Kurzweil and McShane product integration in Banach algebras

We study product integrability of functions with values in unital Banach algebras. The product integrals are understood in the sense of Kurzweil, McShane, or Riemann. In particular, we introduce new concepts of strong Kurzweil and McShane product integrals, and investigate their properties. We also provide necessary and sufficient conditions for product integrability of functions with countably many discontinuities.