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 Gardnerâ€™s theorem on â€˜the third definitionâ€™ 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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