Shifts of group-like projections and contractive idempotent functionals for locally compact quantum groups

Abstract

A one-to-one correspondence between shifts of group-like projections on a locally compact quantum group [Formula: see text] which are preserved by the scaling group and contractive idempotent functionals on the dual [Formula: see text] is established. This is a generalization of the Illie–Spronk’s correspondence between contractive idempotents in the Fourier–Stieltjes algebra of a locally compact group [Formula: see text] and cosets of open subgroups of [Formula: see text]. We also establish a one-to-one correspondence between nondegenerate, integrable, [Formula: see text]-invariant ternary rings of operators [Formula: see text], preserved by the scaling group and contractive idempotent functionals on [Formula: see text]. Using our results, we characterize coideals in [Formula: see text] admitting an atom preserved by the scaling group in terms of idempotent states on [Formula: see text]. We also establish a one-to-one correspondence between integrable coideals in [Formula: see text] and group-like projections in [Formula: see text] satisfying an extra mild condition. Exploiting this correspondence, we give examples of group-like projections which are not preserved by the scaling group.