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Representations of Hermitian Commutative *-Algebras by Unbounded Operators

Marco Thill

Abstract

We give a spectral theorem for unital representations of Hermitian commutative unital *-algebras by possibly unbounded operators in a pre-Hilbert space. A better result is known for the case in which the *-algebra is countably generated.

Representations of Hermitian Commutative *-Algebras by Unbounded Operators

Abstract

We give a spectral theorem for unital representations of Hermitian commutative unital *-algebras by possibly unbounded operators in a pre-Hilbert space. A better result is known for the case in which the *-algebra is countably generated.

Paper Structure

This paper contains 14 sections, 17 theorems, 34 equations.

Table of Contents

  1. Statement of the Main Result
  2. Terminology and Notation
  3. Representations
  4. Unitisation and Spectrum
  5. Homomorphisms and Spectra
  6. Hermitian 9047073-Algebras
  7. Spectral Measures
  8. Commutativity for Unbounded Operators
  9. Reminders and Preliminaries
  10. The Cayley Transformation in Algebras
  11. The Archetypal Spectral Theorem
  12. A Sufficient Criterion for Essential Normality
  13. Proof of the Main Result
  14. Appendix: Derivation of Some Known Consequences

Key Result

Theorem 1

Let $\pi$ be a unital representation of a Hermitian commutative unital $\ast$-algebra $A$ on a pre-Hilbert space $H \neq \{ 0 \}$. The operators $\pi(a)$$(a \in A)$ are essentially normal in the completion. There is a spectral measure $P$ on a subset of $\Delta^*(A)$, acting on the completion of $H$ We then have that

Theorems & Definitions (33)

  • Theorem 1
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • Corollary 7
  • ...and 23 more