When is multiplication in a Banach algebra open?

0480794 - MÚ 2019 RIV US eng J - Journal Article
Draga, Szymon - Kania, Tomasz
When is multiplication in a Banach algebra open?
Linear Algebra and Its Applications. Roč. 538, 1 February (2018), s. 149-165. ISSN 0024-3795. E-ISSN 1873-1856
R&D Projects: GA ČR GF16-34860L
Institutional support: RVO:67985840
Keywords : Banach algebra * open mapping * uniformly open map
OECD category: Pure mathematics
Impact factor: 0.977, year: 2018
http://www.sciencedirect.com/science/article/pii/S0024379517305761?via%3Dihub

We develop the theory of Banach algebras whose multiplication (regarded as a bilinear map) is open. We demonstrate that such algebras must have topological stable rank 1, however the latter condition is strictly weaker and implies only that products of non-empty open sets have non-empty interior. We then investigate openness of convolution in semigroup algebras resolving in the negative a problem of whether convolution in ...1(N0) is open. By appealing to ultraproduct techniques, we demonstrate that neither in ...1(Z) nor in ...1(Q) convolution is uniformly open. The problem of openness of multiplication in Banach algebras of bounded operators on Banach spaces and their Calkin algebras is also discussed.
Permanent Link: http://hdl.handle.net/11104/0276483