We study the relations between abelian groups B and C that every universal (resp. universal-existential) sentence true in B is also true in C , and give algebraic criteria for these relations to hold. As a consequence we characterize the inductive complete theories of abelian groups and prove that they are exactly the model-complete theories.

Infinite Abelian groups

Some Model Theory of Abelian Groups

Abstract Computability theory is used to evaluate the complexity of classifying various kinds of Lebesgue spaces and associated isometric isomorphism problems.

Additional excerpts

On the complexity of classifying Lebesgue spaces

The double commutant of an algebraic operator on a complex Hilbert space is equal to the algebra (with identity) generated by that operator. Introduction. Let fl be an algebra over some field, and let S? be a subset of t. The commutant of 5 in W is the set S?' of all elements of Qi which commute with every element of Yf. The double commutant of Y? in W is the commutant of Y?' in W. If 9Y consists of a single element s, we shall speak of the commutant and double commutant of s rather than of {s}. Let A be a linear transformation on a vector space. We say that A is algebraic if there exists a polynomial p such that p(A) = 0. In this paper we shall prove that if X' is a complex Hilbert space, and A belonging to B(Q#) (the algebra of all bounded linear' transformations on JX) is algebraic, then the double commutant of A in B(,#') is equal to the algebra generated by A. This generalizes the well-known analogous result for finite-dimensional spaces (see [2, p. 113]), since any linear transformation on a finitedimensional vector space is algebraic. It is already known (see [3 p. 72]) that if T is an algebraic linear transformation on a vector space I then the double commutant of T in L('f) (the algebra of all linear transformations on f) is equal to the algebra generated by T. This result does not imply our result for B(JV') however, since in general L(X#) is far larger than B(Q#). In the forthcoming the word operator will mean bounded linear transformation, and Hilbert space will mean complex Hilbert space. 1. Notation. If A is an algebraic operator on a Hilbert space JX', we denote by WA the algebra (with identity) generated by A in B(J). REMARK. In discussing operators on Hilbert space one often speaks of the weakly closed algebras which they generate. Since the algebra generated by an algebraic operator is finite dimensional it is closed in all of the usual Received by the editors September 29, 1971. AMS 1970 subject classifications. Primary 46L15, 47B99.

Infinite Abelian groups

Citations

Some Model Theory of Abelian Groups

On the complexity of classifying Lebesgue spaces

Double commutants of algebraic operators

Indecomposable modules over Nagata valuation domains

Groups elementarily equivalent to a free nilpotent group of finite rank

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