Journal of Operator Theory

Volume 36, Issue 2, Fall 1996 pp. 201-231.

Upper and lower multiplicity for irreducible representations of C*-algebras. II

Authors: R.J. Archbold (1) and J.S. Spielberg (2)
Author institution:(1) Department of Mathematical Sciences, University of Aberdeen, The Edward Wright Building, Dunbar Street, Aberdeen AB9 2TY, U.K.
(2) Department of Mathematics, Arizona State University, Tempe, Arizona 85287, U.S.A.

Summary: We use the notions of upper and lower multiplicity, $M_U(\pi)$ and $M_L(\pi)$, for an irreducible representation $\pi$ of a C*-algebra A to investigate some of the possible structure in point-strong limits of essentially irreducible representations of A on a large Hilbert space. In particular, this leads to a generalization of Gardner’s theorem on ‘the third definition’ of the topology on the spectrum $\hat A$ of A and also to new characterizations of $M_U(\pi)$ and $M_L(\pi)$. We also investigate the possible gap between $M_U(\pi)$ and $M_L(\pi)$ by introducing upper and lower multiplicities for $\pi$ relative to a net in $\hat A$.

Keywords: Multiplicity, spectrum, C*-algebra, lower semi-continuity, irreducible representation, pure states.

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