The article is devoted to the representation theory of locally compact infinite-dimensional group GLB of almost upper-triangular infinite matrices over the finite field with q elements. This group was defined by S.K., A.V., and Andrei Zelevinsky in 1982 as an adequate n = infinity analogue of general linear groups GL(n, q). It serves as an alternative to GL(infinity, q), whose representation theory is poor.Our most important results are the description of semifinite unipotent traces (characters) of the group GLB. via certain probability measures on the Borel subgroup B and the construction of the corresponding von Neumann factor representations of type H-infinity.As a main tool we use the subalgebra A(GLB) of smooth functions in the group algebra L-1 (GLB). This subalgebra is an inductive limit of the finite-dimensional group algebras C(GL(n, q)) under parabolic embeddings.As in other examples of the asymptotic representation theory we discover remarkable properties of the infinite case which does not take pl