Lower bounds for homological dimensions of Banach algebras

Yu. V. Selivanov

Sbornik. Mathematics, Tome 198 (2007) no. 9, pp. 1351-1377

Résumé

Let $A$ be a commutative unital Banach algebra with infinite spectrum. Then by Helemskiǐ's global dimension theorem the global homological dimension of $A$ is strictly greater than one. This estimate has no analogue for abstract algebras or non-normable topological algebras. It is proved in the present paper that for every unital Banach algebra $B$ the global homological dimensions and the homological bidimensions of the Banach algebras $A\mathbin{\widehat{\otimes}}B$ and $B$ (assuming certain restrictions on $A$) are related by $\operatorname{dg}A\mathbin{\widehat{\otimes}}B\geqslant 2+\operatorname{dg}B$ and $\operatorname{db}A\mathbin{\widehat{\otimes}}B\geqslant 2+\operatorname{db}B$. Thus, a partial extension of Helemskiǐ's theorem to tensor products is obtained. Bibliography: 28 titles.