Cohen–Hewitt factorization theorem

In mathematics, the Cohen–Hewitt factorization theorem states that if ${\displaystyle V}${\displaystyle V} is a left module over a Banach algebra ${\displaystyle B}${\displaystyle B} with a left approximate unit ${\displaystyle (u_{i})_{i\in I}}${\displaystyle (u_{i})_{i\in I}}, then an element ${\displaystyle v}${\displaystyle v} of ${\displaystyle V}${\displaystyle V} can be factorized as a product ${\displaystyle v=bw}${\displaystyle v=bw} (for some ${\displaystyle b\in B}${\displaystyle b\in B} and ${\displaystyle w\in V}${\displaystyle w\in V}) whenever ${\displaystyle \displaystyle \lim _{i\in I}u_{i}v=v}${\displaystyle \displaystyle \lim _{i\in I}u_{i}v=v}. The theorem was introduced by Paul Cohen (1959) and Edwin Hewitt (1964).

• Cohen, Paul J. (1959), "Factorization in group algebras", Duke Mathematical Journal , 26 (2): 199–205, doi:10.1215/s0012-7094-59-02620-1, MR 0104982
• Hewitt, Edwin (1964), "The ranges of certain convolution operators", Mathematica Scandinavica, 15: 147–155, doi:10.7146/math.scand.a-10738 , MR 0187016
• Mortini, Raymond (May 2019), "A Simpler Proof of Cohen's Factorization Theorem", The American Mathematical Monthly, 126 (5): 459–463

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