Those who work in operator theory and the theory of operator algebras know how important geometric ideas and techniques are to success. Here Beltita (mathematics, the Romanian Academy) gives readers the background they need to understand this cycle, and also describes the newest research. He covers topological Lie algebras, Lie groups and their Lie algebras, enlargeability, smooth homogeneous spaces, quasimultiplicative maps, complex structures in homogeneous spaces, equivariant monotone operators, L*-ideals and equivariant monotone operators, homogeneous spaces of pseudo-restricted groups