A turbine assembly (18,218,318,418,518,618) for use with a gas turbine engine, the turbine assembly comprising a bladed wheel assembly (22,222,322,422,520,620) adapted to interact with gases flowing through a gas path of the gas turbine engine such that the gases push the bladed wheel assembly to rotate about an axis during use of the turbine assembly, the bladed wheel assembly including a disk (38,338,438,538,638) arranged around the axis and a plurality of blades that extend radially from the disk, a vane assembly (24,224,324,424,524,624) located upstream of the bladed wheel assembly and adapted to direct the gases at the bladed wheel assembly, the vane assembly being fixed relative to the axis and including a vane (66,266,366,466) and an inner support (70,270,370,470), the vane comprises a ceramic matrix composite material, the vane includes an outer platform, an inner platform (76,276,476) spaced apart radially from the outer platform relative to an axis, and an airfoil (78,278) that extends radially between the outer platform and the inner platform, and the inner support located radially inward of the inner platform and coupled with the vane, and an inner seal (26,226,326,426,526,626) engaged with the inner support and coupled with the disk of the bladed wheel assembly for rotation with the disk about the axis to block gases from passing between the inner seal and the vane assembly during use of the turbine assembly, the inner seal includes a radially and circumferentially extending seal body (30,330,430,530,630) fastened with the disk for rotation with the disk, a rub band (32,332,432,532,632) coupled to a radial outer end (36,336,436,536,636) of the seal body and engaged with the inner support to seal between the rub band and the inner support, and a mount ring (34,334,434,534,634) that extends axially aft and radially inward from the rub band, wherein the mount ring is interlocked with the disk to form a bayonet fitting (42,342) with the disk that blocks axial movement of the mount ring away from the disk and that transmits a portion of the force loads caused by rotation of the inner seal to the disk to reduce a magnitude of the force loads carried by the seal body, and characterized in the following: the rub band includes a hoop, a plurality of forward fins that extend radially outward from the hoop, and a plurality of aft fins that extend radially outward from the hoop, the hoop extends circumferentially around the axis and coupled with a radial terminal end of the seal body, the plurality of aft fins are spaced apart axially from the plurality of forward fins to define an annular chamber therebetween, and the hoop is formed to define a plurality of holes that extends radially through the hoop and opens into the annular chamber. a bladed wheel assembly (22,222,322,422,520,620) adapted to interact with gases flowing through a gas path of the gas turbine engine such that the gases push the bladed wheel assembly to rotate about an axis during use of the turbine assembly, the bladed wheel assembly including a disk (38,338,438,538,638) arranged around the axis and a plurality of blades that extend radially from the disk, a vane assembly (24,224,324,424,524,624) located upstream of the bladed wheel assembly and adapted to direct the gases at the bladed wheel assembly, the vane assembly being fixed relative to the axis and including a vane (66,266,366,466) and an inner support (70,270,370,470), the vane comprises a ceramic matrix composite material, the vane includes an outer platform, an inner platform (76,276,476) spaced apart radially from the outer platform relative to an axis, and an airfoil (78,278) that extends radially between the outer platform and the inner platform, and the inner support located radially inward of the inner platform and coupled with the vane, and an inner seal (26,226,326,426,526,626) engaged with the inner support and coupled with the disk of the bladed wheel assembly for rotation with the disk about the axis to block gases from passing between the inner seal and the vane assembly during use of the turbine assembly, the inner seal includes a radially and circumferentially extending seal body (30,330,430,530,630) fastened with the disk for rotation with the disk, a rub band (32,332,432,532,632) coupled to a radial outer end (36,336,436,536,636) of the seal body and engaged with the inner support to seal between the rub band and the inner support, and a mount ring (34,334,434,534,634) that extends axially aft and radially inward from the rub band, wherein the mount ring is interlocked with the disk to form a bayonet fitting (42,342) with the disk that blocks axial movement of the mount ring away from the disk and that transmits a portion of the force loads caused by rotation of the inner seal to the disk to reduce a magnitude of the force loads carried by the seal body, and characterized in the following: the rub band includes a hoop, a plurality of forward fins that extend radially outward from the hoop, and a plurality of aft fins that extend radially outward from the hoop, the hoop extends circumferentially around the axis and coupled with a radial terminal end of the seal body, the plurality of aft fins are spaced apart axially from the plurality of forward fins to define an annular chamber therebetween, and the hoop is formed to define a plurality of holes that extends radially through the hoop and opens into the annular chamber.