Spectral triples on irreversible C∗-dynamical systems

Given a spectral triple on a [Formula: see text]-algebra [Formula: see text] together with a unital injective endomorphism [Formula: see text], the problem of defining a suitable crossed product [Formula: see text]-algebra endowed with a spectral triple is addressed. The proposed construction is mainly based on the works of Cuntz and [A. Hawkins, A. Skalski, S. White and J. Zacharias, On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262–291], and on our previous papers [V. Aiello, D. Guido and T. Isola, Spectral triples for noncommutative solenoidal spaces from self-coverings, J. Math. Anal. Appl. 448(2) (2017) 1378–1412; V. Aiello, D. Guido and T. Isola, A spectral triple for a solenoid based on the Sierpinski gasket, SIGMA Symmetry Integrability Geom. Methods Appl. 17(20) (2021) 21]. The embedding of [Formula: see text] in [Formula: see text] can be considered as the dual form of a covering projection between noncommutative spaces. A main assumption is the expansiveness of the endomorphism, which takes the form of the local isometricity of the covering projection, and is expressed via the compatibility of the Lip-norms on [Formula: see text] and [Formula: see text].

Irreducible skew polynomials over domains

Let S be a domain and R = S[t; σ, δ] a skew polynomial ring, where σ is an injective endomorphism of S and δ a left σ -derivation. We give criteria for skew polynomials f ∈ R of degree less or equal to four to be irreducible. We apply them to low degree polynomials in quantized Weyl algebras and the quantum planes. We also consider f(t) = tm − a ∈ R.

On Endomorphism Monoids of Finite Kleene Algebras

An endomorphism monoid of an algebra [Formula: see text] is said to be a band if every endomorphism on [Formula: see text] is an idempotent, and it is said to be a demi-band if every non-injective endomorphism on [Formula: see text] is an idempotent. We precisely determine finite Kleene algebras whose endomorphism monoids are demi-bands and bands via Priestley duality.

DILATIONS OF INTERACTION GROUPS THAT EXTEND ACTIONS OF ORE SEMIGROUPS

We show that if a unital injective endomorphism of a $C^{\ast }$-algebra admits a transfer operator, then both of them are compressions of mutually inverse automorphisms of a bigger algebra. More generally, every interaction group – in the sense of Exel – extending an action of an Ore semigroup by injective unital endomorphisms of a $C^{\ast }$-algebra, admits a dilation to an action of the corresponding enveloping group on another unital $C^{\ast }$-algebra, of which the former is a $C^{\ast }$-subalgebra: the interaction group is obtained by composing the action with a conditional expectation. The dilation is essentially unique if a certain natural condition of minimality is imposed, and it is faithful if and only if the interaction group is also faithful.

Grouplike minimal sets in ACFA AND in TA

This paper began as a generalization of a part of the author's PhD thesis about ACFA and ended up with a characterization of groups definable in TA. The thesis concerns minimal formulae of the form x Є A ∧ σ(x) = f(x) for an algebraic curve A and a dominant rational function f: A → σ(A). These are shown to be uniform in the Zilber trichotomy, and the pairs (A, f) that fall into each of the three cases are characterized. These characterizations are definable in families. This paper covers approximately half of the thesis, namely those parts of it which can be made purely model-theoretic by moving from ACFA, the model companion of the class of algebraically closed fields with an endomorphism, to TA, the model companion of the class of models of an arbitrary totally-transcendental theory T with an injective endomorphism, if this model-companion exists. A TA analog of the characterization of groups definable in ACFA is obtained in the process. The full characterization of the cases of the Zilber trichotomy in the thesis is obtained from these intermediate results with heavy use of algebraic geometry.

The Preimage of a Coordinate

Let K be a field of characteristic zero. Based on the degree estimate of Makar-Limanov and Yu, we prove that the preimage of a coordinate under an injective endomorphism of K〈x, y〉 is also a coordinate. As by-products, we give new proofs of the following results: (1) the preimage of a coordinate under an injective endomorphism of K[x,y] is also a coordinate; (2) any automorphism of K[x,y] or K〈x, y〉 is tame.

ORE EXTENSIONS SATISFYING A POLYNOMIAL IDENTITY

Necessary and sufficient conditions for an Ore extension S = R[x;σ,δ] to be a PI ring are given in the case σ is an injective endomorphism of a semiprime ring R satisfying the ACC on annihilators. Also, for an arbitrary endomorphism τ of R, a characterization of Ore extensions R[x;τ] which are PI rings is given, provided the coefficient ring R is noetherian.

PRIMITIVE ELEMENTS OF FREE METABELIAN ALGEBRAS OF RANK TWO

Let F(x,y) be a relatively free algebra of rank 2 in some variety of algebras over a field K of characteristic 0. In this paper we consider the problem whether p(x,y) ∈ F(x,y) is a primitive element (i.e. an automorphic image of x): (i) If F(x,y)/(p(x,y)) ≅ F(z), the relatively free algebra of rank 1 (ii) If p(f,g) is primitive for some injective endomorphism (f,g) of F(x,y) (iii) If p(x,y) is primitive in a relatively free algebra of larger rank. These problems have positive solutions for polynomial algebras in two variables. We give the complete answer for the free metabelian associative and Lie algebras and some partial results for free associative algebras.