Duality Theory for Covariant Systems

If <!-- MATH $(A,\rho ,G)$ --> is a covariant system over a locally compact group G, i.e. is a homomorphism from G into the group of -automorphisms of an operator algebra A, there is a new operator algebra <!-- MATH $\mathfrak{A}$ --> called the covariance algebra associated with <!-- MATH $(A,\rho ,G)$ --> . If A is a von Neumann algebra and is -weakly continuous, <!-- MATH $\mathfrak{A}$ --> is defined such that it is a von Neumann algebra. If A is a <!-- MATH ${C^{\ast}}$ --> -algebra and is norm-continuous <!-- MATH $\mathfrak{A}$ --> will be a <!-- MATH ${C^{\ast}}$ --> -algebra. The following problems are studied in these two different settings: 1. If <!-- MATH $\mathfrak{A}$ --> is a covariance algebra, how do we recover A and ? 2. When is an operator algebra <!-- MATH $\mathfrak{A}$ --> the covariance algebra for some covariant system over a given locally compact group G?