TY - JOUR
T1 - Approximately invertible elements in non-unital normed algebras
AU - Esmeral, Kevin
AU - Feichtinger, Hans G.
AU - Hutník, Ondrej
AU - Maximenko, Egor A.
N1 - Publisher Copyright:
© 2023 Elsevier Inc.
PY - 2023/7/1
Y1 - 2023/7/1
N2 - We introduce the concept of approximately invertible elements in non-unital normed algebras which is, on one side, a natural generalization of invertibility when having approximate identities at hand, and, on the other side, it is a direct extension of topological invertibility to non-unital algebras. Basic observations relate approximate invertibility with concepts of topological divisors of zero and density of (modular) ideals. We exemplify approximate invertibility in the group algebra, Wiener algebras, and operator ideals. For Wiener algebras with approximate identities (in particular, for the Fourier image of the convolution algebra), the approximate invertibility of an algebra element is equivalent to the property that it does not vanish. We also study approximate invertibility and its deeper connection with the Gelfand and representation theory in non-unital abelian Banach algebras as well as abelian and non-abelian C*-algebras.
AB - We introduce the concept of approximately invertible elements in non-unital normed algebras which is, on one side, a natural generalization of invertibility when having approximate identities at hand, and, on the other side, it is a direct extension of topological invertibility to non-unital algebras. Basic observations relate approximate invertibility with concepts of topological divisors of zero and density of (modular) ideals. We exemplify approximate invertibility in the group algebra, Wiener algebras, and operator ideals. For Wiener algebras with approximate identities (in particular, for the Fourier image of the convolution algebra), the approximate invertibility of an algebra element is equivalent to the property that it does not vanish. We also study approximate invertibility and its deeper connection with the Gelfand and representation theory in non-unital abelian Banach algebras as well as abelian and non-abelian C*-algebras.
KW - Approximate identity
KW - Approximate invertibility
KW - Maximal modular ideal
KW - Non-unital Banach algebra
KW - Non-unital topological algebra
KW - Principal ideal
UR - http://www.scopus.com/inward/record.url?scp=85149679643&partnerID=8YFLogxK
U2 - 10.1016/j.jmaa.2022.126986
DO - 10.1016/j.jmaa.2022.126986
M3 - Artículo
AN - SCOPUS:85149679643
SN - 0022-247X
VL - 523
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 1
M1 - 126986
ER -