C*-Isomorphisms Associated with Two Projections on a Hilbert C*-Module

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 $${\cal L}(H)$$ , where $${\cal L}(H)$$ denotes the set of all adjointable operators on H. The C*-subalgebras of $${\cal L}(H)$$ 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 $$(\tilde \pi ,X)$$ of i(P, Q, H) such that $$\matrix{{\tilde \pi (P - P \wedge Q) = \pi (P) - \pi (P) \wedge \pi (Q),} \hfill \cr {\tilde \pi (Q - P \wedge Q) = \pi (Q) - \pi (P) \wedge \pi (Q).} \hfill \cr } $$ When (P, Q) is semi-harmonious, that is, $$\overline {{\cal R}(P + Q)} $$ and $$\overline {{\cal R}(2I - P - Q)} $$ 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 $${\cal L}(H)$$ . 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.