We study the stability of certain spectra under some algebraic conditions weaker than commutativity and we generalize many known commutative perturbation results. In particular, in a complex unital Banach algebra $$\mathcal {A},$$ prove that if $$x \in \hbox {comm}_{w}(a)$$ is nilpotent, then $$\sigma (x)=\sigma (x+a).$$ Among other things, {comm}(xy)\cup {comm}(yx)$$ or $$y\in for all $$x,y \m...

We introduce notions of dimension and dynamical entropy for unital Calgebras “metrized” by means of cLip-norms, which are complex-scalar versions of the Lip-norms constitutive of Rieffel’s compact quantum metric spaces. Our examples involve the UHF algebras Mp∞ and noncommutative tori. In particular we show that the entropy of a noncommutative toral automorphism with respect to the canonical cL...

Let A be a complex, commutative unital Banach algebra. We introduce two notions of exponential reducibility of Banach algebra tuples and present an analogue to the Corach-Suárez result on the connection between reducibility in A and in C(M(A)). Our methods are of an analytical nature. Necessary and sufficient geometric/topological conditions are given for reducibility (respectively reducibility...

Let A be a commutative unital C-algebra and let  denote its Gelfand spectrum. We find some necessary and sufficient conditions for a nondegenerate representation of A to be unitarily equivalent to a multiplicative representation on a space L(Â, μ), where μ is a positive measure on the Baire sets of Â. We also compare these conditions with the multiplicity-free property of a representation.

Suppose that G is an abelian p-separable group and R is a commutative unital ring of prime characteristic p. It is proved that if Rp i possesses nilpotent elements for each natural number i, then the normalized Sylow p-subgroup S(RG) in the group ring RG is torsion-complete if and only if G is p-bounded. This supplies our recent results from Acta Math. Hung. (1997), Ric. Mat. (2002), Tamkang J....

Recall that a rng is a ring which is possibly non-unital. In this note, we address the problem whether every finitely generated idempotent rng (abbreviated as irng) is singly generated as an ideal. It is well-known that it is the case for a commutative irng. We prove here it is also the case for a free rng on finitely many idempotents and for a finite irng. A relation to the Wiegold problem for...

Let A be a unital Banach algebra. We give a characterization of the left Banach A-modules X for which there exists a commutative unital C-algebra C(K), a linear isometry i:X → C(K), and a contractive unital homomorphism θ:A → C(K) such that i(a· x) = θ(a)i(x) for any a ∈ A, x ∈ X . We then deduce a “commutative” version of the Christensen-Effros-Sinclair characterization of operator bimodules. ...

For a twisted partial action Θ of a group G on an (associative nonnecessarily unital) algebra A over a commutative unital ring k, the crossed product A⋊ΘG is proved to be associative. Given a G-graded k-algebra B = ⊕g∈GBg with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B1⋊ΘG for some twisted partial action of G on ...

We characterize derivations and 2-local derivations from Mn(A) into Mn(M), n ≥ 2, where A is a unital algebra over C and M is a unital A-bimodule. We show that every derivation D : Mn(A) → Mn(M), n ≥ 2, is the sum of an inner derivation and a derivation induced by a derivation from A to M. We say that A commutes with M if am = ma for every a ∈ A and m ∈ M. If A commutes with M we prove that eve...

Let n be an ordinal number and let R be a non trivial zero structure. One can verify that there exists a monomial of n, R which is non-zero. Let us observe that there exists a field which is non trivial. Let us note that every left zeroed add-right-cancelable right distributive left unital commutative associative non empty double loop structure which is fieldlike is also integral domain-like. L...