Dependence of strict homological dimension of \(C(\Omega)\) on the topology of \(\Omega\)
Throughout the present article \(\Omega\) will be a compact Hausdorff space and \(C(\Omega)\) the Banach algebra with uniform norm and pointwise multiplication of the complex-valued functions that are continuous on this space. The dependence of the values of the homological dimensions of \(C(\Omega)\) on the topology of \(\Omega\) has long been a subject of study. Examples of compacta \(\Omega\) for which the global homological dimension of \(C(\Omega)\) is equal to two or is infinite and, of course, zero, if \(\Omega\) is finite have been constructed. It is known that the global homological dimension of \(C(\Omega)\), like that of any functional Banach algebra, cannot be equal to one. The value of the global homological dimension of \(C(\Omega)\) is not known even in the case in which \(\Omega\) is a line segment. The main reason for the difficulties that arise when attempting to compute the homological dimensions is due to the complex structure of the projective tensor product of the algebras \(C(\Omega)\) (called Varopoulus algebras). We will consider this problem in a somewhat different situation, that is, in the theory of so-called strict algebras (in other words, a theory with injective tensor product). In this case the computations are substantially simplified as a consequence of the natural apparatus of a tensor product of algebras \(C(\Omega)\). We will show that (except for the trivial case of finite \(\Omega\)), for metrizable spaces, and only for metrizable spaces, the strict bidimension (cohomological dimension) of \(C(\Omega)\) is equal to one.
Banach algebra with uniform norm and pointwise multiplication of the complex-valued functions that are continuous
scientific article; zbMATH DE number 771991