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Ideals and hereditary subalgebras in operator algebras

Melahat Almus, David P. Blecher, Charles John Read (2012)

Studia Mathematica

This paper may be viewed as having two aims. First, we continue our study of algebras of operators on a Hilbert space which have a contractive approximate identity, this time from a more Banach-algebraic point of view. Namely, we mainly investigate topics concerned with the ideal structure, and hereditary subalgebras (or HSA's, which are in some sense a generalization of ideals). Second, we study properties of operator algebras which are hereditary subalgebras in their bidual, or equivalently which...

Idempotent completion of pretriangulated categories

Jichun Liu, Longgang Sun (2014)

Czechoslovak Mathematical Journal

A pretriangulated category is an additive category with left and right triangulations such that these two triangulations are compatible. In this paper, we first show that the idempotent completion of a left triangulated category admits a unique structure of left triangulated category and dually this is true for a right triangulated category. We then prove that the idempotent completion of a pretriangulated category has a natural structure of pretriangulated category. As an application, we show that...

Idempotent States and the Inner Linearity Property

Teodor Banica, Uwe Franz, Adam Skalski (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

We find an analytic formulation of the notion of Hopf image, in terms of the associated idempotent state. More precisely, if π:A → Mₙ(ℂ) is a finite-dimensional representation of a Hopf C*-algebra, we prove that the idempotent state associated to its Hopf image A' must be the convolution Cesàro limit of the linear functional φ = tr ∘ π. We then discuss some consequences of this result, notably to inner linearity questions.

Idempotents and the multiplicative group of some totally bounded rings

Mohamed A. Salim, Adela Tripe (2011)

Czechoslovak Mathematical Journal

In this paper, we extend some results of D. Dolzan on finite rings to profinite rings, a complete classification of profinite commutative rings with a monothetic group of units is given. We also prove the metrizability of commutative profinite rings with monothetic group of units and without nonzero Boolean ideals. Using a property of Mersenne numbers, we construct a family of power 2 0 commutative non-isomorphic profinite semiprimitive rings with monothetic group of units.

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