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THE STATE-SPACE OF THE LATTICE OF ORTHOGONALLY CLOSED SUBSPACES

Published online by Cambridge University Press:  31 January 2005

E. CHETCUTI  and
A. DVUREČENSKIJ
E. CHETCUTI
Affiliation:
Mathematical Institute Slovak Academy of Sciences Stefanikova, SK-81473 Bratislava e-mail: chetcuti@mat.savba.sk, dvurecen@mat.savba.sk
A. DVUREČENSKIJ
Affiliation:
Mathematical Institute Slovak Academy of Sciences Stefanikova, SK-81473 Bratislava e-mail: chetcuti@mat.savba.sk, dvurecen@mat.savba.sk
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Abstract

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The notion of a strongly dense inner product space is introduced and it is shown that, for such an incomplete space $S$ (in particular, for each incomplete hyperplane of a Hilbert space), the system $F(S)$ of all orthogonally closed subspaces of $S$ is not stateless, and the state-space of $F(S)$ is affinely homeomorphic to the face consisting of the free states on the projection lattice corresponding to the completion of $S$. The homeomorphism is determined by the extension of the states. In particular, when $S$ is complex, the state-space of $F(S)$ is affinely homeomorphic to the state-space of the Calkin algebra associated with $\skew3\overline S$.

Keywords

Primary 46C0546L30Secondary 03G12
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 47 , Issue 1 , January 2005 , pp. 213 - 220
Copyright
2005 Glasgow Mathematical Journal Trust