Graded C*-algebras and twisted C*-tensor products

We present an explicit construction of a twisted tensor product of a pair of graded C∗-algebras. The grading is inherited by actions of compact abelian groups on such algebras, and a fixed bicharacter is responsible for the arising involutive algebra struc ture (i.e. commutation rules and ∗-operation). This generalise the usual tensor product associated to the action of the trivial group (and, consequently, the trivial bicharacter), and the Fermi case, the action of which is associated to the group Z2 and the involved bicharacter is the Fermi one describing the Canonical Anti-Commutation Relations. Weinvestigate its completions w.r.t. the minimal and maximal C∗-norms in detail. In particular, we prove that the minimal norm is indeed minimal among the compatible ones, that is those for which the product action of the involved groups on the twisted tensor product extends to a continuous one. We also show that the minimal norm is spatial as happens for the usual tensor product.