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The Toms–Winter conjecture is verified for those separable, unital, nuclear, infinite-dimensional real C*-algebras for which the complexification has a tracial state space with compact extreme boundary of finite covering dimension.
One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups.
We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient-
${\mathcal {S}}_2$
condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).
For a given inverse semigroup, one can associate an étale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated étale groupoids. In this paper, we focus on congruences of inverse semigroups, which is a fundamental concept for considering quotients of inverse semigroups. We prove that a congruence of an inverse semigroup induces a closed invariant set and a normal subgroupoid of the universal groupoid. Then we show that the universal groupoid associated to a quotient inverse semigroup is described by the restriction and quotient of the original universal groupoid. Finally we compute invariant sets and normal subgroupoids induced by special congruences including abelianization.
Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let
$\alpha \colon G \to {\text{Aut}} (A)$ be an action of G on A which has the weak tracial Rokhlin property. Let
$A^{\alpha}$ be the fixed point algebra. Then the radius of comparison satisfies
${\text{rc}} (A^{\alpha }) \leq {\text{rc}} (A)$ and
${\text{rc}} ( C^* (G, A, \alpha ) ) \leq ({1}/{\text{card} (G))} \cdot {\text{rc}} (A)$. The inclusion of
$A^{\alpha }$ in A induces an isomorphism from the purely positive part of the Cuntz semigroup
${\text{Cu}} (A^{\alpha })$ to the fixed points of the purely positive part of
${\text{Cu}} (A)$, and the purely positive part of
${\text{Cu}} ( C^* (G, A, \alpha ) )$ is isomorphic to this semigroup. We construct an example in which
$G \,{=}\, {\mathbb {Z}} / 2 {\mathbb {Z}}$, A is a simple unital AH algebra,
$\alpha $ has the Rokhlin property,
${\text{rc}} (A)> 0$,
${\text{rc}} (A^{\alpha }) = {\text{rc}} (A)$, and
${\text{rc}} (C^* (G, A, \alpha ) = ( {1}/{2}) {\text{rc}} (A)$.
Let
$\phi :X\to X$ be a homeomorphism of a compact metric space X. For any continuous function
$F:X\to \mathbb {R}$ there is a one-parameter group
$\alpha ^{F}$ of automorphisms (or a flow) on the crossed product
$C^*$-algebra
$C(X)\rtimes _{\phi }\mathbb {Z}$ defined such that
$\alpha ^{F}_{t}(fU)=fUe^{-itF}$ when
$f \in C(X)$ and U is the canonical unitary in the construction of the crossed product. In this paper we study the Kubo--Martin--Schwinger (KMS) states for these flows by developing an intimate relation to the ergodic theory of non-singular transformations and show that the structure of KMS states can be very rich and complicated. Our results are complete concerning the set of possible inverse temperatures; in particular, we show that when
$C(X) \rtimes _{\phi } \mathbb Z$ is simple this set is either
$\{0\}$ or the whole line
$\mathbb R$.
We introduce an index for symmetry-protected topological (SPT) phases of infinite fermionic chains with an on-site symmetry given by a finite group G. This index takes values in
$\mathbb {Z}_2 \times H^1(G,\mathbb {Z}_2) \times H^2(G, U(1)_{\mathfrak {p}})$ with a generalised Wall group law under stacking. We show that this index is an invariant of the classification of SPT phases. When the ground state is translation invariant and has reduced density matrices with uniformly bounded rank on finite intervals, we derive a fermionic matrix product representative of this state with on-site symmetry.
In this paper, we prove the equivalence between logarithmic Sobolev inequality and hypercontractivity of a class of quantum Markov semigroup and its associated Dirichlet form based on a probability gage space.
Since their inception in the 1930s by von Neumann, operator algebras have been used to shed light on many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two has been sought since their emergence in the late 1960s. We connect these seemingly separate types of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and $C^{*}$-algebras with additional $C^{*}$-algebraic structure. Our approach naturally applies to algebras arising from $C^{*}$-correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs.
We construct a Baum–Connes assembly map localised at the unit element of a discrete group $\Gamma$. This morphism, called $\mu _\tau$, is defined in $KK$-theory with coefficients in $\mathbb {R}$ by means of the action of the idempotent $[\tau ]\in KK_{\mathbin {{\mathbb {R}}}}^\Gamma (\mathbb {C},\mathbb {C})$ canonically associated to the group trace of $\Gamma$. We show that the corresponding $\tau$-Baum–Connes conjecture is weaker than the classical version, but still implies the strong Novikov conjecture. The right-hand side of $\mu _\tau$ is functorial with respect to the group $\Gamma$.
Let
$ H $
be a compact subgroup of a locally compact group
$ G $
. We first investigate some (operator) (co)homological properties of the Fourier algebra
$A(G/H)$
of the homogeneous space
$G/H$
such as (operator) approximate biprojectivity and pseudo-contractibility. In particular, we show that
$ A(G/H) $
is operator approximately biprojective if and only if
$ G/H $
is discrete. We also show that
$A(G/H)^{**}$
is boundedly approximately amenable if and only if G is compact and H is open. Finally, we consider the question of existence of weakly compact multipliers on
$A(G/H)$
.
Suppose that
$\mathcal {A}$
is a unital subhomogeneous C*-algebra. We show that every central sequence in
$\mathcal {A}$
is hypercentral if and only if every pointwise limit of a sequence of irreducible representations is multiplicity free. We also show that every central sequence in
$\mathcal {A}$
is trivial if and only if every pointwise limit of irreducible representations is irreducible. Finally, we give a nice representation of the latter algebras.
Bożejko and Speicher associated a finite von Neumann algebra MT to a self-adjoint operator T on a complex Hilbert space of the form $\mathcal {H}\otimes \mathcal {H}$ which satisfies the Yang–Baxter relation and $ \left\| T \right\| < 1$. We show that if dim$(\mathcal {H})$ ⩾ 2, then MT is a factor when T admits an eigenvector of some special form.
In this paper, we characterize the multiple operator integrals mappings that are bounded on the Haagerup tensor product of spaces of compact operators. We show that such maps are automatically completely bounded and prove that this is equivalent to a certain factorization property of the symbol associated with the operator integral mapping. This generalizes a result by Juschenko-Todorov-Turowska on the boundedness of measurable multilinear Schur multipliers.
For a locally compact Hausdorff space X and a C*-algebra A with only finitely many closed ideals, we discuss a characterization of closed ideals of C0(X,A) in terms of closed ideals of A and a class of closed subspaces of X. We further use this result to prove that a closed ideal of C0(X)⊗minA is a finite sum of product ideals. We also establish that for a unital C*-algebra A, C0(X,A) has the centre-quotient property if and only if A has the centre-quotient property. As an application, we characterize the closed Lie ideals of C0(X,A) and identify all the closed Lie ideals of HC0(X)⊗minB(H), H being a separable Hilbert space.
Let
$(G,\unicode[STIX]{x1D6EC})$
be a self-similar
$k$
-graph with a possibly infinite vertex set
$\unicode[STIX]{x1D6EC}^{0}$
. We associate a universal C*-algebra
${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$
to
$(G,\unicode[STIX]{x1D6EC})$
. The main purpose of this paper is to investigate the ideal structures of
${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$
. We prove that there exists a one-to-one correspondence between the set of all
$G$
-hereditary and
$G$
-saturated subsets of
$\unicode[STIX]{x1D6EC}^{0}$
and the set of all gauge-invariant and diagonal-invariant ideals of
${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$
. Under some conditions, we characterize all primitive ideals of
${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$
. Moreover, we describe the Jacobson topology of some concrete examples, which includes the C*-algebra of the product of odometers. On the way to our main results, we study self-similar
$P$
-graph C*-algebras in depth.
We correct an error in the statement in a proposition and a theorem in Jiang–Su absorption for inclusions of unital C*-algebras. Canad. J. Math. 70(2018), 400–425. This error was found by Dr. M. Ali Asadi-Vasfi and communicated to the authors by N. Christopher Phillips of the University of Oregon, who also suggested the outline for the following correct proofs.
We prove that Voiculescu’s noncommutative version of the Weyl-von Neumann Theorem can be extended to all unital, separably representable
$\mathrm {C}^\ast $
-algebras whose density character is strictly smaller than the (uncountable) cardinal invariant
$\mathfrak {p}$
. We show moreover that Voiculescu’s Theorem consistently fails for
$\mathrm {C}^\ast $
-algebras of larger density character.
Let
$\Theta =(\theta _{j,k})_{3\times 3}$
be a nondegenerate real skew-symmetric
$3\times 3$
matrix, where
$\theta _{j,k}\in [0,1).$
For any
$\varepsilon>0$
, we prove that there exists
$\delta>0$
satisfying the following: if
$v_1,v_2,v_3$
are three unitaries in any unital simple separable
$C^*$
-algebra A with tracial rank at most one, such that
$\begin{align*}\|v_kv_j-e^{2\pi i \theta_{j,k}}v_jv_k\|<\delta \,\,\,\, \mbox{and}\,\,\,\, \frac{1}{2\pi i}\tau(\log_{\theta}(v_kv_jv_k^*v_j^*))=\theta_{j,k}\end{align*}$
for all
$\tau \in T(A)$
and
$j,k=1,2,3,$
where
$\log _{\theta }$
is a continuous branch of logarithm (see Definition 4.13) for some real number
$\theta \in [0, 1)$
, then there exists a triple of unitaries
$\tilde {v}_1,\tilde {v}_2,\tilde {v}_3\in A$
such that
The same conclusion holds if
$\Theta $
is rational or nondegenerate and A is a nuclear purely infinite simple
$C^*$
-algebra (where the trace condition is vacuous).
If
$\Theta $
is degenerate and A has tracial rank at most one or is nuclear purely infinite simple, we provide some additional injectivity conditions to get the above conclusion.
We provide a detailed study of actions of the integers on compact quantum metric spaces, which includes general criteria ensuring that the associated crossed product algebra is again a compact quantum metric space in a natural way. Moreover, we provide a flexible set of assumptions ensuring that a continuous family of
$\ast$
-automorphisms of a compact quantum metric space yields a field of crossed product algebras which varies continuously in Rieffel’s quantum Gromov–Hausdorff distance. Finally, we show how our results apply to continuous families of Lip-isometric actions on compact quantum metric spaces and to families of diffeomorphisms of compact Riemannian manifolds which vary continuously in the Whitney
$C^{1}$
-topology.