Continuous Fields of Postliminal C*-algebras

The classification ofseparable simple C*-algebras which are inductive limits of continuous-trace C*-algebraswith spectrum homeomorphic to the closed interval [0,1]

A classification is given of certain separable nuclear C*-algebras not necessarily of real rank zero, namely, the class of separable simple C*-algebras which are inductive limits of continuous-trace C*-algebras whose building blocks have spectrum homeomorphic to the closed interval [0,1], or to a disjoint union of copies of this space. Also, the range of the invariant is calculated.

Injective Envelopes and Local Multiplier Algebras of Some Spatial Continuous Trace C*-Algebras

A precise description of the injective envelope of a spatial continuous trace C*-algebra A over a Stonean space Delta is given. The description is based on the notion of a weakly continuous Hilbert bundle, which we show to be a Kaplansky--Hilbert module over the abelian AW*-algebra C(Delta). We then use the description of the injective envelope of A to study the first- and second-order local multiplier algebras of A. In particular, we show that the second-order local multiplier algebra of A is precisely the injective envelope of A.

Continuous fields of properly infinite C*-algebras

Any unital separable continuous C(X)-algebra with properly infinite fibres is properly infinite as soon as the compact Hausdorff space X has finite topological dimension. We study conditions under which this is still the case in the infinite dimensional case.

Subtriviality of continuous fields of nuclear C*--algebras

We extend in this paper the characterisation of a separable nuclear \cst-algebra given by Kirchberg proving that given a unital separable continuous field of nuclear C*-algebras A over a compact metrizable space X, the C(X)-algebra A is isomorphic to a unital C(X)-subalgebra of the trivial continuous field C(X;O_2), image of C(X;O_2) by a norm one projection.

Endomorphisms of Stable Continuous-Trace C^*-algebras

We classify spectrum-preserving endomorphisms of stable continuous-trace C^*-algebras up to inner automorphism by a surjective multiplicative invariant taking values in finite dimensional vector bundles over the spectrum. Specializing to automorphisms, this gives a different approach to results of Phillips and Raeburn.

Rational $K$-Stability of Continuous $C(X)$-Algebras

We show that the property of being rationally $K$-stable passes from the fibers of a continuous $C(X)$-algebra to the ambient algebra, under the assumption that the underlying space $X$ is compact, metrizable, and of finite covering dimension. As an application, we show that a crossed product C*-algebra is (rationally) $K$-stable provided the underlying C*-algebra is (rationally) $K$-stable, and the action has finite Rokhlin dimension with commuting towers.

K-stability of continuous C(X)-algebras

A C*-algebra is said to be K-stable if its nonstable K-groups are naturally isomorphic to the usual K-theory groups. We study continuous $C(X)$-algebras, each of whose fibers are K-stable. We show that such an algebra is itself K-stable under the assumption that the underlying space $X$ is compact, metrizable, and of finite covering dimension.

Measure continuous derivations on von Neumann algebras and applications to L^2-cohomology

We prove that norm continuous derivations from a von Neumann algebra into the algebra of operators affiliated with its tensor square are automatically continuous for both the strong operator topology and the measure topology. Furthermore, we prove that the first continuous L^2-Betti number scales quadratically when passing to corner algebras and derive an upper bound given by Shen's generator invariant. This, in turn, yields vanishing of the first continuous L^2-Betti number for II_1 factors with property (T), for finitely generated factors with non-trivial fundamental group and for factors wi…

连续区间代数的场

Continuous fields of interval algebras

本文通过局部到全局策略，利用Cuntz半群和拼接技术，对连续区间代数场进行分类，扩展了AI代数的概念。

Fiberwise KK-equivalence of continuous fields of C*-algebras

Let A and B be separable nuclear continuous C(X)-algebras over a finite dimensional compact metrizable space X. It is shown that an element $σ$ of the parametrized Kasparov group KK_X(A,B) is invertible if and only if all its fiberwise components $σ_x\in KK(A(x),B(x))$ are invertible. This criterion does not extend to infinite dimensional spaces since there exist nontrivial unital separable continuous fields over the Hilbert cube with all fibers isomorphic to the Cuntz algebra O_2. Several applications to continuous fields of Kirchberg algebras are given. It is also shown that if each fiber of…

A topological invariant for continuous fields of the Cuntz algebras

For a continuous field of the Cuntz algebra over a finite CW complex, we introduce a topological invariant, which is an element in Dadarlat-Pennig's generalized cohomology group, and prove that the invariant is trivial if and only if the field comes from a vector bundle via Pimsner's construction.

Continuous families of properly infinite C*-algebras

Any unital separable continuous C(X)-algebra with properly infinite fibres is properly infinite as soon as the compact Hausdorff space X has finite topolog-ical dimension. We study conditions under which this is still the case if the compact space X has infinite topological dimension.

Innerness of continuous derivations on algebras of measurable operators affiliated with finite von Neumann algebras

This paper is devoted to derivations on the algebra $S(M)$ of all measurable operators affiliated with a finite von Neumann algebra $M.$ We prove that if $M$ is a finite von Neumann algebra with a faithful normal semi-finite trace $τ$, equipped with the locally measure topology $t,$ then every $t$-continuous derivation $D:S(M)\rightarrow S(M)$ is inner. A similar result is valid for derivation on the algebra $S(M,τ)$ of $τ$-measurable operators equipped with the measure topology $t_τ$.

Continuous Fields of $C^*$-Algebras Arising from Extensions of Tensor $C^*$-Categories

The notion of extension of a given $C^*$-category $C$ by a $C^*$-algebra $A$ is introduced. In the commutative case $A = C(Ω)$, the objects of the extension category are interpreted as fiber bundles over $Ω$ of objects belonging to the initial category. It is shown that the Doplicher-Roberts algebra (DR-algebra in the following) associated to an object in the extension of a strict tensor $C^*$-category is a continuous field of DR-algebras coming from the initial one. In the case of the category of the hermitian vector bundles over $Ω$ the general result implies that the DR-algebra of a vector …

Absence of Cartan subalgebras in continuous cores of free product von Neumann algebras

We show that the continuous core of any type III free product factor has no Cartan subalgebra. This is a complement to previous works due to Houdayer--Ricard and Boutonnet--Houdayer--Raum.

Continuous spectral decompositions of Abelian group actions on C*-algebras

Let G be a locally compact Abelian group. Following Ruy Exel, we view Fell bundles over the Pontrjagin dual of G as continuous spectral decompositions of G-actions on C*-algebras. We classify such spectral decompositions using certain dense subspaces related to Marc Rieffel's theory of square-integrability. There is a unique continuous spectral decomposition if the group acts properly on the primitive ideal space of the C*-algebra. But there are also examples of group actions without or with several inequivalent spectral decompositions.

Groupoid Crossed Products of Continuous-Trace C*-Algebras

We show that if $(A,G,α)$ is a groupoid dynamical system with $A$ continuous trace, then the crossed product $A\rtimes_αG$ is Morita equivalent to the C*-algebra $C*(\underline G,\underline E)$ of a twist $\underline E$ over a groupoid $\underline G$ equivalent to $G$. This is a groupoid analogue of the well known result for the crossed product of a group acting on an elementary C*-algebra.

The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with Continuous Trace

Given a locally compact abelian group $G$, we give an explicit formula for the Dixmier--Douady invariant of the $C^*$-algebra of the groupoid extension associated to a Čech $2$-cocycle in the sheaf of germs of continuous $G$-valued functions. We then exploit the blow-up construction for groupoids to extend this to some more general central extensions of étale equivalence relations.

The Stable Ideals of a Continuous Nest Algebra II

We continue the study of the rich family of norm-closed, automorphism invariant ideals of a continuous nest algebra. First we present a unified framework which captures all stable ideals as the kernels of limits of diagonal compressions. We then characterize when two such limits give rise to the same ideal, and we obtain detailed information of the structure of sums and intersections of ideals.

Innerness of continuous derivations on algebras of locally measurable operators

It is established that every derivation continuous with respect to the local measure topology acting on the *-algebra $LS(\mathcal{M})$ of all locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$ is necessary inner. If $\mathcal{M}$ is a properly infinite von Neumann algebra, then every derivation on $LS(\mathcal{M})$ is inner. In addition, it is proved that any derivation on $\mathcal{M}$ with values in Banach $\mathcal{M}$-bimodule of locally measurable operators is inner.

Continuous fields of C*-algebras over finite dimensional spaces

Let $X$ be a finite dimensional compact metrizable space. We study a technique which employs semiprojectivity as a tool to produce approximations of $C(X)$-algebras by $C(X)$-subalgebras with controlled complexity. The following applications are given. All unital separable continuous fields of C*-algebras over $X$ with fibers isomorphic to a fixed Cuntz algebra $\mathcal{O}_n$, $n\in\{2,3,...,\infty\}$ are locally trivial. They are trivial if $n=2$ or $n=\infty$. For $n\geq 3$ finite, such a field is trivial if and only if $(n-1)[1_A]=0$ in $K_0(A)$, where $A$ is the C*-algebra of continuous s…

Roe代数的连续场的一个例子

An example of a continuous field of Roe algebras

本文在最简单的非平凡例子$X=\mathbb R$中，通过在$[0,1]$上构造$C^*$-代数的连续场，探索了空间Roe代数与其离散化一致Roe代数之间的关系。

Structure of crossed products by strictly proper actions on continuous-trace algebras

We examine the ideal structure of crossed products B\rtimes G where B is a continuous-trace C*-algebra and the induced action of G on the spectrum of B is proper. In particular, we are able to obtain a concrete description of the topology on the spectrum of the crossed product in the cases where either G is discrete or G is a Lie group acting smoothly on the spectrum of B.

A short note on the continuous Rokhlin property and the universal coefficient theorem

Let $G$ be a metrizable compact group, $A$ a separable C*-algebra and $α$ a strongly continuous action of $G$ on $A$. Provided that $α$ satisfies the continuous Rokhlin property, we show that the property of satisfying the UCT in E-theory passes from $A$ to the crossed product C*-algebra $A\rtimes_αG$ and the fixed point algebra $A^α$. This extends a result by Gardella in the case that $G$ is the circle and $A$ is nuclear. For circle actions on separable, unital C*-algebras with the continuous Rokhlin property, we establish a connection between the $E$-theory equivalence class of the coefficie…

Continuous linear maps on reflexive algebras behaving like Jordan left derivations at idempotent-product elements

Let $\A$ be a Banach algebra with unity $\textbf{1}$ and $ \M $ be a unital Banach left $ \A $-module. let $ δ: \A \rightarrow \M$ be a continuous linear map with the property that \[ a,b\in \A, \quad ab+ba=z \Rightarrow 2aδ(b)+2bδ(a)=δ(z), \] where $z\in \A$. In this article, first we characterize $δ$ for $z=\textbf{1}$. Then we consider the case $\A=\M=Alg \mathcal{L}$, where $Alg \mathcal{L}$ is areflexive algebra on a Hilbert space $ \Hh $ and $z=P$ is a non-triavial idempotent in $\A$ with $P(\Hh) \in \mathcal{L}$ and describe $δ$. Finally we apply the main results to $CSL$-algebras, irre…

A topological invariant for continuous fields of Cuntz algebras II

We investigate an invariant for continuous fields of the Cuntz algebra $\mathcal{O}_{n+1}$ introduced in our previous work, and find a way to obtain a continuous field of $\mathbb{M}_n(\mathcal{O}_\infty)$ from that of $\mathcal{O}_{n+1}$ using the construction of the invariant. By Brown's representability theorem, this gives a bijection from the set of the isomorphism classes of continuous fields of $\mathcal{O}_{n+1}$ to those of $\mathbb{M}_n(\mathcal{O}_\infty)$. As a consequence, we obtain a new proof for M. Dadarlat's classification result of continuous fields of $\mathcal{O}_{n+1}$ aris…

Absolutely Continuous Representations and a Kaplansky Density Theorem for Free Semigroup Algebras

We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra $A_n$. Absolutely continuous functionals are used to help identify the type L part of the free semigroup algebra associated to a $*$-extendible representation $σ$. A $*$-extendible representation of $A_n$ is ``regular'' if the absolutely continuous part coincides with the type L part. All known examples are regular. Absolutely continuous functionals are intimately related to maps which intertwine a given $*$-extendible representation with the left regular representation. A simple a…

On the universal property of Pimsner-Toeplitz C*-algebras and their continuous analogues

We consider C*-algebras generated by a single Hilbert bimodule (Pimsner-Toeplitz algebras) and by a product systems of Hilbert bimodules. We give a new proof of a theorem of Pimsner, which states that any representation of the generating bimodule gives rise to a representation of the Pimsner-Toeplitz algebra. Our proof does not make use of the conditional expectation onto the subalgebra invariant under the dual action of the circle group. We then prove the analogous statement for the case of product systems, generalizing a theorem of Arveson from the case of product systems of Hilbert spaces.

Z-stability, finite dimensional tracial boundaries and continuous rank functions

We observe that a recent theorem of Sato, Toms-White-Winter and Kirchberg-Rordam also holds for certain nonunital C*-algebras. Namely, we show that an algebraically simple, separable, nuclear, nonelementary C*-algebra with strict comparison, whose cone of densely finite traces has as a base a Choquet simplex with compact, finite dimensional extreme boundary, and which admits a continuous rank function, tensorially absorbs the Jiang-Su algebra Z.

Continuous fields of C*-algebras and Lesieur's Quantum Groupoids

In two articles ([L2], [L3]), Franck Lesieur had introduced a notion of quantum groupoid, in the setting of von Neumann algebras, using intensively the notion of pseudo-multiplicative unitary, which had been introduced in a previous article of the author, in collaboration with Jean-Michel Vallin [EV]. We are here studying the analog of Lesieur's construction in a C*-framework, when the basis of the quantum groupoid is central; in this case, the C* structure obtained can be described using the tools of continuous fields of C*-algebras. This allows us to re-use as quantum groupoids some examples…

Orthogonal forms and orthogonality preservers on real function algebras revisited

In 2014, we determine the precise form of a continuous orthogonal form on a commutative real C$^*$-algebra. We also describe the general form of a (not-necessarily continuous) orthogonality preserving linear map between commutative unital real C$^*$-algebras. Among the consequences, we show that every orthogonality preserving linear bijection between commutative unital real C$^*$-algebras is continuous. In this note we revisit these results and their proofs with the idea of filling a gap in the arguments, and to extend the original conclusions.

The Cuntz semigroup of continuous functions into certain simple C*-algebras

This paper contains computations of the Cuntz semigroup of separable C*-algebras of the form C_0(X,A), where A is a unital, simple, Z-stable ASH algebra. The computations describe the Cuntz semigroup in terms of Murray-von Neumann semigroups of C(K,A) for compact subsets K of X. In particular, the computation shows that the Elliott invariant is functorially equivalent to the invariant given by the Cuntz semigroup of C(T,A). These results are a contribution towards the goal of using the Cuntz semigroup in the classification of well-behaved non-simple C*-algebras.

A few remarks on exact C(X)-algebras

We extend in this paper several results of E. Kirchberg, S. Wassermann and the author dealing with continuous fields of C*--algebras to the semi-continuous case. We provide a new characterisation of separable lower semi-continuity C*--bundles and we present exactness criteria for C(X)-algebras and unital bisimplifiable Hopf \cst-algebras.

纯 C*-代数上的连续函数

Continuous functions over a pure C*-algebra

研究纯 C*-代数上连续函数的纯性，并证明其与 ASH 代数张量积的纯性性质。

Orthogonal forms and orthogonality preservers on real function algebras

We initiate the study of orthogonal forms on a real C$^*$-algebra. Motivated by previous contributions, due to Ylinen, Jajte, Paszkiewicz and Goldstein, we prove that for every continuous orthogonal form $V$ on a commutative real C$^*$-algebra, $A$, there exist functionals $φ_1$ and $φ_2$ in $A^{*}$ satisfying $$V(x,y) = φ_1 (x y) + φ_2 (x y^*),$$ for every $x,y$ in $A$. We describe the general form of a (not-necessarily continuous) orthogonality preserving linear map between unital commutative real C$^*$-algebras. As a consequence, we show that every orthogonality preserving linear bijection …

Continous-trace $k$-graph $C^*$-algebras

A characterization is given for directed graphs that yield graph $C^*$-algebras with continuous trace. This is established for row-finite graphs with no sources first using a groupoid approach, and extended to the general case via the Drinen-Tomforde desingularization. Partial results are given to characterize higher-rank graphs that yield $C^*$-algebras with continuous trace.

Joint Majorization in Continuous Matrix Algebras

Various notions of joint majorization are examined in continuous matrix algebras. The relative strengths of these notions are established via proofs and examples. In addition, the closed convex hulls of joint unitary orbits are completely characterized in continuous matrix algebras via notions of joint majorization. Some of these characterizations are extended to subhomogeneous C$^*$-algebras.

On continuous fields of JB-algebras

We introduce and study continuous fields of JB-algebras (which are real non-associate analogues of C*-algebras). In particular, we show that for the universal enveloping C*-algebra C*sub-u(B) for the JB-algebra B defined by a continuous field of JB-algebras A-sub-t, t belongs to T, on a locally compact space T, there exists a decomposition of C*-sub-u(B) into a continuous field of C*-algebras C*u(A-sub-t), t belongs to T, on the same space T, composed entirely of the universal enveloping C*-algebras of the corresponding JB-algebras from the aforementioned decomposition of the algebra B.

连续逆代数的完备化未必是连续逆代数

The completion of a continuous inverse algebra need not be a continuous inverse algebra

该研究解决了Neeb2006年提出的连续逆代数完备化问题，构造了一个非交换反例，证明其Hausdorff完备化虽保持逆运算连续性但可逆元集不开，破坏了单位元处的局部谱稳定性，明确了交换完备化定理的边界。

Continuous Fields of Kirchberg C*-algebras

In this paper we study the C*-algebras associated to continuous fields over locally compact metrisable zero dimensional spaces whose fibers are Kirchberg C*-algebras satisfying the UCT. We show that these algebras are inductive limits of finite direct sums of Kirchberg algebras and they are classified up to isomorphism by topological invariants.

One-sided topological conjugacy of topological Markov shifts, continuous full groups and Cuntz--Krieger algebras

We will characterize topological conjugate one-sided topological Markov shifts in terms of their subgroups of continuous full groups and subalgebras of Cuntz--Krieger algebras.

Embedding of exact C*-algebras and continuous fields in the Cuntz algebra O_2

We prove that any separable exact C*-algebra is isomorphic to a subalgebra of the Cuntz algebra ${\cal O}_2.$ We further prove that if $A$ is a simple separable unital nuclear C*-algebra, then ${\cal O}_2 \otimes A \cong {\cal O}_2,$ and if, in addition, $A$ is purely infinite, then ${\cal O}_{\infty} \otimes A \cong A.$ The embedding of exact C*-algebras in $\OA{2}$ is continuous in the following sense. If $A$ is a continuous field of C*-algebras over a compact manifold or finite CW complex $X$ with fiber $A (x)$ over $x \in X,$ such that the algebra of continuous sections of $A$ is separable…

A Continuous Field of C*-algebras and the Tangent Groupoid for Manifolds with Boundary

For a smooth manifold with boundary we construct a semigroupoid and a continuous field of C*-algebras which extend Connes' construction of the tangent groupoid. We show the asymptotic multiplicativity of \hbar-scaled truncated pseudodifferential operators with smoothing symbols and compute the K-theory of the associated symbol algebra.

One-parameter continuous fields of Kirchberg algebras with rational K-theory

We show that separable continuous fields over the unit interval whose fibers are stable Kirchberg algebras that satisfy the universal coefficient theorem in KK-theory and have rational K-theory groups are classified up to isomorphism by filtrated K-theory.

Continuous orbit equivalence of topological Markov shifts and KMS states on Cuntz--Krieger algebras

We study KMS states for gauge actions with potential functions on Cuntz--Krieger algebras whose underlying one-sided topological Markov shifts are continuous orbit equivalent. As a result, we have a certain relationship between topological entropy of continuous orbit equivalent one-sided topological Markov shifts.

Asymptotic continuous orbit equivalence of Smale spaces and Ruelle algebras

In the first part of the paper, we introduce notions of asymptotic continuous orbit equivalence and asymptotic conjugacy in Smale spaces and characterize them in terms of their asymptotic Ruelle algebras with their dual actions. In the second part, we introduce a groupoid $C^*$-algebra which is an extended version of the asymptotic Ruelle algebra from a Smale space and study the extended Ruelle algebras from the view points of Cuntz--Krieger algebras. As a result, the asymptotic Ruelle algebra is realized as a fixed point algebra of the extended Ruelle algebra under certain circle action.

Projectivity of Banach and $C^*$-algebras of continuous fields

We give necessary and sufficient conditions for the left projectivity and biprojectivity of Banach algebras defined by locally trivial continuous fields of Banach algebras. We identify projective $C^*$-algebras $\A$ defined by locally trivial continuous fields $\mathcal{U} = \{Ω,(A_t)_{t \in Ω},Θ\}$ such that each $C^*$-algebra $ A_{t}$ has a strictly positive element. For a commutative $C^*$-algebra $\D$ contained in ${\cal B}(H)$, where $H$ is a separable Hilbert space, we show that the condition of left projectivity of $\D$ is equivalent to the existence of a strictly positive element in $\…

Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz--Krieger algebras

We will study torus actions on Cuntz--Krieger algebras trivially acting on its canonical maximal abelian $C^*$-subalgebras from the view points of continuous orbit equivalence of one-sided topological Markov shifts and flow equivalence of two-sided topological Markov shifts.

Classification of certain continuous fields of Kirchberg algebras

We show that the K-theory cosheaf is a complete invariant for separable continuous fields with vanishing boundary maps over a finite-dimensional compact metrizable topological space whose fibers are stable Kirchberg algebras with rational K-theory groups satisfying the universal coefficient theorem. We provide a range result for fields in this class with finite-dimensional K-theory. There are versions of both results for unital continuous fields.

Lie group extensions associated to projective modules of continuous inverse algebras

We call a unital locally convex algebra $A$ a continuous inverse algebra if its unit group $A^\times$ is open and inversion is a continuous map. For any smooth action of a, possibly infinite-dimensional, connected Lie group $G$ on a continuous inverse algebra $A$ by automorphisms and any finitely generated projective right $A$-module $E$, we construct a Lie group extension $\hat G$ of $G$ by the group $\GL_A(E)$ of automorphisms of the $A$-module $E$. This Lie group extension is a ``non-commutative'' version of the group $\Aut(\V)$ of automorphism of a vector bundle over a compact manifold $M$…

On some applications of representable and continuous functionals of Banach quasi *-algebras

This survey aims to highlight some of the consequences that representable (and continuous) functionals have in the framework of Banach quasi *-algebras. In particular, we look at the link between the notions of *-semisimplicity and full representability in which representable functionals are involved. Then, we emphasize their essential role in studying *-derivations and representability properties for the tensor product of Hilbert quasi *-algebras, a special class of Banach quasi *-algebras.

The Mackey bijection for complex reductive groups and continuous fields of reduced group C*-algebras

The purpose of this paper is to make a further contribution to the Mackey bijection for a complex reductive group G, between the tempered dual of G and the unitary dual of the associated Cartan motion group. We shall construct an embedding of the C*-algebra of the motion group into the reduced C*-algebra of G, and use it to characterize the continuous field of reduced group C*-algebras that is associated to the Mackey bijection. We shall also obtain a new characterization of the Mackey bijection using the same embedding.

Representable and continuous functionals on Banach quasi *-algebras

In the study of locally convex quasi *-algebras an important role is played by representable linear functionals; i.e., functionals which allow a GNS-construction. This paper is mainly devoted to the study of the continuity of representable functionals in Banach and Hilbert quasi *-algebras. Some other concepts related to representable functionals (full-representability, *-semisimplicity, etc) are revisited in these special cases. In particular, in the case of Hilbert quasi *-algebras, which are shown to be fully representable, the existence of a 1-1 correspondence between positive, bounded ele…

Universal Continuous Calculus for Su*-Algebras

Universal continuous calculi are defined and it is shown that for every finite tuple of pairwise commuting Hermitian elements of a Su*-algebra (an ordered *-algebra that is symmetric, i.e. "strictly" positive elements are invertible, and uniformly complete), such a universal continuous calculus exists. This generalizes the continuous calculus for C*-algebras to a class of generally unbounded ordered *-algebras. On the way, some results about *-algebras of continuous functions on locally compact spaces are obtained. The approach used throughout is rather elementary and especially avoids any rep…