The spatial form of antiautomorphisms of von Neumann algebras (Q1173652)

An antiautomorphism \(\alpha\) of a von Neumann algebra \(M\) is called central if it leaves the center of \(M\) elementwise fixed. An involution is an antiautomorphism whose square is the identity. In this paper the author investigates when an automorphism is spatial, i.e., of the form \(x\in M\to w^*xw\) with \(w\) a conjugate linear isometry of a prescribed type. We write \(e\sim f\), for two projections in \(M\), if there is a partial isometry \(v\) in \(M\) satisfying \(v^*v=e\) and \(vv^*=f\). We have Theorem 1. Let \(M\) be a von Neumann algebra and \(\alpha\) an antiautomorphism such that \(\alpha(e)\sim e\) for all projections \(e\in M\). Then \(\alpha\) is spatial. We also have Theorem 2. Let \(M\) be a von Neumann algebra with no direct summand of type \(II_ \infty\) with finite commutant. Then each central antiautomorphism of \(M\) is spatial. Theorem 3. Let \(M\) be a von Neumann algebra and \(\alpha\) a periodic central antiautomorphism. Then \(\alpha\) is spatial. Furthermore, if each normal state on \(M\) is a vector state, then there exists a conjugate linear isometry \(w\) such that \(\alpha(x)=w^*xw\) with \(w^{2n}=1\), where \(2n\) is the period of \(\alpha\). Recall that a conjugation is a conjugate linear isometry \(J\) such that \(J^ 2=1\). Then we have Theorem 4. Let \(M\) be a von Neumann algebra whose commutant has no direct summand of type \(I_ n\) with \(n\) and odd integer. If \(\alpha\) is a central involution on \(M\) then there exists a conjugation \(J\) such that \(\alpha(x)=JxJ^*\), \(x\) in \(M\). The paper concludes with an example which shows that Theorem 4 is false if the commutant of \(M\) is of type \(I_ n\) with \(n\) odd.