Abstract
Departing from Fuglede-Putnam-Rosenblum's theorem, we examine several commutativity conditions in involutive algebras with $C^{\ast }$-equalities. Among questions considered are Ogasawara's theorem on operator algebras and Radjavi-Rosenthal's result on an algebra of normal operators. In the frame of $C^{\ast }$-algebras, conditions of apparently different natures turn out to be equivalent. Also, remarks are made about Hirshfeld-Zelazko's problem.
Highlights
Ogasawara considered a commutativity condition in relation with the order, that is the monotonicity of the square map [21]
The result of that author is still valid in locally C∗-convex algebras (Proposition V.4) and in C∗-bornological algebras (Proposition V.6)
Let (E, τ) be a locally convex algebra (l.c.a.), with a separately continuous multiplication, whose topology τ is given by a familyλ∈Λ of seminorms
Summary
Ogasawara considered a commutativity condition in relation with the order, that is the monotonicity of the square map [21]. The result of that author is still valid in locally C∗-convex algebras (Proposition V.4) and in C∗-bornological algebras (Proposition V.6). This is the content of Section V. All of them imply commutativity ( modulo the Jacobson radical) They appear to be equivalent (Proposition VI.3) though being of diferent natures It appears that in involutive algebras, Hirschfeld-Zelazko’s problem can not be reducd to its involutive version These are partial answers, but the problem remains open
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