background on group extensions and factor systems we refer the reader to [19,Chap. IV].

Crossed products and crossed systems
Let H be a group and let R = h∈H R h be a unital H -graded ring, i.e., R h R h ⊆ R hh for all h, h ∈ G. We write R × for the group of invertible elements of R and , C A (r )(s) := rsr −1 denotes the canonical conjugation action. It is not hard to check that each crossed product gives rise to a crossed system and vice versa. For details we refer the reader to [24].

Complex group rings
The complex group ring C[G] of a group G is the space of all functions f : G → C with finite support endowed with the usual convolution product of functions which we shall denote by . Each element in C[G] can be uniquely written as a sum g∈G f g δ g with only finitely many non-zero coefficients f g ∈ C and the Dirac functions has a natural involution given by * : and may be equipped with several appropriate norms. Interesting to us is the 1-norm into a normed * -algebra. The corresponding universal enveloping C * -algebra is the full group C *algebra C * (G).

Representation via crossed products
Throughout this section, let 1 → N → G q → H → 1 be a short exact sequence of discrete groups. Furthermore, let σ : H → G be a section of q and let (S, ω) be the corresponding factor system for N and H .
We wish to give a description of the complex group ring C[G] in terms of data associated with the groups N and H . In fact, since N is a normal subgroup of G, we may also consider C[G] as an H -graded ring (h) . In this representation of C[G], each element can be uniquely written as a sum h∈H f h δ σ (h) with only finitely many non-zero coefficients f h ∈ C[N ]. Furthermore, each homogeneous component contains an invertible element, and consequently which is based on the factor system (S, ω). To this end, we first introduce the mapσ where C[G] × denotes the group of invertible elements of C[G]. Then we definē where ) denotes the canonical conjugation action, and

Lemma 3.1 The pair (S,ω) is a (C[N ], H )-crossed system.
Proof It is easily seen thatS(e) = id C[N ] and thatω(g, e) =ω(e, g) = δ e for all g ∈ G. Next, we establish (4). For this, let h, h ∈ H and let f ∈ C[N ]. Then a few moments of thought show that To verify (5), we choose h, h , h ∈ H . Then a short computation yields On the other hand, it is straightforwardly checked that Consequently, (5) follows from the classical cocycle identity (3). where is an isomorphism of (C [N ], H )-crossed products.
We have just seen that the factor system (S, ω) gives rise to a (C[N ], H )-crossed product that is isomorphic to C[G]. Conversely, keeping in mind that N ⊆ C[N ] × via n → δ n we have the following result:

Corollary 3.3 Suppose that (S,ω) is an abstract (C[N ], H )-crossed system. Then the corresponding (C[N ], H )-crossed product C[N ] × (S,ω) H is isomorphic to C[G] for some extension G of H by N ifS(H )(N ) ⊆ N and image(ω) ⊆ N . If this holds, then the factor system (S, ω) for G is defined by S(h) :=S(h) |N , h ∈ H , and the corestriction ofω to N .
We now proceed to investigate the structure of the crossed product C[N ] × (S,ω) H . Recall that a group H is a unique product group if for any two non-empty finite subsets A, B ⊆ H there exists at least one element h ∈ H which has a unique representation of the form h = ab with a ∈ A and b ∈ B. In the next proof, given an element for the corresponding support.

Theorem 3.4 Suppose that C[N ] is a domain and that H is a unique product group. Then C[N ] × (S,ω) H is a domain.
where A := Supp( f ) and B := Supp( f ). Using that H is a unique product group, we find h ∈ AB, a ∈ A, and b ∈ B, such that

Corollary 3.6 Suppose that N satisfies Kaplansky's complex zero-divisor conjecture and that H is a unique product group. Then G satisfies Kaplansky's complex zerodivisor conjecture and the complex idempotent conjecture.
We continue with a series of examples and remarks.

Example 3.7
The discrete Heisenberg group H 3 is abstractly defined as the group generated by elements a and b such that the commutator c = aba −1 b −1 is central. It can be realized as the multiplicative group of upper-triangular matrices Moreover, a short computation shows that H 3 is isomorphic (as a group) to the semidirect product Z 2 S Z, where the semidirect product is defined by the group homomorphism S : Z → Aut(Z 2 ), S(k)((m, n)) := (m, km + n).
Consequently, Theorem 3.2 implies that the complex group ring C[H 3 ] is isomorphic to C[Z 2 ] ×S Z. Since C[Z 2 ] is a domain and Z is an orderable group, and hence a unique product group, it follows from Corollary 3.6 that H 3 satisfies Kaplansky's zero-divisor conjecture. In particular, C[H 3 ] has no non-trivial idempotents.

Example 3.9
Let P be Passman's fours group [5,11], which is a non-split extension 1 → Z 3 → P → Z/2Z × Z/2Z → 1. Note that P is torsion-free. It is easy to see that P is a solvable group, and hence C[P] is a domain by Remark 3.8. Now, let us consider the group G := P × F 2 where F 2 is the free group on two generators. First of all, we notice that G is not a unique product group because P is not (cf. [11]). Indeed, there are non-empty finite subsets A, B of P witnessing that P is not a unique product group. The subsets A × {e} and B × {e} of P × F 2 are witnessing that G is not a unique product group. Secondly, solvable groups are amenable, and subgroups of amenable groups are amenable. But G contains a copy of the non-amenable group F 2 as a subgroup. Hence, G cannot be solvable.
In light of Remark 3.8, we cannot immediately see that C[G] is a domain. However, using Corollary 3.6, we are able to conclude that C[G] is a domain. Remark 3. 10 We emphasize that it is not known whether every unit in C[P] is trivial (cf. [11]). By Theorem 3.2, C[P] ∼ = C[Z 3 ] × (S,ω) (Z/2Z × Z/2Z), meaning that we may study the units of C[P] inside the crossed product on the right-hand side. Note that every unit in C[Z 3 ] is, as a matter of fact, trivial.

Remark 3.11
The purpose of this remark is to illustrate how to obtain families of groups satisfying the conditions of Corollary 3.6. 2. In the situation of Example 3.7, the set Ext(Z, Z 2 ) s consists of a single element, where s := q • S and q : Aut(Z 2 ) → Out(Z 2 ) denotes the canonical projection map. Indeed, it is a well-known fact that H n gr (Z, Z 2 ) s = 0 for all n > 1. 3. It is also possible to realize H 3 as a central group extension of Z by Z 2 with respect to the group 2-cocycle ω : Z 2 × Z 2 → Z defined by ω (k, k ), (l, l ) := k + l . Moreover, the set Ext(Z 2 , Z) is parametrized by H 2 gr (Z 2 , Z) ∼ = Z. Consequently, we obtain an infinite family of groups satisfying Kaplansky's complex zero-divisor conjecture.

Remark 3.12
The map from Theorem 3.2 may also be used to turn C[N ] × (S,ω) H into a * -algebra. The purpose of this remark is to demonstrate that if H is amenable, then the full group C * -algebra C * (G) is isomorphic to a suitable C * -completion of C[N ] × (S,ω) H . For this, we regard C[G] as a dense subset of C * (G) and note that the mapσ in (6)

Representation via global sections
Let N and H be torsion-free and countable discrete groups with N Abelian. Furthermore, let G be a central extension of H by N . Our aim is to analyze the structure of the group C * -algebra C * (G) in terms of data associated with N and H . For technical reasons, we additionally assume that H is amenable. Then G is amenable, and hence [25, Thm. 1.2] implies that C * (G) is isomorphic to the C * -algebra (E) of global continuous sections of a C * -algebraic bundle q : E → N , N being the dual group of N endowed with its natural topology turning it into a compact Hausdorff space. Moreover, its fibre E ε := q −1 ({ε}) at the trivial character ε ∈ N is * -isomorphic to C * (H ). The proof of the next statement is very much inspired by the proof of [ Proof Let p be a projection in C * (G). We write s p for the corresponding continuous global section in (E). By assumption, we either have s p (ε) = 0 ∈ E ε or s p (ε) = 1 ∈ E ε . For now, assume that s p (ε) = 0 ∈ E ε . Since N is torsion-free, its dual group N is connected, and therefore the function where · χ denotes the C * -norm on E χ := q −1 ({χ }), is a continuous {0, 1}-valued function on a connected space with f p (ε) = 0. It follows that f p must be 0 everywhere which in turn implies that s p ≡ 0. That is, we have p = 0. Analogously, the case s p (ε) = 1 ∈ E ε leads to p = 1.