Banach algebras arising from quivers

A quiver is a directed graph. The path algebras of quivers encompass many well-known algebras: the simplest examples are:

- the polynomials in 1 variable,
- the higher rank free semigroup algebras, and
- the upper triangular matrices.

All of these have Banach algebra completions. Quivers also have associated $C^*$-algebras, which are generated by partial isometries corresponding to forward paths and adjoints corresponding to the reversed paths. This class of $C^*$-algebra includes the Cuntz-Krieger algebras, which
are amenable.

In this talk we will consider analogous $\ell^1$-algebras, which can similarly be constructed from the inverse semigroup of paths on a quiver. These $\ell^1$-algebras are rarely amenable, but we will show that they have relatively simple cyclic cohomology. This cohomology is reminiscent of the elegant $K$-theory of the Cuntz-Krieger algebras, given by the idempotent structure in the $C^*$-algebras.