  We give a definition of partition C*-algebras: To any partition of a finite
set, we assign algebraic relations for a matrix of generators of a universal
C*-algebra. We then prove how certain relations may be deduced from others and
we explain a partition calculus for simplifying such computations. This article
is a small note for C*-algebraists having no background in compact quantum
groups, although our partition C*-algebras are motivated from those underlying
Banica-Speicher quantum groups (also called easy quantum groups). We list many
open questions about partition C*-algebras that may be tackled by purely
C*-algebraic means, ranging from ideal structures and representations on
Hilbert spaces to K-theory and isomorphism questions. In a follow up article,
we deal with the quantum algebraic structure associated to partition
C*-algebras.
