$SU$-linear operations and $c_1$-spherical bordism theory

The $c_1$-spherical bordism theory $W^*$ is an intermediate theory between complex and $SU$-bordisms and it plays an important role in calculations of the $SU$-bordism coefficient ring. There is no natural choice of multiplication on the theory $W^*$ (since a direct product of two $c_1$-spherical manifolds doesn't have to be $c_1$-spherical). However, it turns out that the theory $W^*$ is a direct summand in complex cobordism theory $MU^*$, and one can define a multiplication on $W^*$ via projections. I will talk about $SU$-linear projections and multiplications on the theory $W^*$, as well as complex orientations of this theory, corresponding formal group laws and its Landweber exactness.