On the decomposition of states of some<sup>∗</sup>-algebras

Mathot, Françoise

doi:10.2140/pjm.1980.90.411

Cited by 6 publications

(6 citation statements)

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“…This last result about decomposition of Op*-algebras generalizes the result we got in [18] where we made the stronger assumption that 21 was separable for some quasi-uniform topology [22] which is actually a finer topology than the j*-topology we consider here. 5o6.…”

Section: 3 Topology On 2t 0 (^)supporting

confidence: 81%

“…The proof of 1) and 2) is exactly the same as what we did in [18] for Op*-algebras, excepted that we consider here the j*-topology instead of a quasi-uniform one. This proof consists in showing that the maximal extension [15] ft on ^C^ possesses the same properties as TT on ^c^f, essentially that 2f is metrizable (this is because SI 0 is dominating) and that ?r(2I) is separable in the /"-topology (see [18] for the details). Finally, we apply the decomposition of the previous sections to #( §!)…”

Section: 3 Topology On 2t 0 (^)mentioning

confidence: 84%

“…For Op*-algebras we considered in [18] a quasi-uniform topology [4], [22] and then proved the result that separable Op*-algebras could be decomposed.…”

Section: Construction Of a Common Domain @ (X) For A Countable Set Ofmentioning

confidence: 99%

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Integral Decomposition of Partial *-Algebras of Closed Operators

Mathot¹

1986

Publ. Res. Inst. Math. Sci.

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Section: 3 Topology On 2t 0 (^)supporting

confidence: 81%

Section: 3 Topology On 2t 0 (^)mentioning

confidence: 84%

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Integral Decomposition of Partial *-Algebras of Closed Operators

Mathot¹

1986

Publ. Res. Inst. Math. Sci.

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“…Finally, connections with the work of Mathot [14] on the disintegration of representations is discussed. Now let / = rKfyl/e P(A)}.…”

Section: Representation (It D(tt) H) Of a »-Algebra A On A Hubert Smentioning

confidence: 95%

An irreducible representation of a symmetric star algebra is bounded

Bhatt¹

1985

Trans. Amer. Math. Soc.

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Abstract.A »-algebra A is called symmetric if (1 + x*x) is invertible in A for each x in A. An irreducible hermitian representation of a symmetric »-algebra A maps A onto an algebra of bounded operators.1. Theorem. Let A be a symmetric * -algebra with identity 1. Let (it, D(ir), H) be a closed *-representation of A on a Hubert space H. If the only tr-invariant self adjoint subspaces of D(tt) are (0) and D(w), then m is a bounded representation. Corollary.Every closed (algebraically) irreducible *-representation of a symmetric * -algebra is bounded.The purpose of this paper is to prove the above theorem. A * -algebra A is a linear associative algebra with identity 1 over the complex field C such that A admits an involution a & A -» a* e A satisfying the usual axioms. If (1 + a*a)~x exists in A, for every a e A, then A is called symmetric.A The analysis of the representations of abstract »-algebras has been motivated in Quantum Field Theory to avoid starting with (and staying within) a specific Hilbert space (the Fock space) scheme and rather to stress that the basic objects of the theory are observables considered as purely algebraic quantities forming a * -algebra. Realizations of these algebraic objects as Hilbert space operators naturally lead to unbounded representations defined above. In [15], R. T. Powers developed a basic representation theory for * -algebras admitting unbounded observables. Representations of symmetric »-algebras have been investigated in [11]. On the other hand,

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“…Mathot [14,Theorem 3.2 and §3.3] has proved that if A is a separable locally convex »-algebra dominated in a given selfadjoint strongly continuous representation (77, D(tt), H) by a countable subset B (in the sense that given an a g A, there are b G B and k < 00 such that ||ir(a)x|| < /c||77(¿>)x|¡ for all x), then over a compact space Z with a positive measure ¡u, (77, D(tt), H) can be disintegrated as H = (9 H(t)dn{t), Z>(77) = P D(t)dfi(t), 77= P 77,a>(0…”

Section: Corollarymentioning

confidence: 99%