# A generalization of real characters on a group

Yesterday I understood that I can't live without this construction:

Let $G$ be a group, $A$ an associative algebra over $\mathbb R$ and $n\in{\mathbb N}$. We consider a sequence of maps $\varphi_k:G\to A$, $k=0,...,n$, satisfying the following three identities: $$\varphi_k(1_G)=\begin{cases}1_A,& k=0\\ 0,& k\ne 0\end{cases}$$ $$\varphi_k(a\cdot b)=\sum_{0\le l\le k}\begin{pmatrix}k \\ l\end{pmatrix}\cdot \varphi_l(a)\cdot \varphi_{k-l}(b),\qquad a,b\in G,$$ $$\forall k\ge 1\qquad \varphi_0(a)\cdot\varphi_k(b)=\varphi_k(b)\cdot\varphi_0(a),\qquad a,b\in G,$$

In particular, this means that $\varphi_0:G\to A$ is a homomorphism (of $G$ into the group of invertible elements in $A$): $$\varphi_0(1_G)=1_A,\qquad \varphi_0(a\cdot b)=\varphi_0(a)\cdot\varphi_0(b),\qquad a,b\in G.$$ And in the case of $\varphi_0(a)=1$ ($a\in G$), the map $\varphi_1:G\to A$ becomes a generalization of what is called real character on $G$: $$\varphi_1(1_G)=0,\qquad \varphi_1(a\cdot b)=\varphi_1(a)+\varphi_1(b),\qquad a,b\in G.$$

In what I consider $G$ is a locally compact group, $A$ is a $C^*$-algebra, and the maps $\varphi_k:G\to A$ are continuous and satisfy the following supplementary identities: $$\varphi_0(a)^*=\varphi_0(a^{-1}),\qquad a\in G.$$ $$\varphi_1(a)^*=-\varphi_0(a)^{-2}\cdot\varphi_1(a),\qquad a\in G.$$ $$\varphi_2(a)^*=-\varphi_0(a)^{-2}\cdot\varphi_2(a)+\varphi_0(a)^{-3}\cdot\varphi_1(a),\qquad a\in G.$$ $$...$$

I never saw something like this before, so my main question is

What is this?

Did anybody consider the sequences of maps $\varphi_k:G\to A$ like these? If yes, what is known about them? I am curious, for example, in

How long can these sequences $\varphi_k:G\to A$ (with non-zero $\varphi_k$) be?

It is easy to see, for example, that if $G$ is a compact group, and $A$ is a $C^*$-algebra, then only $\varphi_0$ can be non-zero: $$\forall k\ge 1\qquad \varphi_k=0.$$ This follows from the fact that $\varphi_0$ in this case acts into the set of unitary elements, therefore $$\|\varphi_0(a)\|=1,$$ and $\varphi_1$ satisfies the identity $$\varphi_1(a^m)=m\cdot\varphi_0(a)^{m-1}\cdot\varphi_1(a)$$ So if $\varphi_1(a)\ne 0$, then $\|\varphi_1(a^m)\|\to\infty$ as $m\to\infty$, but this is impossible, since $\varphi_1:G\to A$ is a continuous map of a compact space $G$. As a corollary, $\varphi_1=0$, and the same reasoning for other $\varphi_k$ with $k>0$.

Any references, thoughts, feelings will be appreciated.

• If you extend by linearity to the group algebra, you get a sequence of linear maps between two associative algebras satisfying the same conditions. So these axioms make sense in this context, with no group on the left side. Maybe "ra.rings-and-algebras" would be a natural tag then. – YCor Sep 24 '14 at 15:31
• @YCor, yes initially $G$ and $A$ both were algebras. I reformulated this for the case when $G$ is a group. – Sergei Akbarov Sep 24 '14 at 15:36
• +1 for the first sentence. – Nick Gill Sep 24 '14 at 16:30
• I think you mean $\varphi_n(a.b)$ rather than $\varphi_k(a.b)$ in the second condition. – Neil Strickland Sep 24 '14 at 17:17
• The first two conditions just mean that the map $\psi(u)=\sum_k\varphi_k(u)t^k/k!$ gives a ring homomorphism $\psi\colon\mathbb{R}[G]\to A[t]$. – Neil Strickland Sep 24 '14 at 17:18