Actions of locally compact (quantum) groups on ternary rings of operators, their crossed products, and generalized Poisson boundaries

Abstract

Actions of locally compact groups and quantum groups on W${}^{\ast }$-ternary rings of operators are discussed, and related crossed products are introduced. The results generalize those for von Neumann algebraic actions with proofs based mostly on passing to the linking von Neumann algebra. They are motivated by the study of fixed-point spaces for convolution operators generated by contractive, not necessarily positive measures, both in the classical and in the quantum context.