On Kelley's multiplicity function of an abelian von Neumann algebra

Year, pages: 2001, 598 - 605

Let $A$ be an abelian von Neumann algebra of operators on a Hilbert space $H$ and let $G(·)$ be its canonical spectral measure (see Definition 5) on the Borel subsets of its maximal ideal space $M$. By describing Kelley's multiplicity function $φ$ of $A$ in terms of the uniform multiplicity function of Halmos, the basic structure theorem of Kelley [KELLEY, J. L.: Commutative operator algebras, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 598–605] is deduced from the theory of orthogonal spectral representations applied to $G(·)$. When the commutant $A'$ is countably decomposable, $G(·)$ has CGS@-property in $H$ and in this case, $φ$ is also described in terms of the multiplicity functions $m_p$ and $m_c$ of $G(·)$ (see Definition 4).

Panchapagesan, T. 2001. On Kelley's multiplicity function of an abelian von Neumann algebra. In Mathematica Slovaca, vol. 51, no.5, pp. 598-605. 0139-9918.

Panchapagesan, T. (2001). On Kelley's multiplicity function of an abelian von Neumann algebra. Mathematica Slovaca, 51(5), 598-605. 0139-9918.