Hereditary Subalgebras Research Articles

The Fast Recurrent Subspace on an N-Level Quantum Energy Transport Model

The fast recurrent subspace (the biggest support of all invariant states) of a weak coupling limit type quantum Markov semigroup modeling a quantum transport open system of [Formula: see text]-energy levels is determined. This is achieved by characterizing the structure of all the invariant states and their spectra in terms of a natural generalization of the discrete Fourier transform operator. Finally, the attraction domains and long-time behaviour of the evolution are studied on hereditary subalgebras where faithful invariant states exist.

Hereditary subalgebras of a JB-algebra and the Jordan Pedersen ideal

We study (not necessarily closed) hereditary subalgebras of a JB-algebra. It is proved that in every JB-algebra A there exists a minimal hereditary dense ideal which is called the Jordan Pedersen ideal. If A is a JC-algebra the Jordan Pedersen ideal is minimal among all dense ideals. The hereditary properties of the Jordan Pedersen ideal are also studied.

General Uniform Roe algebra rigidity

We generalize all known results on rigidity of uniform Roe algebras to the setting of arbitrary uniformly locally finite coarse spaces. For instance, we show that isomorphism between uniform Roe algebras of uniformly locally finite coarse spaces whose uniform Roe algebras contain only compact ghost projections implies that the base spaces are coarsely equivalent. Moreover, if one of the spaces has property A, then the base spaces are bijectively coarsely equivalent. We also provide a characterization for the existence of an embedding onto hereditary subalgebra in terms of the underlying spaces. As an application, we partially answer a question of White and Willett about Cartan subalgebras of uniform Roe algebras.

Stable elements and property (S)

We study the relation (and differences) between stability and Property (S) in the simple and stably finite framework. This leads us to characterize stable elements in terms of their support, and study these concepts from different sides: hereditary subalgebras, projections in the multiplier algebra and order properties in the Cuntz semigroup. We use these approaches to show both that cancellation at infinity on the Cuntz semigroup just holds when its Cuntz equivalence is given by isomorphism at the level of Hilbert right-modules, and that different notions such as Regularity, ω-comparison, Corona Factorization Property, property R, etc. are equivalent under mild assumptions.

A unitary Cuntz semigroup for C⁎-algebras of stable rank one

We introduce a new invariant for C⁎-algebras of stable rank one that merges the Cuntz semigroup information together with the K1-group information. This semigroup, termed the Cu1-semigroup, is constructed as equivalence classes of pairs consisting of a positive element in the stabilization of the given C⁎-algebra together with a unitary element of the unitization of the hereditary subalgebra generated by the given positive element. We show that the Cu1-semigroup is a well-defined continuous functor from the category of C⁎-algebras of stable rank one to a suitable codomain category that we write Cu∼. Furthermore, we compute the Cu1-semigroup of some specific classes of C⁎-algebras. Finally, in the course of our investigation, we show that we can recover functorially Cu, K1 and K⁎:=K0⊕K1 from Cu1.

Jordan operator algebras revisited

AbstractJordan operator algebras are norm‐closed spaces of operators on a Hilbert space with for all . In two recent papers by the authors and Neal, a theory for these spaces was developed. It was shown there that much of the theory of associative operator algebras, in particular, surprisingly much of the associative theory from several recent papers of the first author and coauthors, generalizes to Jordan operator algebras. In the present paper we complete this task, giving several results which generalize the associative case in these papers, relating to unitizations, real positivity, hereditary subalgebras, and a couple of other topics. We also solve one of the three open problems stated at the end of our earlier joint paper on Jordan operator algebras.

Noncommutative topology and Jordan operator algebras

AbstractJordan operator algebras are norm‐closed spaces of operators on a Hilbert space with for all . We study noncommutative topology, noncommutative peak sets and peak interpolation, and hereditary subalgebras of Jordan operator algebras. We show that Jordan operator algebras present perhaps the most general setting for a “full” noncommutative topology in the ‐algebraic sense of Akemann, L. G. Brown, Pedersen, etc, and as modified for not necessarily selfadjoint algebras by the authors with Read, Hay and other coauthors. Our breakthrough relies in part on establishing several strong variants of ‐algebraic results of Brown relating to hereditary subalgebras, proximinality, deeper facts about for a left ideal L in a ‐algebra, noncommutative Urysohn lemmas, etc. We also prove several other approximation results in ‐algebras and various subspaces of ‐algebras, related to open and closed projections and technical ‐algebraic results of Brown.

Purely infinite labeled graph -algebras

In this paper, we consider pure infiniteness of generalized Cuntz–Krieger algebras associated to labeled spaces $(E,{\mathcal{L}},{\mathcal{E}})$. It is shown that a $C^{\ast }$-algebra $C^{\ast }(E,{\mathcal{L}},{\mathcal{E}})$ is purely infinite in the sense that every non-zero hereditary subalgebra contains an infinite projection (we call this property (IH)) if $(E,{\mathcal{L}},{\mathcal{E}})$ is disagreeable and every vertex connects to a loop. We also prove that under the condition analogous to (K) for usual graphs, $C^{\ast }(E,{\mathcal{L}},{\mathcal{E}})=C^{\ast }(p_{A},s_{a})$ is purely infinite in the sense of Kirchberg and Rørdam if and only if every generating projection $p_{A}$, $A\in {\mathcal{E}}$, is properly infinite, and also if and only if every quotient of $C^{\ast }(E,{\mathcal{L}},{\mathcal{E}})$ has property (IH).

Polynomial Representations of $$C^*$$ C ∗ -Algebras and their Applications

This is a sequel to our paper on nonlinear completely positive maps and dilation theory for real involutive algebras, where we have reduced all classification problems to the passage from a \(C^*\)-algebra \(\mathcal {A}\) to its symmetric powers \(S^n(\mathcal {A})\), resp., to holomorphic representations of the multiplicative \(*\)-semigroup \((\mathcal {A},\cdot )\). Here we study the correspondence between representations of \(\mathcal {A}\) and of \(S^n(\mathcal {A})\) in detail. As \(S^n(\mathcal {A})\) is the fixed point algebra for the natural action of the symmetric group \(S_n\) on \(\mathcal {A}^{\otimes n}\), this is done by relating representations of \(S^n(\mathcal {A})\) to those of the crossed product \(\mathcal {A}^{\otimes n} \rtimes S_n\) in which it is a hereditary subalgebra. For \(C^*\)-algebras of type I, we obtain a rather complete description of the equivalence classes of the irreducible representations of \(S^n(\mathcal {A})\) and we relate this to the Schur–Weyl theory for \(C^*\)-algebras. Finally we show that if \(\mathcal {A}\subseteq B(\mathcal {H})\) is a factor of type II or III, then its corresponding multiplicative representation on \(\mathcal {H}^{\otimes n}\) is a factor representation of the same type, unlike the classical case \(\mathcal {A}=B(\mathcal {H})\).

Crossed Products for Interactions and Graph Algebras

We consider Exel's interaction $(V,H)$ over a unital $C^*$-algebra $A$, such that $V(A)$ and $H(A)$ are hereditary subalgebras of $A$. For the associated crossed product, we obtain a uniqueness theorem, ideal lattice description, simplicity criterion and a version of Pimsner-Voiculescu exact sequence. These results cover the case of crossed products by endomorphisms with hereditary ranges and complementary kernels. As model examples of interactions not coming from endomorphisms we introduce and study in detail interactions arising from finite graphs. The interaction $(V,H)$ associated to a graph $E$ acts on the core $F_E$ of the graph algebra $C^*(E)$. By describing a partial homeomorphism of $\widehat{F}_E$ dual to $(V,H)$ we find Cuntz-Krieger uniqueness theorem, criteria for gauge-invariance of all ideals and simplicity of $C^*(E)$ as results concerning reversible noncommutative dynamics. We also provide a new approach to calculation of $K$-theory of $C^*(E)$ using only an induced partial automorphism of $K_0(F_E)$ and the six-term exact sequence.

An algebraic approach to the radius of comparison

The radius of comparison is an invariant for unital C∗-algebras which extends the theory of covering dimension to noncommutative spaces. We extend its definition to general C∗-algebras, and give an algebraic (as opposed to functional-theoretic) reformulation. This yields new permanence properties for the radius of comparison which strengthen its analogy with covering dimension for commutative spaces. We then give several applications of these results. New examples of C∗-algebras with finite radius of comparison are given, and the question of when the Cuntz classes of finitely generated Hilbert modules form a hereditary subset of the Cuntz semigroup is addressed. Most interestingly, perhaps, we treat the question of when a full hereditary subalgebra B of a stable C∗-algebra A is itself stable, giving a characterization in terms of the radius of comparison. We also use the radius of comparison to quantify the least n for which a C∗-algebra D without bounded 2-quasitraces or unital quotients has the property that Mn(D) is stable.

Ideals and hereditary subalgebras in operator algebras

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 (HSA's), which are in some sense generalization of ideals. Second, we study properties of operator algebras which are hereditary subalgebras in their bidual, or equivalently which are `weakly compact'. We also give several examples answering natural questions that arise in such an investigation.

Multipliers and hereditary subalgebras of operator algebras

We generalize some technical results of Glicksberg to the realm of general operator algebras and use them to give a characterization of open and closed projections in terms of certain multiplier algebras. This generalizes a theorem of J. Wells characterizing an important class of ideals in uniform algebras. The difficult implication in our main theorem is that if a projection is open in an operator algebra, then the multiplier algebra of the associated hereditary subalgebra arises as the closure of the subalgebra with respect to the strict topology of the multiplier algebra of a naturally associated hereditary C*-subalgebra. This immediately implies that the multiplier algebra of an operator algebra A may be obtained as the strict closure of A in the multiplier algebra of the C*-algebra generated by A.

Hereditary C*-subalgebras and comparison of positive elements

Characterizations of hereditary subalgebras generated by subsets of a C*-algebra are given through open projections. Using these results, we give some equivalent conditions of comparison of positive elements.

The nuclear dimension of [formula omitted]-algebras

We introduce the nuclear dimension of a C ∗ -algebra; this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank. Our notion behaves well with respect to inductive limits, tensor products, hereditary subalgebras (hence ideals), quotients, and even extensions. It can be computed for many examples; in particular, it is finite for all UCT Kirchberg algebras. In fact, all classes of nuclear C ∗ -algebras which have so far been successfully classified consist of examples with finite nuclear dimension, and it turns out that finite nuclear dimension implies many properties relevant for the classification program. Surprisingly, the concept is also linked to coarse geometry, since for a discrete metric space of bounded geometry the nuclear dimension of the associated uniform Roe algebra is dominated by the asymptotic dimension of the underlying space.

On the classification of nonsimple graph C *-algebras

We prove that a graph C *-algebra with exactly one proper nontrivial ideal is classified up to stable isomorphism by its associated six-term exact sequence in K-theory. We prove that a similar classification also holds for a graph C *-algebra with a largest proper ideal that is an AF-algebra. Our results are based on a general method developed by the first named author with Restorff and Ruiz. As a key step in the argument, we show how to produce stability for certain full hereditary subalgebras associated to such graph C *-algebras. We further prove that, except under trivial circumstances, a unique proper nontrivial ideal in a graph C *-algebra is stable.

On ideals and quotients of A % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBqj3BWbIqubWexLMBb50ujbqegm0B % 1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqr % Ffpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0F % irpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaa % GcbaWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgaiuaacqWF % tepvaaa!44CF! $$ \mathcal{T} $$ -algebras

Some results on A\( \mathcal{T} \) -algebras are given. We study the problem when ideals, quotients and hereditary subalgebras of A\( \mathcal{T} \) -algebras are A\( \mathcal{T} \) -algebras or A\( \mathcal{T} \) -algebras, and give a necessary and sufficient condition of a hereditary subalgebra of an A\( \mathcal{T} \) -algebra being an A\( \mathcal{T} \) -algebra.

$S$-regularity and the corona factorization property

Stability is an important and fundamental property of $C^{*}$-algebras. Given a short exact sequence of $C^{*}$-algebras $0\longrightarrow B\longrightarrow E\longrightarrow A\longrightarrow 0$ where the ends are stable, the middle algebra may or may not be stable. We say that the first algebra, $B$, is $S$-regular if every extension of $B$ by a stable algebra $A$ has a stable extension algebra, $E$. Rördam has given a sufficient condition for $S$-regularity. We define a new condition, weaker than Rördam's, which we call the corona factorization property, and we show that the corona factorization property implies $S$-regularity. The corona factorization property originated in a study of the Kasparov $KK^1(A,B)$ group of extensions, however, we obtain our results without explicit reference to $KK$-theory. Our main result is that for a separable stable $C^{*}$-algebra $B$ the first two of the following properties (which we define later) are equivalent, and both imply the third. With additional hypotheses on the $C^{*}$-algebra, all three properties are equivalent. $B$ has the corona factorization property. Stability is a stable property for full hereditary subalgebras of $B$. $B$ is $S$-regular. We also show that extensions of separable stable $C^{*}$-algebras with the corona factorization property give extension algebras with the corona factorization property, extending the results of [9].

Double commutants of hereditary subalgebras of a corona

We show that in certain corona algebras, σ -unital hereditary subalgebras have the property that their double relative commutant in the corona algebra is equal to the unitization of the given hereditary subalgebra. We also give several results on polar decomposition in such algebras.

Geometry of the unit ball and representation theory for operator algebras

We investigate the relationship between the facial structure of the unit ball of an operator algebra A and its algebraic structure, including the hereditary subalgebras and the socle of A. Many questions about the facial structure of A are studied with the aid of representation theory. For that purpose we establish the existence of reduced atomic type representations for certain nonselfadjoint operator algebras. Our results are applicable to C*-algebras, strongly maximal TAF algebras, free semigroup algebras and various semicrossed products.