Primitive Ideal Space Research Articles

Classification of equivariantly [formula omitted]-stable amenable actions on nuclear C⁎-algebras

Given a second-countable, locally compact group G, we consider amenable G-actions on separable, stable, nuclear C⁎-algebras that are isometrically shift-absorbing and tensorially absorb the trivial action on the Cuntz algebra O2. We show that such actions are classified up to cocycle conjugacy by the induced G-action on the primitive ideal space. In the special case when G is exact, we prove a unital version of our classification theorem. For compact groups, we obtain a classification up to conjugacy.

The Primitive Ideal Space of the Partial-isometric Crossed Product by Automorphic Actions of the Semigroup $\mathbb{N}^{2}$

The Primitive Ideal Space of the Partial-isometric Crossed Product by Automorphic Actions of the Semigroup $\mathbb{N}^{2}$

Square-integrable representations and the coadjoint action of solvable Lie groups

Abstract We characterize the square-integrable representations of (connected, simply connected) solvable Lie groups in terms of the generalized orbits of the coadjoint action. We prove that the normal representations corresponding, via the Pukánszky correspondence, to open coadjoint orbits are type I, not necessarily square-integrable representations. We show that the quasi-equivalence classes of type I square-integrable representations are in bijection with the simply connected open coadjoint orbits, and the existence of an open coadjoint orbit guarantees the existence of a compact open subset of the space of primitive ideals of the group. When the nilradical has codimension 1, we prove that the isolated points of the primitive ideal space are always of type I. This is not always true for codimension greater than 2, as shown by specific examples of solvable Lie groups that have dense, but not locally closed, coadjoint orbits.

A tracial characterization of Furstenberg’s conjecture

AbstractWe investigate almost minimal actions of abelian groups and their crossed products. As an application, given multiplicatively independent integers p and q, we show that Furstenberg’s $\times p,\times q$ conjecture holds if and only if the canonical trace is the only faithful extreme tracial state on the $C^*$ -algebra of the group $\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2$ . We also compute the primitive ideal space and K-theory of $C^*(\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2)$ .

R⁎-algebras and C⁎-uniqueness of pre-C⁎-algebras

In 1967, Kadison raised the following question, if a self-adjoint operator algebra on a Hilbert space is finitely generated (algebraically) and each self-adjoint operator in it has finite spectrum, is it finite dimensional? In 2021, M. Mori connected the Kadison's question to von Neumann's concept of regular rings given in 1930's, and introduced a notion of R⁎-algebras. In this paper, we study R⁎-algebras, the Kadison's question, and a related question of uniqueness of C⁎-norm on pre-C⁎-algebras. Precisely, we study atomic R⁎-algebras, and find a maximal ultramatricial R⁎-subalgebra of an atomic R⁎-algebra, which properly contains the maximal purely atomic R⁎-subalgebra generated by all finite projections of the algebra. We prove that the set of all ⁎-algebras for which the Kadison's question has an affirmative answer is closed under direct sum, tensor product, and crossed product, but it is not closed under free product. From the aforementioned stable properties of the algebra in the Kadison's question, we find plentiful examples of ⁎-algebras for which the Kadison's question has an affirmative answer, for instance, the crossed product ⁎-algebra B⋊αG, where G is a locally finite group and B is one of the following algebras: an algebra of finite rank operators, an abelian algebra, a finite dimensional algebra, a twisted group algebra satisfying a certain condition, a rotation algebra, or a group measure space construction algebra. We characterize uniqueness of C⁎-norm on a pre-C⁎-algebras in terms of its ⁎-representations, weak containment of its ⁎-representations, ideals of its enveloping C⁎-algebra, or the primitive ideal space of the C⁎-algebra. We also prove that a unital pre-C⁎-algebra has a unique C⁎-norm if and only if, for a faithful ⁎-representation of the algebra, invertibility of a self-adjoint element of the algebra, as an operator on the representation Hilbert space, is independent of the choice of the ⁎-representation. These characterizations provide answers to the question raised by M. Mori in 2021 on finding a sufficient and necessary condition for a ⁎-algebra to have a unique C⁎-norm. For discrete groups, we give a characterization for a group algebra to have a unique C⁎-norm in terms of the group's unitary representations, providing an answer to the question raised by Alekseev in 2019 on characterizing C⁎-unique discrete groups.

The generator rank of subhomogeneous -algebras

Abstract We compute the generator rank of a subhomogeneous $C^*\!$ -algebra in terms of the covering dimension of the pieces of its primitive ideal space corresponding to irreducible representations of a fixed dimension. We deduce that every $\mathcal {Z}$ -stable approximately subhomogeneous algebra has generator rank one, which means that a generic element in such an algebra is a generator. This leads to a strong solution of the generator problem for classifiable, simple, nuclear $C^*\!$ -algebras: a generic element in each such algebra is a generator. Examples of Villadsen show that this is not the case for all separable, simple, nuclear $C^*\!$ -algebras.

The primitive ideal space of groupoid C^*-algebras for groupoids with Abelian isotropy

The primitive ideal space of groupoid C^*-algebras for groupoids with Abelian isotropy

CONDITION (K) FOR BOOLEAN DYNAMICAL SYSTEMS

AbstractWe generalize Condition (K) from directed graphs to Boolean dynamical systems and show that a locally finite Boolean dynamical system $({{\mathcal {B}}},{{\mathcal {L}}},\theta )$ with countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ satisfies Condition (K) if and only if every ideal of its $C^*$ -algebra is gauge-invariant, if and only if its $C^*$ -algebra has the (weak) ideal property, and if and only if its $C^*$ -algebra has topological dimension zero. As a corollary we prove that if the $C^*$ -algebra of a locally finite Boolean dynamical system with ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ countable either has real rank zero or is purely infinite, then $({{\mathcal {B}}}, {{\mathcal {L}}}, \theta )$ satisfies Condition (K). We also generalize the notion of maximal tails from directed graph to Boolean dynamical systems and use this to give a complete description of the primitive ideal space of the $C^*$ -algebra of a locally finite Boolean dynamical system that satisfies Condition (K) and has countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ .

Jacobson topology of the primitive ideal space of self-similar k-graph C*-algebras

We describe the Jacobson topology of the primitive ideal space of self-similar k-graph C*-algebras under certain conditions.

Gluing Hilbert C⁎-modules over the primitive ideal space

We show that the gluing construction for Hilbert modules introduced by Raeburn in his computation of the Picard group of a continuous-trace C⁎-algebra (1981) [14] can be applied to arbitrary C⁎-algebras, via an algebraic argument with the Haagerup tensor product. We put this result into the context of descent theory by identifying categories of gluing data for Hilbert modules over C⁎-algebras with categories of comodules over C⁎-coalgebras, giving a Hilbert-module version of a standard construction from algebraic geometry. As a consequence we show that if two C⁎-algebras have the same primitive ideal space T, and are Morita equivalent up to a 2-cocycle on T, then their Picard groups relative to T are isomorphic.

AF‐embeddability for Lie groups with T1 primitive ideal spaces

We study simply connected Lie groups $G$ for which the hull-kernel topology of the primitive ideal space $\text{Prim}(G)$ of the group $C^*$-algebra $C^*(G)$ is $T_1$, that is, the finite subsets of $\text{Prim}(G)$ are closed. Thus, we prove that $C^*(G)$ is AF-embeddable. To this end, we show that if $G$ is solvable and its action on the centre of $[G, G]$ has at least one imaginary weight, then $\text{Prim}(G)$ has no nonempty quasi-compact open subsets. We prove in addition that connected locally compact groups with $T_1$ ideal spaces are strongly quasi-diagonal.

Stone duality and quasi‐orbit spaces for generalised C∗‐inclusions

Let $A$ and $B$ be $C^*$-algebras with $A\subseteq M(B)$. Exploiting the duality between sober spaces and spatial locales, and the adjunction between restriction and induction for ideals in $A$ and $B$, we identify conditions that allow to define a quasi-orbit space and a quasi-orbit map for $A\subseteq M(B)$. These objects generalise classical notions for group actions. We characterise when the quasi-orbit space is an open quotient of the primitive ideal space of $A$ and when the quasi-orbit map is open and surjective. We apply these results to cross section $C^*$-algebras of Fell bundles over locally compact groups, regular $C^*$-inclusions, tensor products, relative Cuntz--Pimsner algebras, and crossed products for actions of locally compact Hausdorff groupoids and quantum groups.

Morita equivalence for $k$-algebras

We review Morita equivalence for finite type k-algebras A and also a weakening of Morita equivalence which we call stratified equivalence. The spectrum of A is the set of equivalence classes of irreducible A-modules. For any finite type k-algebra A, the spectrum of A is in bijection with the set of primitive ideals of A. The stratified equivalence relation preserves the spectrum of A and also preserves the periodic cyclic homology of A. However, the stratified equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence. A key example illustrating the distinction between Morita equivalence and stratified equivalence is provided by affine Hecke algebras associated to affine Weyl groups.

Phase transitions on C∗-algebras arising from number fields and the generalized Furstenberg conjecture

We describe the low-temperature extremal KMS states of the semigroup C∗-algebra of the ax+b semigroup of algebraic integers in a number field in terms of ergodic invariant measures for certain groups of linear toral automorphisms. We classify different behaviours in terms of the ideal class group, the degree, and the unit rank of the field and we also obtain an explicit description of the primitive ideal space of the associated transformation group C∗-algebra for number fields of unit rank at least 2 that are not complex multiplication fields.

Primitive Ideal Space of Ultragraph $C^*$-algebras

In this paper, we describe the primitive ideal space of the $C^*$-algebra $C^*(mathcal G)$ associated to the ultragraph $mathcal{G}$. We investigate the structure of the closed ideals of the quotient ultragraph $ C^* $-algebra $C^*left(mathcal G/(H,S)right)$ which contain no nonzero set projections and then we characterize all non gauge-invariant primitive ideals. Our results generalize the Hong and Szyma$ acute{ mathrm { n } } $ski's description of the primitive ideal space of a graph $ C ^ * $-algebra by a simpler method.

Primitive ideal space of higher-rank graph C⁎-algebras and decomposability

In this paper, we describe primitive ideal space of the C⁎-algebra C⁎(Λ) associated to any locally convex row-finite k-graph Λ. To do this, we will apply the Farthing's desourcifying method on a recent result of Carlsen, Kang, Shotwell, and Sims. We also characterize certain maximal ideals of C⁎(Λ). Furthermore, we study the decomposability of C⁎(Λ). We apply the description of primitive ideals to show that if I is a direct summand of C⁎(Λ), then it is gauge-invariant and isomorphic to a certain k-graph C⁎-algebra. So, we may characterize decomposable higher-rank C⁎-algebras by giving necessary and sufficient conditions for the underlying k-graphs. Moreover, we determine all such C⁎-algebras which can be decomposed into a direct sum of finitely many indecomposable C⁎-algebras.

A new proof of Kirchberg's 𝒪2-stable classification

Abstract I present a new proof of Kirchberg’s 𝒪 2 \mathcal{O}_{2} -stable classification theorem: two separable, nuclear, stable/unital, 𝒪 2 \mathcal{O}_{2} -stable C ∗ C^{\ast} -algebras are isomorphic if and only if their ideal lattices are order isomorphic, or equivalently, their primitive ideal spaces are homeomorphic. Many intermediate results do not depend on pure infiniteness of any sort.

Just-infinite $C^*$-algebras

By analogy with the well-established notions of just-infinite groups and just-infinite (abstract) algebras, we initiate a systematic study of just-infinite C^* -algebras, i.e. infinite dimensional C^* -algebras for which all proper quotients are finite dimensional. We give a classification of such C^* -algebras in terms of their primitive ideal space, that leads to a trichotomy. We show that just-infinite, residually finite dimensional C^* -algebras do exist by giving an explicit example of (the Bratteli diagram of) an AF-algebra with these properties. Further, we discuss when C^* -algebras and ^* -algebras associated with a discrete group are just-infinite. If \mathcal G is the Burnside-type group of intermediate growth discovered by the first-named author, which is known to be just-infinite, then its group algebra \mathbb C[\mathcal G] and its group C^* -algebra C^*(\mathcal G) are not just-infinite. Furthermore, we show that the algebra B = \pi(\mathbb C[\mathcal G]) under the Koopman representation \pi of \mathcal G associated with its canonical action on a binary rooted tree is just-infinite. It remains an open problem whether the residually finite dimensional C^* -algebra C^*_\pi(\mathcal G) is just-infinite.

The primitive ideal space of the partial-isometric crossed product of a system by a single automorphism

Let $(A,\alpha )$ be a system consisting of a $C^*$-algebra $A$ and an automorphism $\alpha $ of~$A$. We describe the primitive ideal space of the partial-isometric crossed product $A\times _{\alpha }^{piso }\mathbb{N} $ of the system by using its realization as a full corner of a classical crossed product and applying some results of Williams and Echterhoff.

On the decomposition into Discrete, Type II and Type III C*-algebras

AbstractWe obtained a “decomposition scheme” of C*-algebras. We show that the classes of discrete C*-algebras (as defined by Peligard and Zsidó), type II C*-algebras and type III C*-algebras (both defined by Cuntz and Pedersen) form a good framework to “classify” C*-algebras. In particular, we found that these classes are closed under strong Morita equivalence, hereditary C*-subalgebras as well as taking “essential extension” and “normal quotient”. Furthermore, there exist the largest discrete finite ideal Ad,1, the largest discrete essentially infinite ideal Ad,∞, the largest type II finite ideal AII,1, the largest type II essentially infinite ideal AII,∞, and the largest type III ideal AIII of any C*-algebra A such that Ad,1 + Ad,∞ + AII,1 + AII,∞ + AIII is an essential ideal of A. This “decomposition” extends the corresponding one for W*-algebras.We also give a closer look at C*-algebras with Hausdorff primitive ideal spaces, AW*-algebras as well as local multiplier algebras of C*-algebras. We find that these algebras can be decomposed into continuous fields of prime C*-algebras over a locally compact Hausdorff space, with each fiber being non-zero and of one of the five types mentioned above.