KMS states on the crossed product -algebra of a homeomorphism

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 $\mathbb R$ .

The block Schur product is a Hadamard product

Given two $n \times n $ matrices $A = (a_{ij})$ and $B=(b_{ij}) $ with entries in $B(H)$ for some Hilbert space $H$, their block Schur product is the $n \times n$ matrix $ A\square B := (a_{ij}b_{ij})$. Given two continuous functions $f$ and $g$ on the torus with Fourier coefficients $(f_n)$ and $(g_n)$ their convolution product $f \star g$ has Fourier coefficients $(f_n g_n)$. Based on this, the Schur product on scalar matrices is also known as the Hadamard product. We show that for a C*-algebra $\mathcal{A} $, and a discrete group $G$ with an action $\alpha _g$ of $G$ on $\mathcal{A} $ by *-automorphisms, the reduced crossed product C*-algebra $\mathrm {C}^*_r(\mathcal{A} , \alpha , G)$ possesses a natural generalization of the convolution product, which we suggest should be named the Hadamard product. We show that this product has a natural Stinespring representation and we lift some known results on block Schur products to this setting, but we also show that the block Schur product is a special case of the Hadamard product in a crossed product algebra.

Dynamics of compact quantum metric spaces

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.

Smooth crossed product of minimal unique ergodic diffeomorphisms of a manifold and cyclic cohomology

Different diffeomorphisms can give the same [Formula: see text] crossed product algebra. Our main purpose is to show that we can still classify dynamical systems with some appropriate smooth crossed product algebras when their corresponding [Formula: see text] crossed product algebras are isomorphic. For this purpose, we construct two minimal unique ergodic diffeomorphisms [Formula: see text] and [Formula: see text] of [Formula: see text]. The [Formula: see text] algebras classification theory, smooth crossed product algebras considered by R. Nest and cyclic cohomology are used to show that [Formula: see text] and [Formula: see text] give the same [Formula: see text] algebra and induce different smooth crossed product algebras.

SOLVABLE CROSSED PRODUCT ALGEBRAS REVISITED

For any central simple algebra over a field F which contains a maximal subfield M with non-trivial automorphism group G = AutF(M), G is solvable if and only if the algebra contains a finite chain of subalgebras which are generalized cyclic algebras over their centers (field extensions of F) satisfying certain conditions. These subalgebras are related to a normal subseries of G. A crossed product algebra F is hence solvable if and only if it can be constructed out of such a finite chain of subalgebras. This result was stated for division crossed product algebras by Petit and overlaps with a similar result by Albert which, however, was not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F.

The Jiang–Su Absorption for Inclusions of Unital C*-algebras

We introduce the tracial Rokhlin property for a conditional expectation for an inclusion of unital C*-algebras P ⊂ A with index finite, and show that an action α from a finite group G on a simple unital C*- algebra A has the tracial Rokhlin property in the sense of N. C. Phillips if and only if the canonical conditional expectation E: A → AG has the tracial Rokhlin property. Let be a class of infinite dimensional stably finite separable unital C*-algebras that is closed under the following conditions:(1) If A ∊ and B ≅ A, then B ∊ .(2) If A ∊ and n ∊ ℕ, then Mn(A) ∊ .(3) If A ∊ and p ∊ A is a nonzero projection, then pAp ∊ .Suppose that any C*-algebra in is weakly semiprojective. We prove that if A is a local tracial -algebra in the sense of Fan and Fang and a conditional expectation E: A → P is of index-finite type with the tracial Rokhlin property, then P is a unital local tracial -algebra.The main result is that if A is simple, separable, unital nuclear, Jiang–Su absorbing and E: A → P has the tracial Rokhlin property, then P is Jiang–Su absorbing. As an application, when an action α from a finite group G on a simple unital C*-algebra A has the tracial Rokhlin property, then for any subgroup H of G the fixed point algebra AH and the crossed product algebra H is Jiang–Su absorbing. We also show that the strict comparison property for a Cuntz semigroup W(A) is hereditary to W(P) if A is simple, separable, exact, unital, and E: A → P has the tracial Rokhlin property.

TILTED ALGEBRAS AND CROSSED PRODUCTS

We consider an artin algebra A and its crossed product algebra Aα#σG, where G is a finite group with its order invertible in A. Then, we prove that A is a tilted algebra if and only if so is Aα#σG.

ON CROSSED DOUBLE BIPRODUCT

Let H be a bialgebra. Let σ : H ⊗ H → A be a linear map, where A is a left H-comodule coalgebra, and an algebra with a left H-weak action. Let B be a right H-module algebra and also a comodule coalgebra. In this paper, we provide necessary and sufficient conditions for the one-sided crossed product algebra A#σ H # B and the two-sided smash coproduct coalgebra A × H × B to form a bialgebra, which we call the crossed double biproduct. Majid's double biproduct is recovered from this. Moreover, necessary and sufficient conditions are given for Brzeziński's crossed product equipped with the smash coproduct coalgebra structure to be a bialgebra. The celebrated Radford's biproduct in [The structure of Hopf algebra with a projection, J. Algebra92 (1985) 322–347], the unified product defined by Agore and Militaru in [Extending structures II: The quantum version, J. Algebra336 (2011) 321–341] and the Wang–Jiao–Zhao's crossed product in [Hopf algebra structures on crossed products Comm. Algebra26 (1998) 1293–1303] are all derived as special cases.

The Linear Span of Projections in AH Algebras and for Inclusions ofC*-Algebras

In the first part of this paper, we show that an AH algebraA=lim→(Ai,ϕi)has the LP property if and only if every element of the centre ofAibelongs to the closure of the linear span of projections inA. As a consequence, a diagonal AH-algebra has the LP property if it has small eigenvalue variation in the sense of Bratteli and Elliott. The second contribution of this paper is that for an inclusion of unitalC*-algebrasP⊂Awith a finite Watatani index, if a faithful conditional expectationE:A→Phas the Rokhlin property in the sense of Kodaka et al., thenPhas the LP property under the condition thatAhas the LP property. As an application, letAbe a simple unitalC*-algebra with the LP property,αan action of a finite groupGontoAut(A). Ifαhas the Rokhlin property in the sense of Izumi, then the fixed point algebraAGand the crossed product algebraA ⋊α Ghave the LP property. We also point out that there is a symmetry on the CAR algebra such that its fixed point algebra does not have the LP property.

Certain properties for crossed products by automorphisms with a certain non-simple tracial Rokhlin property

Let $\Omega $ be a class of unital $C^*$-algebras. Then any simple unital $C^*$-algebra $A\in \mathrm {TA}(\mathrm {TA}\Omega )$ is a $\mathrm {TA}\Omega $ algebra. Let $A\in \mathrm {TA}\Omega $ be an infinite-dimensional $\alpha $-simple unital $C^*$-algebra with the property SP. Suppose that $\alpha :G\to \mathrm {Aut}(A)$ is an action of a finite group $G$ on $A$ which has a certain non-simple tracial Rokhlin property. Then the crossed product algebra $C^*(G,A,\alpha )$ belongs to $\mathrm {TA}\Omega $.