Trace and higher traces on an algebra Let $A$ be a $k$-algebra for some field $k$. In Loday 2.4.5 he calls maps of the form
$$\{f:A\to k|f(a_1a_2) = f(a_2a_1)\}$$
a trace on $A$, which he identifies as the $0$-th cyclic cohomology of $A$, that is
$$HC^0(A) = \text{set of traces on }A$$
I'm studying cyclic cohomology, so I've been trying to find more information about this to better understand what it means, but I've been struggling to find other resources that talk explicitly about traces in this way. I've found this entry here that gives the same definition for $C^*$-algebras, but that lead me to Dixmier's $C^*$-algebras, which, if you can believe it, only talks about $C^*$-algebras, and not general algebras, so that's a dead end.
Everywhere else I've looked talks about the more common notions of trace, like the trace of a matrix or the field trace. I know that the trace function on a matrix with entries in $k$ is an example of a trace on the $k$-algebra $M_n(k)$, and I've not checked but I wouldn't be surprised if the field trace was an example of a trace on $L/k$ as a $k$-algebra. The wikipedia page for the trace mentions that on a general algebra $A$ the trace is "often" defined in the way I'm interested in, but frustratingly provides no reference for this claim.
The paper Cyclic cohomology and algebra extensions by Quillen states that "In a general way, we can describe cyclic cohomology as the study of higher traces on an
algebra", so I tried reading about higher traces and finding how that might relate to the definition that Loday gives. However, I've so far struggled to find a decent explanation of what higher traces are, most of what I've found has been pretty impenetrable. And I've either been unable to get hold of the references mentioned in Quillen's paper or they've been a dead end.
Would anyone be able direct me to a decent introduction to higher traces and/or a reference that talks about traces on an algebra? Thanks!
Edit: It seems that really what I'm after is understanding what is meant by the terms "trace" and "higher trace" as they appear in Quillen's paper that I link above.
 A: A trace is just a $k$-linear map $f:A\to k$ which is zero on the subspace $[A,A]$ generated by the commutators $[a_1,a_2]=a_1a_2-a_2a_1$. In other words, this is an element of the vector space $(A/[A,A])^*$. Nothing more, nothing less.
Note that if $A$ is a commutative $k$-algebra, a higher trace is just a linear map $f:A\to k$, so you really can't expect to say anything interesting in full generality.
I think you attach too much importance to the word "trace".
The name "trace" is just chosen as an analogy to the case $A=M_n(k)$, and that there is nothing deeper about it (because of the commutative case).
Indeed, I claim that the vector space of traces of $M_n(k)$ is generated by the usual trace map.
Let $f:M_n(k)\to k$ be a trace.
For all $i\neq j,$ $f(E_{ij})=f(E_{ij} E_{jj})=f(E_{jj}E_{ij})=f(0)=0$.
We also have $f(E_{ii})=f(E_{ij}E_{ji})=f(E_{ji})E_{ij})=
f(E_{jj})$. Let $\lambda\in k$ the common value of the $f(E_{ii})'s$.
Now if $M=(a_{ij})$, then $f(M)=\sum_{ij} a_{ij}f(E_{ij})=\lambda\sum_i a_{ii}=\lambda tr(M)$. QED
Hence a higher trace of $M_n(k)$ is just a scalar multiple of the usul trace map.
It is worth pointing out you can also derive from standard theorems that, if $A$ is a central simple $k$-algebra, then a higher trace is a scalar multiple of the reduced trace map $Trd_A$ (which is a generalisation of the usual trace map).
