2006 Fiscal Year Final Research Report Summary
Research on algebraic equations with coefficients in Banach algebras
| Project/Area Number |
17540151
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Basic analysis
|
|---|
| Research Institution | Niigata University |
|---|
Principal Investigator |
HATORI Osamu Niigata University, Institute of Science and Technology, Professor, 自然科学系, 教授 (70156363)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
MIURA Takeshi Yamagata University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (90333989)
TAKAHASI Sin-Ei Yamagata University, Faculty of Engineering, Professor, 工学部, 教授 (50007762)
IZHUCHI Kei Ji Niigata University, Institute of Science and Technology, Professor, 自然科学系, 教授 (80120963)
SAITO Kichi-Suke Niigata University, Institute of Science and Technology, Professor, 自然科学系, 教授 (30018949)
WATANABE Keiichi Niigata University, Institute of Science and Technology, Associate Professor, 自然科学系, 助教授 (50210894)
|
|---|
| Project Period (FY) |
2005 – 2006
|
| Keywords | commutative Banach algebras / algebraic equations / spectrum |
|---|
| Research Abstract |
Algebraic structure of commutative Banach algebra and thier maximal ideal spaces are related each other. To investigate the spectrum of the algebra element is important for the study of algebraic structure including of algebraically closedness of the Banach algebra C(X) of all complex-valued continuous functions on a given compact Hausdorff space X. Thus it is important to study spectrum-preserving maps between two Banach algebras for the research on algebraic structure of C(X). We have proved several results on spectrum-preserving maps between commutative Banach algebras without assuming linearity on the maps. We investigate the existence of n-th root in the abstract sense, which was introduced by Karahanjan, for COO, and we proved the condition is equivalent for algebraically closedness if X is locally connected or first countable. We show an example of the map T which is not linear nor multiplicative while the norm of TfTg coincides with the norm of fg for every element f and g in C(X) for the case where C(X) is square-root closed at the first time, and Finally reduce the condition on the square-root closedness. We study the maps T such that the norm of TfTg+1 coincides with the norm of fg+1 for every f and g and show that T is linear and multiplicative in certain cases. This research is not complete but give new and interesting feature. We prove a Takesaki duality theorem for Arveson's spectral subspaces if a locally compact abelian group acts on the von Neumann algebra.
|
