Research of a lifting theorem of uniform algebras and analytic operator algebras
Project/Area Number 
63540082

Research Category 
GrantinAid for General Scientific Research (C)

Allocation Type  Singleyear Grants 
Research Field 
代数学・幾何学
代数学・幾何学

Research Institution  Hokkaido University 
Principal Investigator 
NAKAZI Takahiko Hokkaido University, Faculty of Science, Associate Professor, 理学部, 助教授 (30002174)

CoInvestigator(Kenkyūbuntansha) 
HAYASHI Mikihiro Hokkaido University, Faculty of Science, Associate Professor, 理学部, 助教授 (40007828)
KATUMATA Osamu Hokkaido University, Faculty of Science, Associate Professor, 理学部, 助教授 (40032825)
INOUE Junji Hokkaido University, Faculty of Science, Professor, 理学部, 教授 (40000856)
KOSHI Shozo Hokkaido University, Faculty of Science, Professor, 理学部, 教授 (40032792)

Project Period (FY) 
1988 – 1989

Project Status 
Completed (Fiscal Year 1989)

Budget Amount *help 
¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1989: ¥700,000 (Direct Cost: ¥700,000)

Keywords  2 x 2 operator matrix / Lifting theorem / Uniform algebra / Analytic operator algebra / Hardy space / Hibert operator / Weighted norm inequality / Hankel operator / Idardy空間 / Idilbert作用素 / Idankel作用素 
Research Abstract 
Let K be a Hilbert space, B a von Neumann algebra on K and A a weakly closed subalgebra of B. T = (T_<ij>) denotes 2 x 2 operator matrices on K <symmetry> K where T_<ij> epsilon B (i, j = 1, 2), T_<11> <greater than or equal> 0, T_<22> <greater than or equal> 0 and T^*_ = T_<12>. [B] denotes the set of such operator matrices T and [A]_0 denotes the subset of [B] such that T_<12> epsilon A and T_<11> = T_<22> = 0. Let F be the subset of lat A, the lattice of all Ainvariant projections. We studied whether if T epsilon [B] is positive on PK <symmetry> (1  P)K for every P epsilon F then there exists T^^ in T + [A]_0 which is positive on K <symmetry> K. We say that (B, A, F). has a lifting property when there exists such a T^^. When B is a commutative von Neumann algebra L^", if A is an abstract Hardy space H^", we could show that {B = L^", A = H^", F} has a lifting property even if F is very small. Using this result we could generalize a theorem of norms of Hankel operators and two theorems of weighted norm inequalities in the disc algebra to those in a general uniform algebra. When B is noncommutative, if A and F satisfy two conditions which are satisfied in many important examples, we could that {B, A, F} has a lifting property. Using the distanck formula of Arveson or the theorem of Neharils type. We could generalize two theorems of weighted norm inequalities in the vectorvalued Hardy space to those in nest algebras and noncommutative Hardy spaces.

Report
(3 results)
Research Products
(12 results)