Let F be a field. An F-algebra is called an ultramatricial algebra if it is isomorphic to the union of an increasing chain of a finite product of matrix algebras over F. When F is the field of complex numbers, these algebras are also called locally semisimple algebras (or LS-algebras for short), as they are isomorphic to a union of a chain of semisimple ℂ-algebras. An important example of such rings is the group ring ℂ [S∞], where S∞ is the infinite symmetric group. These rings appeared in the setting of C*-algebras and then von Neumann regular algebras in the work of Grimm, Bratteli, Elliott, Goodearl, Handelman and many others after them.

Despite their simple constructions, the study of ultramatricial algebras is far from over. As is noted in [91]: “The current state of the theory of LS-algebras and its applications should be considered as the initial one; one has discovered the first fundamental facts and noted a general circle of questions. To estimate it in perspective, one must consider the enormous number of diverse and profound examples of such algebras. In addition one can observe the connections with a large number of areas of mathematics.”

One of the sparkling examples of the Grothendieck group as a complete invariant is in the setting of ultramatricial algebras (and AF C*-algebras). It is by now a classical result that the K0-group along with its positive cone K0+ (the dimension group) and the position of identity is a complete invariant for such algebras ([40, Bratteli–Elliott Theorem 15.26]). To be precise, let R and S be (unital) ultramatricial algebras. Then R ≅ S if and only if there is an order isomorphism ≅ K0(R), [R]) ≅ (K0(S), [S]), that is, an isomorphism from K0(R) to K0(S) which sends the positive cone onto the positive cone and [R] to [S]. To emphasise the ordering, this isomorphism is also denoted by K0(R), K0(R)+, [R]) ≅(K0(S), K0(S)+, [S]) (see §3.6.1).

The theory has also been worked out for the nonunital ultramatricial algebras (see Remark 5.2.7 and [41, Chapter XII]). Two valuable surveys on ultramatricial algebras and their relations with other branches of mathematics are [91, 98].