Discussiones Mathematicae - General Algebra and Applications

Solution to a problem of FitzGerald

Semigroup Forum ◽

10.1007/s00233-021-10170-5 ◽

2021 ◽

Author(s):

Jens Hemelaer ◽

Morgan Rogers

Keyword(s):

Affirmative Answer ◽

General Algebra

AbstractFitzGerald identified four conditions (RI), (UR), (RI*) and (UR*) that are necessarily satisfied by an algebra if its monoid of endomorphisms has commuting idempotents. We show that these conditions are not sufficient, by giving an example of an algebra satisfying the four properties, such that its monoid of endomorphisms does not have commuting idempotents. This settles a problem presented by FitzGerald at the Conference and Workshop on General Algebra and Its Applications in 2013 and more recently at the workshop NCS 2018. After giving the counterexample, we show that the properties (UR), (RI*) and (UR*) depend only on the monoid of endomorphisms of the algebra, and that the counterexample we gave is in some sense the easiest possible. Finally, we list some categories in which FitzGerald’s question has an affirmative answer.

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Simultaneous Triangularization of Algebras of Polynomially Compact Operators

Canadian Mathematical Bulletin ◽

10.4153/cmb-1991-042-6 ◽

1991 ◽

Vol 34 (2) ◽

pp. 260-264 ◽

Cited By ~ 1

Author(s):

M. Radjabalipour

Keyword(s):

Hilbert Space ◽

Jacobson Radical ◽

Compact Operators ◽

General Algebra ◽

Closed Algebra ◽

Quasinilpotent Operators ◽

Simultaneous Triangularization

AbstractIf A is a norm closed algebra of compact operators on a Hilbert space and if its Jacobson radical J(A) consists of all quasinilpotent operators in A then A/ J(A) is commutative. The result is not valid for a general algebra of polynomially compact operators.

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