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In this paper, we construct uncountably many examples of multiparameter CCR flows, which are not pullbacks of $1$-parameter CCR flows, with any given index. Moreover, the constructed CCR flows are type I in the sense that the associated product system is the smallest subsystem containing its units.
We investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterize the topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and C*-algebras. This significantly generalizes recent work of Matsumoto and of the second- and third-named authors.
We introduce Poisson boundaries of II$_1$ factors with respect to density operators that give the traces. The Poisson boundary is a von Neumann algebra that contains the II$_1$ factor and is a particular example of the boundary of a unital completely positive map as introduced by Izumi. Studying the inclusion of the II$_1$ factor into its boundary, we develop a number of notions, such as double ergodicity and entropy, that can be seen as natural analogues of results regarding the Poisson boundaries introduced by Furstenberg. We use the techniques developed to answer a problem of Popa by showing that all finite factors satisfy his MV property. We also extend a result of Nevo by showing that property (T) factors give rise to an entropy gap.
For a state $\omega$ on a C$^{*}$-algebra $A$, we characterize all states $\rho$ in the weak* closure of the set of all states of the form $\omega \circ \varphi$, where $\varphi$ is a map on $A$ of the form $\varphi (x)=\sum \nolimits _{i=1}^{n}a_i^{*}xa_i,$$\sum \nolimits _{i=1}^{n}a_i^{*}a_i=1$ ($a_i\in A$, $n\in \mathbb {N}$). These are precisely the states $\rho$ that satisfy $\|\rho |J\|\leq \|\omega |J\|$ for each ideal $J$ of $A$. The corresponding question for normal states on a von Neumann algebra $\mathcal {R}$ (with the weak* closure replaced by the norm closure) is also considered. All normal states of the form $\omega \circ \psi$, where $\psi$ is a quantum channel on $\mathcal {R}$ (that is, a map of the form $\psi (x)=\sum \nolimits _ja_j^{*}xa_j$, where $a_j\in \mathcal {R}$ are such that the sum $\sum \nolimits _ja_j^{*}a_j$ converge to $1$ in the weak operator topology) are characterized. A variant of this topic for hermitian functionals instead of states is investigated. Maximally mixed states are shown to vanish on the strong radical of a C$^{*}$-algebra and for properly infinite von Neumann algebras the converse also holds.
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.
We prove that many, but not all, injective factors arise as crossed products by nonsingular Bernoulli actions of the group
$\mathbb {Z}$
. We obtain this result by proving a completely general result on the ergodicity, type and Krieger’s associated flow for Bernoulli shifts with arbitrary base spaces. We prove that the associated flow must satisfy a structural property of infinite divisibility. Conversely, we prove that all almost periodic flows, as well as many other ergodic flows, do arise as associated flow of a weakly mixing Bernoulli action of any infinite amenable group. As a byproduct, we prove that all injective factors with almost periodic flow of weights are infinite tensor products of
$2 \times 2$
matrices. Finally, we construct Poisson suspension actions with prescribed associated flow for any locally compact second countable group that does not have property (T).
We establish a theory of noncommutative (NC) functions on a class of von Neumann algebras with a particular direct sum property, e.g., $B({\mathcal H})$. In contrast to the theory’s origins, we do not rely on appealing to results from the matricial case. We prove that the $k{\mathrm {th}}$ directional derivative of any NC function at a scalar point is a k-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g., the NC polydisk and NC row ball.
We show that the properties of being rationally K-stable passes from the fibres 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.
A
$C^{*}$
-algebra A is said to detect nuclearity if, whenever a
$C^{*}$
-algebra B satisfies
$A\otimes _{\mathrm{min}} B = A\otimes _{\mathrm{max}} B,$
it follows that B is nuclear. In this note, we survey the main results associated with this topic and present the background and tools necessary for proving the main results. In particular, we show that the
$C^{*}$
-algebra
$A = C^{*}(\mathbb {F}_{\infty })\otimes _{\mathrm{min}} B(\ell ^{2})/K(\ell ^{2})$
detects nuclearity. This result is known to experts, but has never appeared in the literature.
Scarparo has constructed counterexamples to Matui’s HK-conjecture. These counterexamples and other known counterexamples are essentially principal but not principal. In the present paper, a counterexample to the HK-conjecture that is principal is given. Like Scarparo’s original counterexample, our counterexample is the transformation groupoid associated to a particular odometer. However, the relevant group is the fundamental group of a flat manifold (and hence is torsion-free) and the associated odometer action is free. The examples discussed here do satisfy the rational version of the HK-conjecture.
We initiate the study of C*-algebras and groupoids arising from left regular representations of Garside categories, a notion which originated from the study of Braid groups. Every higher rank graph is a Garside category in a natural way. We develop a general classification result for closed invariant subspaces of our groupoids as well as criteria for topological freeness and local contractiveness, properties which are relevant for the structure of the corresponding C*-algebras. Our results provide a conceptual explanation for previous results on gauge-invariant ideals of higher rank graph C*-algebras. As another application, we give a complete analysis of the ideal structures of C*-algebras generated by left regular representations of Artin–Tits monoids.
We examine a semigroup analogue of the Kumjian–Renault representation of C*-algebras with Cartan subalgebras on twisted groupoids. Specifically, we represent semigroups with distinguished normal subsemigroups as ‘slice-sections’ of groupoid bundles.
We introduce certain
$C^*$
-algebras and k-graphs associated to k finite-dimensional unitary representations
$\rho _1,\ldots ,\rho _k$
of a compact group G. We define a higher rank Doplicher-Roberts algebra
$\mathcal {O}_{\rho _1,\ldots ,\rho _k}$
, constructed from intertwiners of tensor powers of these representations. Under certain conditions, we show that this
$C^*$
-algebra is isomorphic to a corner in the
$C^*$
-algebra of a row-finite rank k graph
$\Lambda $
with no sources. For G finite and
$\rho _i$
faithful of dimension at least two, this graph is irreducible, it has vertices
$\hat {G}$
and the edges are determined by k commuting matrices obtained from the character table of the group. We illustrate this with some examples when
$\mathcal {O}_{\rho _1,\ldots ,\rho _k}$
is simple and purely infinite, and with some K-theory computations.
Given a self-similar set K defined from an iterated function system
$\Gamma =(\gamma _{1},\ldots ,\gamma _{d})$
and a set of functions
$H=\{h_{i}:K\to \mathbb {R}\}_{i=1}^{d}$
satisfying suitable conditions, we define a generalized gauge action on Kajiwara–Watatani algebras
$\mathcal {O}_{\Gamma }$
and their Toeplitz extensions
$\mathcal {T}_{\Gamma }$
. We then characterize the KMS states for this action. For each
$\beta \in (0,\infty )$
, there is a Ruelle operator
$\mathcal {L}_{H,\beta }$
, and the existence of KMS states at inverse temperature
$\beta $
is related to this operator. The critical inverse temperature
$\beta _{c}$
is such that
$\mathcal {L}_{H,\beta _{c}}$
has spectral radius 1. If
$\beta <\beta _{c}$
, there are no KMS states on
$\mathcal {O}_{\Gamma }$
and
$\mathcal {T}_{\Gamma }$
; if
$\beta =\beta _{c}$
, there is a unique KMS state on
$\mathcal {O}_{\Gamma }$
and
$\mathcal {T}_{\Gamma }$
which is given by the eigenmeasure of
$\mathcal {L}_{H,\beta _{c}}$
; and if
$\beta>\beta _{c}$
, including
$\beta =\infty $
, the extreme points of the set of KMS states on
$\mathcal {T}_{\Gamma }$
are parametrized by the elements of K and on
$\mathcal {O}_{\Gamma }$
by the set of branched points.
We consider two inclusions of $C^{*}$-algebras whose small $C^{*}$-algebras have approximate units of the large $C^{*}$-algebras and their two spaces of all bounded bimodule linear maps. We suppose that the two inclusions of $C^{*}$-algebras are strongly Morita equivalent. In this paper, we shall show that there exists an isometric isomorphism from one of the spaces of all bounded bimodule linear maps to the other space and we shall study the basic properties about the isometric isomorphism. And, using this isometric isomorphism, we define the Picard group for a bimodule linear map and discuss the Picard group for a bimodule linear map.
We demonstrate how exact structures can be placed on the additive category of right operator modules over an operator algebra in order to discuss global dimension for operator algebras. The properties of the Haagerup tensor product play a decisive role in this.
In this paper, let A be an infinite-dimensional stably finite unital simple separable
$\mathrm {C^*}$
-algebra. Let
$B\subset A$
be a centrally large subalgebra in A such that B has uniform property
$\Gamma $
. Then we prove that A has uniform property
$\Gamma $
. Let
$\Omega $
be a class of stably finite unital
$\mathrm {C^*}$
-algebras such that for any
$B\in \Omega $
, B has uniform property
$\Gamma $
. Then we show that A has uniform property
$\Gamma $
for any simple unital
$\mathrm {C^*}$
-algebra
$A\in \rm {TA}\Omega $
.
We show that if G is an amenable group and H is a hyperbolic group, then the free product
$G\ast H$
is weakly amenable. A key ingredient in the proof is the fact that
$G\ast H$
is orbit equivalent to
$\mathbb{Z}\ast H$
.
We resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically, we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis, Inter. Math. Res. Not.2014 (2014), 1289–1311 relating to work of Arveson, Acta Math.118 (1967), 95–109 from the 1960s, and extends related work of Kakariadis and Katsoulis, J. Noncommut. Geom.8 (2014), 771–787.
We construct two types of unital separable simple
$C^*$
-algebras:
$A_z^{C_1}$
and
$A_z^{C_2}$
, one exact but not amenable, the other nonexact. Both have the same Elliott invariant as the Jiang–Su algebra – namely,
$A_z^{C_i}$
has a unique tracial state,
and
$K_{1}\left (A_z^{C_i}\right )=\{0\}$
(
$i=1,2$
). We show that
$A_z^{C_i}$
(
$i=1,2$
) is essentially tracially in the class of separable
${\mathscr Z}$
-stable
$C^*$
-algebras of nuclear dimension
$1$
.
$A_z^{C_i}$
has stable rank one, strict comparison for positive elements and no
$2$
-quasitrace other than the unique tracial state. We also produce models of unital separable simple nonexact (exact but not nuclear)
$C^*$
-algebras which are essentially tracially in the class of simple separable nuclear
${\mathscr Z}$
-stable
$C^*$
-algebras, and the models exhaust all possible weakly unperforated Elliott invariants. We also discuss some basic properties of essential tracial approximation.