STbimodule

STbimodule is a mathematical structure studied in algebra and representation theory. Roughly, it refers to a bimodule equipped with compatible actions of two algebras, often denoted A and B. An STbimodule X over a pair (A,B) consists of an abelian group (or module) X with a left action of A and a right action of B such that the two actions commute: (a x) b = a (x b) for all a in A, x in X, b in B. Equivalently, X is a module over the tensor product A ⊗ B^op. In many texts the term STbimodule highlights additional structure or symmetry conditions that may be imposed, but the essential content is the commuting left and right actions.

In category-theoretic terms, STbimodules are the objects of the category of A–B bimodules, with morphisms given by maps that are bilinear with respect to the left A-action and the right B-action. This framework generalizes simpler notions such as left A-modules (when B is the ground field) or right B-modules (when A is the ground field).

In operator algebra theory, STbimodules often appear as correspondences or Hilbert C*-modules bridging two C*-algebras. Such correspondences are central to constructions in Morita theory, the study of equivalences between module categories, and the composition of bimodules via tensor products.

Applications of STbimodules include establishing Morita equivalences, transporting representations between algebras, and forming functors between module categories. See also bimodule, Morita theory, tensor product, and C*-correspondence.