Duality theory for covariant systems

Abstract:

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