Global and internal structure of operator algebras

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My research is in an area of Pure Mathematics known as Operator Theory and Operator Algebras. Very loosely speaking, this is an area which studies sets of linear transformations, of which rotations, reflections, and dilations in our three-dimensional space R^3 are but three examples. My interest is to study linear transformations in the infinite-dimensional analogues of R^3 known as Hilbert spaces. Some of these collections of linear transformations (or operators) possess certain algebraic properties, allowing us to scale the members, to compose two transformations, or add two transformations, and yet still remain in the collection. For example, if we compose two rotations, we simply obtain another rotation. If we add two translations, we obtain another translation. Such collections are referred to as algebras.