J. Synnatzschke
Zur Erzeugung von Algebren durch Unteralgebren mit Quadrat Null
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 50, 53-64 (1997)
16S15 Finite generation, finite presentability, normal forms
08A05 Structure theory
Abstract: For a simple algebra ${\cal A}$ with unit, a condition depending on $n \in
\N$ is given being equivalent to the possibility to generate ${\cal A}$ by $n$
subalgebras having square zero and all except one of them having dimension 1.
As a corollary under this condition, the algebra can be generated by two
subalgebras with square zero such that one of them has dimension 1 or 2 in
dependence on whether $n$ is even or odd. In the case of the algebra ${\cal
B}(E)$ of all continuous linear operators on a Banach space $E$, the
condition is fulfilled if and only if $E$ is the $n$th power $E = E_0^n$ of a Banach
space $E_0$. This way by elementary considerations, not only a problem
considered by W.\,{\. Z}elazko and afterwards by P.\,{\v S}emrl is finished
but also completely extended to arbitrary simple algebras with unit.

Keywords: Generation of algebras by subalgebras, subalgebras with square zero, commutative subalgebras, operators in powers of Banach spaces