A necessary condition for zero divisors in complex group algebra of torsion-free groups

TL;DR

This paper establishes a necessary condition for the existence of zero divisors in the complex group algebra of torsion-free groups, advancing understanding of algebraic structures in group theory.

Contribution

It introduces a new necessary condition for zero divisors specifically in complex group algebras of torsion-free groups, which was previously unknown.

Findings

Identifies a specific necessary condition for zero divisors.

Provides insights into the algebraic structure of torsion-free groups.

Advances theoretical understanding of group algebra properties.

Abstract

We find a necessary condition for zero divisors in complex group algebras of torsion-free groups.

Equations

$$\alpha \beta =\sum _{g,h\in G}{a}_g{b}_{h}gh=\sum _{g\in G}\left(\sum _{x\in G}{a}_{g{x}^{-1}}{b}_x\right)x$$

$$⟨\alpha ,\beta ⟩:=\sum _{g\in G}{a}_g{\bar{b}}_g(\alpha ,\beta \in \mathbb{C}G)$$

$${\ell }^2(G)=\left\{\alpha :G\to \mathbb{C}:\sum _{g\in G}\Vert \alpha (g){\Vert }^2<\infty \right\}.$$

$$\Vert \alpha {\Vert }_1=\sum _{g\in G}|a_g|,\text{for α=∑g∈Gagg in CG}.$$

$$⟨\alpha \beta ,\gamma ⟩=⟨\beta ,{\alpha }^{\ast }\gamma ⟩.$$

$$(\alpha +\beta t)\sum _{k=0}^n{\gamma }_k{t}^k=0.$$

$$0=⟨\tilde{\alpha }\beta ,\beta ⟩=\sum _{\alpha \in \mathcal{F}}⟨{\alpha }^{\ast }\alpha \beta ,\beta ⟩=\sum _{\alpha \in \mathcal{F}}⟨\alpha \beta ,\alpha \beta ⟩=\sum _{\alpha \in \mathcal{F}}\Vert \alpha \beta {\Vert }_2^2,$$

$${\rm Y}(\alpha ):=a_1-\sum _{g\ne 1}|a_g|.$$

$$\alpha$$

$$=a_1+\frac{1}{2}\sum _{g\ne 1}\left({\bar{a}}_g{g}^{-1}+a_{g}g\right)$$

$$={\rm Y}(\alpha )+\frac{1}{2}\sum _{g\ne 1}\left(2|a_g|+{\bar{a}}_g{g}^{-1}+a_{g}g\right)$$

$$={\rm Y}(\alpha )+\frac{1}{2}\sum _{g\ne 1}|a_g|{\left(\frac{{\bar{a}}_g}{|a_g|}+g\right)}^{\ast }\left(\frac{a_g}{|a_g|}+g\right)$$

$${\rm Y}({\alpha }^{\ast }\alpha )\ge 2\Vert \alpha {\Vert }_2^2-\Vert \alpha {\Vert }_1^2.$$

Full text

1. Introduction

Let $G$ be any group and $\mathbb{C}G$ be the complex group algebra of $G$, i.e. the set of finitely supported complex functions on $G$. We may represent an element $\alpha$ in $\mathbb{C}G$ as a formal sum $\sum_{g\in G}a_{g}g$, where $a_{g}\in\mathbb{C}$ is the value of $\alpha$ in $g$. The multiplication in $\mathbb{C}G$ is defined by

[TABLE]

for $\alpha=\sum_{g\in G}a_{g}g$ and $\beta=\sum_{g\in G}b_{g}g$ in $\mathbb{C}G$. We shall say that $\alpha$ is a zero divisor if there exists $0\neq\beta\in\mathbb{C}G$ such that $\alpha\beta=0$. If there is a non-zero $\beta\in\ell^{2}(G)$ such that $\alpha\beta=0$, then we may say that $\alpha$ is analytical zero divisor. If $\alpha\beta\neq 0$ for all $0\neq\beta\in\mathbb{C}G$, then we say that $\alpha$ is regular. The following conjecture is called the zero divisor conjecture.

Conjecture 1.1**.**

Let $G$ be a torsion-free group. Then all elements in $\mathbb{C}G$ are regular.

Amazingly, this conjecture has held up for many years. The conjecture 1.1 has been proven affirmative when $G$ belongs to special classes of groups; ordered groups ([10] and [11]), supersolvable groups ([6]), polycyclic-by-finite groups ([1] and [5]) and uniqe product groups ([2]). Delzant [3] deals with group rings of word-hyperbolic groups and proves the conjecture for certain word-hyperbolic groups. Let $\mathcal{C}$ (Linnell’s class of groups) be the smallest class of groups which contains all free groups and is closed under directed unions and extensions with elementary amenable quotients. Let $G$ be a group in $\mathcal{C}$ such that there is an upper bound on the orders of finite subgroups, then $G$ satisfies the above conjecture ([8]).

The map $\left\langle\cdot,\cdot\right\rangle:\mathbb{C}G\times\mathbb{C}G\rightarrow\mathbb{R}$ defined by

[TABLE]

is an inner product on $\mathbb{C}G$, so $\|\alpha\|_{2}=\left\langle\alpha,\alpha\right\rangle^{\frac{1}{2}}$ becomes a norm, called 2-norm; the completion of $\mathbb{C}G$ w.r.t. 2-norm is the Hilbert space $\ell^{2}(G)$. Indeed, we have

[TABLE]

In [9], Linnell formulated an analytic version of the zero divisor conjecture.

Conjecture 1.2**.**

Let $G$ be a torsion-free group. If $0\neq\alpha\in\mathbb{C}G$ and $0\neq\beta\in\ell^{2}(G)$, then $\alpha\beta\neq 0$.

In [7], it is shown that Since $\mathbb{C}G\subset\ell^{2}(G)$, the second conjecture implies the first one. In [4], it is proved that for finitely generated amenable groups, the two conjectures are actually equivalent. We prove this is true for all amenable torsion-free groups.

The so-called 1-norm is defined on $\mathbb{C}G$ by

[TABLE]

The adjoint of an element $\alpha=\sum_{g\in G}a_{g}g$ in $\mathbb{C}G$, denoted by $\alpha^{*}$, is $\alpha^{*}=\sum_{g\in G}\bar{a}_{g}g^{-1}$. We call an element $\alpha\in\mathbb{C}G$ self-adjoint if $\alpha^{*}=\alpha$, and use $(\mathbb{C}G)_{\textbf{s}}$ to denote the set of self-adjoint elements of $\mathbb{C}G$. It is worthy of mention that if $\alpha=\sum_{g\in G}a_{g}g$ is self-adjoint then $a_{1}$ should be a real number. For $\alpha\in\mathbb{C}G$, $\beta$ and $\gamma$ in $\ell^{2}(G)$, the following equalities hold:

[TABLE]

The goal of this paper is to give a criterion for an element in a complex group algebra to be regular:

Theorem 1.1**.**

Let $G$ be a torsion free group. Then $\alpha\in\mathbb{C}G$ is regular if $2\|\alpha\|_{2}^{2}\geq\|\alpha\|_{1}^{2}$.

2. Preliminaries

In this section we provide some preliminaries needed in the following.

Let $G$ be a group. The support of an element $\alpha=\sum_{g\in G}a_{g}g$ in $\mathbb{C}G$, $\text{supp}\left(\alpha\right)$, is the finite subset $\left\{g\in G:a_{g}\neq 0\right\}$ of $G$.

Let $H$ be a subgroup of $G$, and $T$ be a right transversal for $H$ in $G$. Then every element $\alpha\in\mathbb{C}G$ (resp. $\alpha\in\ell^{2}(G)$) can be written uniquely as a finite sum of the form $\sum_{t\in T}\alpha_{t}t$ with $\alpha_{t}\in\mathbb{C}H$ (resp. $\alpha_{t}\in\ell^{2}(H)$).

For $S\subset G$, we denote by $\langle S\rangle$, the subgroup of $G$ generated by $S$. We have the following key lemma:

Lemma 2.1**.**

Let $G$ be a group, $\alpha\in\mathbb{C}G$ and $H=\langle\text{supp}\left(\alpha\right)\rangle$. Then $\alpha$ is regular in $\mathbb{C}G$ iff $\alpha$ is regular in $\mathbb{C}H$.

Proof.

Suppose that $\alpha$ is a zero divisor. Among elements $0\neq\gamma$ in $\mathbb{C}G$ which satisfy $\alpha\gamma=0$ consider an element $\beta$ such that $1\in\text{supp}\left(\beta\right)$ and $|\text{supp}\left(\beta\right)|$ is minimal, then one can easily show that $\beta\in\mathbb{C}H$, and this proves the result of the lemma. ∎

An immediate consequence of this lemma is:

Corollary 2.1**.**

A group $G$ satisfies the Conjecture 1.1 iff all its finitely generated subgroups satisfy the Conjecture 1.1.

By Lemma 2.1 in hand, we can generalize the main theorem of [4]:

Theorem 2.1**.**

Let $G$ be an amenable group. If $0\neq\alpha\in\mathbb{C}G$, $0\neq\beta\in\ell^{2}(G)$ and $\alpha\beta=0$, then there exists $0\neq\gamma\in\mathbb{C}G$ such that $\alpha\gamma=0$.

The above theorem along with results in [12] provides another proof for [7, Theorem 2].

For a normal subgroup $N$ of a group $G$, we denote the natural quotient map by $q_{N}:G\rightarrow G/N$. We continue to show that:

Lemma 2.2**.**

Let $N$ be a normal subgroup of a group $G$ satisfying Conjecture 1.1. Consider a non-torsion element $q_{N}(t)$, $t\in G$, in the quotient group. Then $\alpha+\beta t$ is regular, for all $\alpha,\beta\in\mathbb{C}N\setminus\left\{0\right\}$.

Proof.

Suppose that $\alpha+\beta t$ is a zero divisor for non zero elements $\alpha,\beta\in\mathbb{C}N$. Applying Lemma 2.1 and multiplying by a suitable power of $t$, we can assume that there are non zero elements $\gamma_{k}$, $k=0,1,\dotsc,n$, such that

[TABLE]

In particular, $0=\beta t\gamma t^{n}=(\beta t\gamma_{n}t^{-1})t^{n+1}$, whence $\beta t\gamma_{n}t^{-1}=0$, a contradiction, because $t\gamma_{n}t^{-1}$ is a non zero element of $\mathbb{C}N$. ∎

Proposition 2.1**.**

Let $N$ be an amenable normal subgroup of a group $G$ satisfying Conjecture 1.1. Consider a non-torsion element $q_{N}(t)$, $t\in G$, in the quotient group. Then there is no $0\neq\gamma\in\ell^{2}(G)$ such that $(\alpha+\beta t)\gamma=0$. In particular, $a+bg$ is an analytical zero divisor, for all non-torsion element $g\in G$ and non zero complex numbers $a,b$.

Proof.

The group $\left\langle N,t\right\rangle$ is amenable. Hence Lemma 2.2 together with Theorem 2.1 yields the result. ∎

3. A cone of regular elements

The result of the Proposition 2.1 is true if we replace $\mathbb{C}$ by an arbitrary field $\mathbb{F}$. The field of complex numbers allows us to define inner product on the group algebra; with the help of inner product, we can construct new regular elements from the ones we have:

Proposition 3.1**.**

Let $G$ be a group and $\mathcal{F}$ be a finite non-empty subset of $\mathbb{C}G$. If $\sum_{\alpha\in\mathcal{F}}\alpha^{*}\alpha$ is an analytical zero divisor then all elements of $\mathcal{F}$ are analytical zero divisors. In particular, $\alpha\in\mathbb{C}G$ is an analytical zero divisor if and only if $\alpha^{*}\alpha$ is an analytical zero divisor.

Proof.

Let $\tilde{\alpha}:=\sum_{\alpha\in\mathcal{F}}\alpha^{*}\alpha$ and $\tilde{\alpha}\beta=0$ for some $\beta\in\ell^{2}(G)$. Then

[TABLE]

whence $\alpha\beta=0$ for all $\alpha\in\mathcal{F}$. This completes the proof. ∎

A cone in a vector space $\mathfrak{X}$ is a subset $\mathfrak{K}$ of $\mathfrak{X}$ such that $\mathfrak{K}+\mathfrak{K}\subset\mathfrak{K}$ and $\mathbb{R}_{+}\mathfrak{K}\subset\mathfrak{K}$. We proceed by introducing a cone of regular elements in $\mathbb{C}G$. First a definition:

Definition 3.1**.**

Let $G$ be a group and $(\mathbb{C}G)_{\text{s}}$ be the set of self adjoint elements $\alpha\in\mathbb{C}G$, we define a function $\Upsilon:(\mathbb{C}G)_{\text{s}}\rightarrow\mathbb{R}$ by

[TABLE]

We call an element $\alpha\in(\mathbb{C}G)_{\text{s}}$ golden if $\Upsilon(\alpha)\geq 0$. The set of all golden elements in $(\mathbb{C}G)_{\text{s}}$ is denoted by $(\mathbb{C}G)_{\text{gold}}$.

What is important about golden elements is:

Proposition 3.2**.**

For a torsion free group $G$, $(\mathbb{C}G)_{\text{gold}}$ is a cone of regular elements.

Proof.

It is obvious that if $\alpha$ is golden then so is $r\alpha$ for any $r>0$. The triangle inequality for $\mathbb{C}$ shows that if $\alpha$ and $\gamma$ are golden then so is $\alpha+\gamma$. For $\alpha\in(\mathbb{C}G)_{\text{s}}$, we have

[TABLE]

Hence, by Lemma 2.1 and Proposition 3.1, $\alpha$ is regular. ∎

Now, we are ready to prove our main result:

Proof of Theorem 1.1.

For $\alpha=\sum_{g\in G}a_{g}g$ in $\mathbb{C}G$, $\alpha^{*}\alpha$ is self-adjoint, and one can easily show that

[TABLE]

Hence, by Proposition 3.2, the result of the Theorem is proved. ∎