For a \(*\) -automorphism group G on a von Neumann algebra, we study the G-quasi-invariant states and their properties. The G-quasi-invariance or G-strongly quasi-invariance is weaker than the G-invariance and has wide applications. We develop several properties for G-strongly quasi-invariant states. Many of them are the extensions of the already developed theories for G-invariant states. Among others, we consider the relationship between the group G and modular automorphism group, invariant subalgebras, ergodicity, modular theory, and abelian subalgebras. We provide with some examples to support the results.
