  Non-commutativity and non-associativity are quite natural in string theory.
For open strings it appear due to the presence of non-vanishing background
two-form in the world volume of Dirichlet brane, while in closed string theory
the flux compactifications with non-vanishing three-form also lead to
non-geometric backgrounds. In this paper, working in the framework of
deformation quantization, we study the violation of associativity imposing the
condition that the associator of three elements should vanish whenever each two
of them are equal. The corresponding star products are called alternative and
satisfy an important for physical applications properties like the Moufang
identities, alternative identities, Artin's theorem, etc. The condition of
alternativity is invariant under the gauge transformations, just like it
happens in the associative case. The price to pay is the restriction on the
non-associative algebra which can be represented by the alternative star
product, it should satisfy the Malcev identity. The example of nontrivial
Malcev algebra is the algebra of imaginary octonions. For this case we
construct an explicit expression of the non-associative and alternative star
product. We also discuss the quantization of Malcev-Poisson algebras of general
form, study its properties and provide the lower order expression for the
alternative star product. To conclude we define the integration on the algebra
of the alternative star products and show that the integrated associator
vanishes.
