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| C*-Isomorphisms Associated with Two Projections on a Hilbert C*-Module* |
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Citation： |
Chunhong FU.C*-Isomorphisms Associated with Two Projections on a Hilbert C*-Module*[J].Chinese Annals of Mathematics B,2023,44(3):325~344 |
| Page view： 865
Net amount： 557 |
Authors： |
Chunhong FU; |
Foundation： |
National Natural Science Foundation of China (No. 11971136) and the Science and Technology Commission of Shanghai Municipality (No. 18590745200). |
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| Abstract： |
Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies C*-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple (P, Q, H) is said to be matched if H is a Hilbert C*-module, P and Q are projections on H such that their infimum P ∧ Q exists as an element of L(H), where L(H) denotes the set of all adjointable operators on H. The C*-subalgebras of L(H) generated by elements in {P - P ∧ Q, Q - P ∧ Q, I} and {P, Q, P ∧ Q, I} are denoted by i(P, Q, H) and o(P, Q, H), respectively. It is proved that each faithful representation (π, X) of o(P, Q, H) can induce a faithful representation (π, X e) of i(P, Q, H) such that e π(P - P ∧ Q) = π(P) - π(P) ∧ π(Q),eπ(Q - P ∧ Q) = π(Q) - π(P) ∧ π(Q).When (P, Q) is semi-harmonious, that is, R(P + Q) and R(2I - P - Q) are both orthogonally complemented in H, it is shown that i(P, Q, H) and i(I - Q, I - P, H) are unitarily equivalent via a unitary operator in L(H). A counterexample is constructed, which shows that the same may be not true when (P, Q) fails to be semi-harmonious. Likewise, a counterexample is constructed such that (P, Q) is semi-harmonious, whereas (P, I - Q) is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert C*-modules are also provided. |
Keywords： |
Hilbert C*-module, Projection, Orthogonal complementarity,C*-Isomorphism |
Classification： |
46L08, 47A05 |
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