Exercise 25, § 6

Let G be a group. Denote by $\mathbf{C}[G]$ the group algebra of G over $\mathbf{C}$. It is a subspace of the Hilbert space $\ell^2(G)$. For $g$ in G, denote by $[g]$ the element of $\mathbf{C}[G]$ corresponding to $g$. The elements $[g]$, for $g\in G$, form an orthonormal basis of $\ell^2(G)$ and $\mathbf{C}[G]$ is dense in $\ell^2(G)$.

a) The mapping

is an involution on $\mathbf{C}[G]$.

b) For $g\in G$, let $\pi (g)$ be the mapping $x\mapsto gx$ of $\mathbf{C}[G]$. The mapping which to $g$ associates $\pi (g)$ extends to a continuous and injective morphism of involutive algebras from $\mathbf{C}[G]$ into $\mathscr{L}(\ell^2(G))$.

Denote by Stell$_r(G)$, and call the reduced stellar algebra of G, the closure in $\mathscr{L}(\ell^2(G))$ of the image of the mapping $\pi$. It is a stellar algebra. We identify $\mathbf{C}[G]$ with its image under $\pi$ in Stell$_r(G)$.

c) The linear trace mapping Tr$:\mathbf{C}[G]\rightarrow \mathbf{C}$ such that Tr(1) = 1 and Tr($g$) $=$ 0 for $g\not= 1$ extends to a continuous linear mapping Stell$_r(G)\rightarrow \mathbf{C}$.

d) One has Tr($xy$) $=$ Tr($yx$) for all $x$ and $y$ in Stell$_r(G)$.

e) One has Tr($xx^*$)$\geqslant 0$ for every $x$ in Stell$_r(G)$, and Tr($xx^*$) $= 0$ if, and only if, $x= 0$.

f) Let $e$ be an idempotent element of $\mathbf{C}[G]$ such that $e\not= 0$ and $e\not= 1$. Now prove that $0<$ Tr($e$)$<1$. (Use exercise 3).

g) Let $e$ be an idempotent element of $\mathbf{Z}[G]$. Now prove that $e= 0$ or $e= 1$. (“Kaplansky’s theorem”).

h) Let $x$ be an element of $\mathbf{C}[G]$. Then $x$ is invertible if and only if it is left invertible, if and only if it is right invertible.

i) Let $n$ be a strictly positive integer. A matrix $x$ of type $(n, n)$ with coefficients in $\mathbf{C}[G]$ is invertible if and only if it is left invertible, if and only if it is right invertible.