Algebraic crossed product by an automorphism

Core idea

For a unital $*$ -algebra $A$ and $*$ -automorphism $\alpha$ , the algebraic crossed product $A\rtimes_{\alpha,\mathrm{alg}}\mathbb Z$ consists of finite formal sums $\sum_n a_nu^n$ , with

(au^m)(bu^n)=a\alpha^m(b)u^{m+n},\qquad (au^m)^*=\alpha^{-m}(a^*)u^{-m}.

Addition and scalar multiplication are coefficientwise. The formal unitary $u$ obeys $uau^*=\alpha(a)$ .

Noncommutativity

The commutator is $ua-au=(\alpha(a)-a)u$ . Thus a commutative function algebra and the abelian group $\mathbb Z$ can produce a noncommutative algebra through a nontrivial action.

Completions

The algebraic object has no specified complete operator norm. Its operator-algebra completions include $C^*$ -crossed products and, for normal actions, von Neumann crossed products. The formulas are the discrete $\mathbb Z$ case of the crossed-product convolution algebra.