C R $CR$ analysis via local uniform completion, a sharp maximum modulus principle and holomorphic extension

Using iterated uniform local completion, we introduce a notion of continuous $CR$ functions on locally closed subsets of reduced complex spaces, generalising both holomorphic functions and $CR$ functions on $CR$ submanifolds. Under additional assumptions of set-theoretical weak pseudo-concavity, we prove optimal maximum modulus principles for these functions, extending classical results for holomorphic functions and ordinary $CR$ functions. Restricting to real submanifolds (possibly with $CR$ singularities) of complex manifolds, we generalise results on holomorphic extension to full neighbourhoods known before only for $CR$ submanifolds. The article is concluded by a study of $CR$ singularities and explicit constructions of submanifolds on which the extension results are valid.