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

A one-to-one correspondence between shifts of group-like projections on a locally compact quantum group G ?? which are preserved by the scaling group and contractive idempotent functionals on the dual Gˆ??̂ 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 GG and cosets of open subgroups of GG. We also establish a one-to-one correspondence between nondegenerate, integrable, G??-invariant ternary rings of operators X⊂L∞(G) X⊂L∞(??), preserved by the scaling group and contractive idempotent functionals on G??. Using our results, we characterize coideals in L∞(Gˆ) L∞(??̂) admitting an atom preserved by the scaling group in terms of idempotent states on G??. We also establish a one-to-one correspondence between integrable coideals in L∞(G)L∞(??) and group-like projections in L∞(Gˆ) L∞(??̂) satisfying an extra mild condition. Exploiting this correspondence, we give examples of group-like projections which are not preserved by the scaling group.