Representations of the group of invertible infinite matrices with coefficients in a finite-dimensional algebra (Q1316831)

Let \(U\) be a finite-dimensional complex topological algebra with a unit. Let \(U\) admit a basis consisting of the elements \(a_ 0 = 1_ U\), \(a_ n\), \(n \in \{1,2,\dots, n\}\), such that the elements \(\sum_{i = 1}^ n c_ i a_ i\), \(c_ i \in \mathbb{C}\), form a subalgebra \(U_ 0\) and \(U_ 0^ p = 0\) for some natural \(p\). Let \(M_ \infty (U)\) be the set of infinite-dimensional matrices with coefficients from \(U\), where only finitely many entries do not vanish. Let \(G(U) = \{I + a\mid a \in M_ \infty(U)\), \((I + a)^{-1} = I + b\), \(b\in M_ \infty(U)\}\). The author considers \(G(U)\) in the corresponding inductive limit topology. Unitary representations of this group continuous in this topology are studied. The main result of the paper is a description of all admissible representations of the group \(G(U)\) in terms of antihomomorphisms of \(U\) into the full matrix algebra \(M_ p(\mathbb{C})\). It is indicated that the method used in this paper can be generalized to some other infinite- dimensional matrix groups. As an example, the author considers the group \(SU^*(2,\infty)\).