On certain structural properties of Banach algebras associated with automorphisms

Abstract

In the Hilbert space setting (i.e., in the framework of C∗-algebra theory), analogous objects are closely related with a crossed product (see, e.g., [4]), and many works are devoted to the description of their structures. In particular, Landstad [3] found necessary and sufficient conditions (in terms of duality theory) under which a C∗-algebra is isomorphic to the crossed product (of a certain algebra and a locally compact automorphism group). In the case of a discrete group, in [1, Chap. 2], there are necessary and sufficient conditions under which a C∗-algebra is isomorphic to a crossed product in terms of the action of the group (the so-called topologically free action (see Sec. 2.8 of the present paper) and also in terms of fulfillment of a certain inequality (Property (∗), (2.1) of the present paper) that guarantees the existence of an expectation of the algebra B(A, Tg) on the algebra A (see (2.2) and (2.3)). The goal of the present paper is the study of interrelations between the properties (topologically free action, Property (∗), and dual action of the group) mentioned above in the general Banach space setting. Along with this, we study property (∗∗) (Sec. 2.2 ) of reconstructing an element of the algebra B(A, Tg) by its “Fourier coefficients,” which naturally arises here.