Extending Ring Topologies

Abstract

Throughout this abstract, R denotes a compact (Hausdorff) topological ring. The authors extend to ring topologies on R which are totally bounded or even pseudocompact; a principal tool is the Bohr compactification of a topological ring. They show inter alia: If the Jacobson radical J R of R satisfies w R/J R > ω then there is a pseudocompact ring topology on R strictly finer than ; if in addition w R = w R/J R = α with cf α > ω then there are exactly 2 2 R -many such topologies. The ring R is said to be a van der Waerden ring if is the only totally bounded ring topology on R. Theorem 4.13 asserts that if R is semisimple, then R is a van der Waerden ring if and only if in the Kaplansky representation R = n<ω R n α n of R as a product of full matrix rings over finite fields each α n is finite. Other classes of van der Waerden rings are constructed, and it is shown that there are non-compact totally bounded rings S such that is the only totally bounded ring topology on S.

A version of van der Waerden's theorem and a proof of Mishchenko's conjecture on homomorphisms of locally compact groups

“…π K,N is continuous on the commutator subgroup of the component (G/N ) 0 of the identity of G/N by Corollary 8. However, the image of G 0 in G/N is dense in (G/N ) 0 by Theorem 7.12 in [8], and therefore this image coincides with (G/N ) 0 because N is compact. Since the commutator subgroup of G 0 is continuously mapped onto the commutator subgroup of (G/N ) 0 = G 0 /(G 0 ∩N ) under the canonical homomorphism ν : G → G/N , it follows that the original homomorphism π defines a continuous homomorphism of the commutator subgroup of G 0 into all possible quotient Lie groups H/K, where K is a compact normal subgroup of H. It is clear from the construction that this family of continuous homomorphisms commutes with the canonical homomorphisms of the quotient Lie groups (the Lie group maps of the form H/K 1 → H/K 2 for K 1 ⊂ K 2 , where K 1 and K 2 are compact normal subgroups of H), and thus a continuous homomorphism of the commutator subgroup of G into H is well defined.…”

“…In this 'unitary' form, the theorem strongly influenced the theory of finite-dimensional representations of semisimple Lie groups (which took shape at this time) and evoked an extensive literature. In particular, this included the 'converse' theorem (on the semisimplicity of compact Lie groups each of whose finite-dimensional locally bounded representations is automatically continuous [2]; we recall that there are many, namely, 2 2 ℵ 0 , discontinuous unitary characters on the real line R and on the one-dimensional torus T (in connection with the existence of Hamel bases, see, for example, [3], (25.6)), and therefore the characters, representations and homomorphisms of commutative and soluble Lie groups need not be continuous) and established relationships between van der Waerden's theorem and properties of Bohr compactifications of topological groups [4]- [7], which led to the introduction and study of classes of so-called van der Waerden groups and algebras (this attribute has several meanings, see [5] and [8]- [10]: for instance, it can mean the automatic continuity of morphisms into compact objects or the automatic continuity of all automorphisms). More than 70 years after the paper [1] was published, the theorem still enters monographs [11] and surveys [12] in diverse forms.…”

Connected locally compact groups: The Hochschild kernel and faithfulness of locally bounded finite-dimensional representations

“…In this case the image of B in G/N , where N is a compact normal divisor in G contained in V , is a nontrivial compact subgroup of the closure in H of the image of rad (G ) in H. On the other hand, since the normal divisor N is compact, it follows from Corollary 3.8 in [7] in combination with §3 in [14] that the image in H of rad (G ) coincides with rad (H ) since the radicals in G and H are automatically closed. Thus, if B is nontrivial, then the Hochschild kernel urk(H) of any quotient Lie group of the form H = G/N is nontrivial since it contains the nontrivial image of B in H. Thus B ⊂ urk(G) and the family of continuous finite-dimensional linear representations of G does not separate points of the group.…”

Contributions to the Bohr topology by W.W. Comfort

“…Wis, H. Szambien and the second author used in [23] Bohr compactifications of topological rings and a modification of Theorem 3.2 (see [23], Theorem 2.6) to study the number of pseudocompact ring topologies on compact rings. Let α be a cardinal considered as a (limit) ordinal.…”