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A consistent description of gauge theories on coordinate dependent non‐commutative (NC) space‐time is a long‐standing problem with a number of solutions, none of which is free from criticism. In this work, we discuss the approach proposed in Blumenhagen et al. [1] based on the conjecture that any consistent gauge theory can be described in terms of the    L  ∞   ‐structure. Starting with a well‐defined commutative gauge theory, we represent it, together with the non‐commutative deformation, as a part of a bigger    L  ∞   ‐algebra by setting some initial brackets ℓ 1 , ℓ 2 , etc. Then, solving the    L  ∞   ‐relations we determine the missing brackets    ℓ  n    and close the    L  ∞   ‐algebra defining the NC gauge theory which reproduces in the commutative limit the original one. We provide the recurrence relations for the construction of the pure gauge algebra    L  ∞   gauge    , using which we find an explicit form of the NC    su  (  2  )   ‐like and non‐associative octonionic‐like deformations of the Abelian gauge transformations. The construction of the    L  ∞   full    ‐algebra describing the dynamics is discussed using the example of the NC Chern–Simons theory. The obtained equations of motion are non‐Lagrangian, which indicates the difference between our approach and the previous ones.

L ∞ ‐Bootstrap Approach to Non‐Commutative Gauge Theories