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Dividing in the algebra of compact operators. (English) Zbl 1070.03019

The author studies Hilbert spaces by model-theoretic methods developed in stability theory. One of the main concepts in that theory is the notion of independence, which is technically defined in terms of dividing of a type over a set. In the context of Hilbert spaces these notions were studied by S. Buechler and the author in Ann. Pure Appl. Logic 128, No. 1–3, 75–101 (2004; Zbl 1047.03025)].
In the paper under review model theory of finite rank and compact operators in Hilbert spaces is developed. Finite rank operators are interpreted as imaginaries, equivalence classes of finite tuples for definable equivalence relations. It is shown that any compact operator is interdefinable with a collection of finite rank operators. The author studies independence and characterizes dividing for finite rank operators. He shows that the Hilbert space enlarged with the imaginaries for the finite rank operators has built-in canonical bases.

MSC:

03C45 Classification theory, stability, and related concepts in model theory
03C98 Applications of model theory
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
47B07 Linear operators defined by compactness properties

Citations:

Zbl 1047.03025
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References:

[1] Archive for Mathematical Logic
[2] DOI: 10.1142/S0219061303000297 · Zbl 1039.03032
[3] DOI: 10.1142/S0219061303000212 · Zbl 1028.03034
[4] Theory of Linear Operators in Hilbert Space, Vol I and Vol II 10 (1981)
[5] Annals of Pure and Applied Logic
[6] Models, Algebras, and Proofs (Bogotá, 1995) 203 pp 77– (1999)
[7] Analysis and Logic (2002)
[8] DOI: 10.1090/S0894-0347-02-00407-1 · Zbl 1010.03025
[9] Algebras and Operator Theory (1990)
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