Examples of (strongly continuous) one-parameter groups of automorphisms of UHF C*-algebras are given. One example has the property that the infinitesimal generator does not have a union of finite-dimensional subalgebras as a core. (This solves the so-called core problem, raised by S. Sakai, in the negative.) The other has the property of not being classical or commutative, i.e., the one-parameter group is not just an inner perturbation of a one-parameter group which has an increasing sequence of invariant finite-dimensional subalgebras with dense union. (This was certainly expected but has not been proven explicitly.)
We study the triangular subalgebras of UHF algebras which provide new examples of algebras with the Dirichlet property and the Ando property. This in turn allows us to describe the semicrossed product by an isometric automorphism. We also study the isometric automorphism group of these algebras and prove that it decomposes into the semidirect product of an abelian group by a torsion free group. Various other structure results are proven as well.
Glimm's theorem says that a UHF algebra is almost embedded in a separable $C^*$-algebra not of type I. Applying his methods we obtain a covariant version of his result; a UHF algebra with a product type automorphism is covariantly embedded in such a $C^*$-algebra equipped with an automorphism with full Connes spectrum.
A UHF flow is an infinite tensor product type action of the reals on a UHF algebra $A$ and the flip automorphism is an automorphism of $A\otimes A$ sending $x\otimes y$ into $y\otimes x$. If $α$ is an inner perturbation of a UHF flow on $A$, there is a sequence $(u_n)$ of unitaries in $A\otimes A$ such that $α_t\otimes α_t(u_n)-u_n$ converges to zero and the flip is the limit of $\Ad u_n$. We consider here whether the converse holds or not and solve it with an additional assumption: If $A\otimes A\cong A$ and $α$ absorbs any UHF flow $β$ (i.e., $α\otimesβ$ is cocycle conjugate to $α$), then th…
Automorphisms of the canonical core UHF-subalgebra F_n of the Cuntz algebra O_n do not necessarily extend to automorphisms of O_n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras M_n. In that case, necessary and sufficient conditions for the extension property are presented. It is also addressed the problem of extending to O_n the automorphisms of the diagonal D_n, which is a regular MASA with Cantor spectrum. In particular, it is shown the existence of product-type automorphisms of D_n that are not extensible to (poss…
We give a complete classification up to cocycle conjugacy of uniformly outer actions of Z^2 on UHF algebras. In particular, it is shown that any two uniformly outer actions of Z^2 on a UHF algebra of infinite type are cocycle conjugate. We also classify them up to outer conjugacy.
We will show that separable unital AF-algebras whose Bratteli diagrams do not allow converging two nodes into one node, can be classified up to the tensor product with the universal UHF-algebra $\Q$ only by their trace spaces. That is, if $\A$ and $\B$ are such AF-algebras, then $T(A)=T(B)$ if and only if $\A\otimes \Q \cong \B\otimes \Q$.
In this paper, we concentrate on the MF property of reduced free products of unital C*-algebras with amalgamation over finite dimensional C*-algebras. More specifically, we give a necessary and sufficient condition for a reduced free product of two UHF algebras amalgamated over a finite-dimensional C*-algebra with respect to trace preserving conditional expectations to be MF.
We show that if $A$ is $\mathcal{Z}$, $\mathcal{O}_2$, $\mathcal{O}_{\infty}$, a UHF algebra of infinite type, or the tensor product of a UHF algebra of infinite type and $\mathcal{O}_{\infty}$, then the conjugation action $\mathrm{Aut}(A) \curvearrowright \mathrm{Aut}(A)$ is generically turbulent for the point-norm topology. We moreover prove that if $A$ is either (i) a separable C*-algebra which is stable under tensoring with $\mathcal{Z}$ or $\mathcal{K}$, or (ii) a separable ${\mathrm{II}}_1$ factor which is McDuff or a free product of ${\mathrm{II}}_1$ factors, then the approximately inne…
We prove that all strongly outer Z^N-actions on a UHF algebra of infinite type are strongly cocycle conjugate to each other. We also prove that all strongly outer, asymptotically representable Z^N-actions on a unital simple AH algebra with real rank zero, slow dimension growth and finitely many extremal tracial states are cocycle conjugate to each other.
We describe the structure of ground states and ceiling states for generalized gauge actions on an UHF algebra. It is shown that both sets are affinely homeomorphic to the state space of a unital AF algebra, and that any pair of unital AF algebras can occur in this way, independently of the field of KMS states. In addtion, we study the KMS-infinity states.
Let A be a unital separable simple C*-algebra with a unique tracial state. We prove that if A is nuclear and quasidiagonal, then A tensored with the universal UHF-algebra has decomposition rank at most one. Then it is proved that A is nuclear, quasidiagonal and has strict comparison if and only if A has finite decomposition rank. For such A, we also give a direct proof that A tensored with a UHF-algebra has tracial rank zero. Applying this characterization, we obtain a counter-example to the Powers-Sakai conjecture.
For any countable discrete group $G$ with a reduced abelian subgroup of finite index, we construct an action $α$ of $G$ on the universal UHF algebra $\Qq$ using an infinite tensor product of permutation representations of $G$ and show that these actions possess some sort of Rokhlin property. The crossed product $\qag$ is then deduced to be tracially AF with a unique tracial state. We also compute the Elliott invariants in the case that $G$ is abelian.
We construct Hopf algebra isomorphisms of discrete multiplier Hopf C*-algebras, and Hopf AF C*-algebras (generalized quantum UHF algebras), from K-theoretical data. Some of the intermediate results are of independent interest, such as a result that Jordan maps of Hopf algebras intertwine antipodes, and the applications to automorphisms of Hopf algebras.
We classify equivariant *-homomorphisms between C*-dynamical systems associated to actions of finite groups on C*-algebras with the Rokhlin property. In addition, the given actions are classified. An obstruction is obtained for the Cuntz semigroup of a C*-algebra allowing such an action. We also obtain an equivariant UHF-absorption result.
In this paper we show that certain simple locally recursive subhomogeneous (RSH) C*-algebras are tracially approximately interval algebras after tensoring with the universal UHF algebra. This involves a linear algebraic encoding of the structure of the local RSH algebra allowing us to find a path through the algebra which looks like a discrete version of [0,1] and exhausts most of the algebra. We produce an actual copy of the interval and use properties of C*-algebras tensored with UHF algebras to move the honest interval underneath the discrete version. It follows from our main result that su…
We prove that every AF-algebra is isomorphic to a crossed product of a commutative AF-algebra by a partial automorphism. The case of UHF-algebras is treated in detail.
We introduce a non-symmetric tensor product of representations of UHF algebras by using Kronecker products of matrices. We prove tensor product formulae of GNS representations by product states and show examples.
It is proved that (a stabilization of) the norm-closure of a self- adjoint representation of the twisted homogeneous coordinate ring of a Tate curve contains a copy of the UHF-algebra.
We define a Rohlin property for actions of $Z^{2}$ on UHF algebras and show a noncommutative Rohlin type theorem. Among those actions with the Rohlin property, we classify product type actions up to outer conjugacy. In particular we present two classes of UHF algebras. For UHF algebras in one class, which includes the CAR algebra, there is one and only one outer conjugacy class of product type actions and for UHF algebras in the other class, contrary to the case of actions of $Z$, there are infinitely many outer conjugacy classesof product type actions.
It was shown recently by Conti, Rørdam and Szymański that there exist endomorphisms $λ_u$ of the Cuntz algebra $\mathcal{O}_n$ such that $λ_u (\mathcal{F}_n)\subseteq\mathcal{F}_n$ but $u\not\in\mathcal{F}_n$, and a question was raised if for such a $u$ there must always exist a unitary $v\in\mathcal{F}_n$ with $λ_u|_{\mathcal{F}_n} = λ_v|_{\mathcal{F}_n}$. In the present paper, we answer this question to the negative. To this end, we analyze the structure of such endomorphisms $λ_u$ for which the relative commutant $λ_u(\mathcal{F}_n)'\cap\mathcal{F}_n$ is finite dimensional.
We prove that simple, separable, monotracial UHF $L^{p}$-operator algebras are not classifiable up to (complete) isomorphism using countable structures, such as K-theoretic data, as invariants. The same assertion holds even if one only considers UHF $L^{p}$-operator algebras of tensor product type obtained from a diagonal system of similarities. For $p=2$, it follows that separable nonselfadjoint UHF operator algebras are not classifiable by countable structures up to (complete) isomorphism. Our results, which answer a question of N. Christopher Phillips, rely on Borel complexity theory, and p…
Let ${\cal A}_{0}(*)$ denote the direct sum of a certain set of UHF algebras and let ${\cal A}(*)\equiv {\bf C}\oplus {\cal A}_{0}(*)$. We introduce a non-cocommutative comultiplication $Δ_ϕ$ on ${\cal A}(*)$, and give an example of comodule-C$^{*}$-algebra of the C$^{*}$-bialgebra $({\cal A}(*),Δ_ϕ)$. With respect to $Δ_ϕ$, we define a non-symmetric tensor product of *-representations of UHF algebras and show tensor product formulas of GNS representations by product states.
This paper serves as a source of examples of Rokhlin actions or locally representable actions of finite groups on C*-algebras satisfying a certain UHF-absorption condition. We show that given any finite group $G$ and a separable, unital C*-algebra $A$ that absorbs $M_{|G|^\infty}$ tensorially, one can lift any group homomorphism $G\to\operatorname{Aut}(A)/{\approx_u}$ to an honest Rokhlin action $γ$ of $G$ on $A$. Unitality may be dropped in favour of stable rank one or being stable. If $A$ belongs to a certain class of C*-algebras that is classifiable by a suitable invariant (e.g. $K$-theory)…
For every uncountable cardinal $κ$ there are $2^κ$ nonisomorphic simple AF algebras of density character $κ$ and $2^κ$ nonisomorphic hyperfinite II$_1$ factors of density character $κ$. These estimates are maximal possible. All C*-algebras that we construct have the same Elliott invariant and Cuntz semigroup as the CAR algebra.
There are three natural ways to define UHF (uniformly hyperfinite) C*-algebras, and all three definitions are equivalent for separable algebras. In 1967 Dixmier asked whether the three definitions remain equivalent for not necessarily separable algebras. We give a complete answer to this question. More precisely, we show that in small cardinality two definitions remain equivalent, and give counterexamples in other cases. Our results do not use any additional set-theoretic axioms beyond the usual axioms, namely ZFC.
Let $1 \in A \subset B$ be an inclusion of C*-algebras of C*-index-finite type with depth 2. We try to compute topological stable rank of $B$ ($= \tsr(B)$) when $A$ has topological stable rank one. We show that $\tsr(B) \leq 2$ when $A$ is a tsr boundedly divisible algebra, in particular, $A$ is a C*-minimal tensor product $UHF \otimes D$ with $\tsr(D) = 1$. When $G$ is a finite group and $α$ is an action of $G$ on UHF, we know that a crossed product algebra $UHF \rtimes_αG$ has topological stable rank less than or equal to two. These results are affirmative datum to a generalization of a ques…
If $A_n$ is a sequence of C*-algebras, then the C*-algebra $\prod A_n / \bigoplus A_n$ is called a reduced product. We prove, assuming Todorcevic's Axiom and Martin's Axiom, that every isomorphism between two reduced products of separable, unital UHF algebras must be definable in a strong sense. As a corollary we deduce that two such reduced products $\prod A_n / \bigoplus A_n$ and $\prod B_n / \bigoplus B_n$ are isomorphic if and only if, up to an almost-permutation of $\mathbb{N}$, $A_n$ is isomorphic to $B_n$.
Unital endomorphisms of the Cuntz algebra O_n which preserve the canonical UHF-subalgebra F_n of O_n are investigated. We give examples of such endomorphisms for which the associated unitary element in O_n does not belong to F_n. One such example, in the case where n=2, arises from a construction of a unital endomorphism on O_2 which preserves the canonical UHF-subalgebra and where the relative commutant of the image in O_2 contains a copy of O_2.
The problem of inner vs outer conjugacy of subalgebras of certain graph C*-algebras is investigated. For a large class of finite graphs E, we show that whenever $α$ is a vertex-fixing quasi-free automorphism of the corresponding graph C*-algebra C*(E) such that α(\D_E)\neq\D_E, where \D_E is the canonical MASA in C*(E), then α(\D_E)\neq w\D_E w^* for all unitaries w\in C*(E). That is, the two MASAs \D_E and α(\D_E) of C*(E) are outer but not inner conjugate. Passing to an isomorphic C*-algebra by changing the underlying graph makes this result applicable to certain non quasi-free automorphisms…
Trace scaling automorphisms of stable AF algebras with dimension group totally ordered are outer conjugate if the scaling factors are the same (not equal to one). This is an adaptation of a similar result for the AFD type II_infty factor by Connes and extends the previous result for stable UHF algebras.
Evans-Hudson flows are constructed for a class of quantum dynamical semigroups with unbounded generator on UHF algebras, which appeared in \cite{Ma}. It is shown that these flows are unital and covariant. Ergodicity of the flows for the semigroups associated with partial states is also discussed.
We prove that the pure state space is homogeneous under the action of the automorphism group (or the subgroup of asymptotically inner automorphisms) for all the separable simple C*-algebras. The first result of this kind was shown by Powers for the UHF algebras some 30 years ago.
We present several equivalent conditions for $C^*$-finitely correlated states defined on the UHF algebras to be factor states and consider the types of factors generated by them. Subfactors generated by generalized quantum Markov chains defined on the gauge-invariant parts of the UHF algebras are also discussed.
We study circle actions with the Rokhlin property, in relation to their restrictions to finite subgroups. We construct examples showing the following: the restriction of a circle action with the Rokhlin property (even on a real rank zero $C^*$-algebra), need not have the Rokhlin property; and even if every restriction of a given circle action has the Rokhlin property, the circle action itself need not have it. As a positive result, we show that the restriction of a circle action with the Rokhlin property to the subgroup $\mathbb{Z}_n$ has the Rokhlin property if the underlying algebra absorbs …
We prove that the pure state space is homogeneous under the action of the automorphism group (or a certain smaller group of approximately inner automorphisms) for a fairly large class of simple separable nuclear C*-algebras, including the approximately homogeneous C*-algebras and the class of purely infinite C*-algebras which has been recently classified by Kirchberg and Phillips. This extends the known results for UHF algebras and AF algebras by Powers and Bratteli.
For a large class of C*-algebras $A$, we calculate the $K$-theory of reduced crossed products $A^{\otimes G}\rtimes_rG$ of Bernoulli shifts by groups satisfying the Baum--Connes conjecture. In particular, we give explicit formulas for finite-dimensional C*-algebras, UHF-algebras, rotation algebras, and several other examples. As an application, we obtain a formula for the $K$-theory of reduced C*-algebras of wreath products $H\wr G$ for large classes of groups $H$ and $G$. Our methods use a generalization of techniques developed by the second named author together with Joachim Cuntz and Xin Li…
We show that for any countable discrete maximally almost periodic group $G$ and any UHF algebra $A$, there exists a strongly outer product type action $α$ of $G$ on $A$. We also show the existence of countable discrete almost abelian group actions with a certain Rokhlin property on the universal UHF algebra. Consequently we get many examples of unital separable simple nuclear $C^*$-algebras with tracial rank zero and a unique tracial state appearing as crossed products.
Let $Λ$ be a countably infinite property (T) group, and let $D$ be UHF-algebra of infinite type. We prove that there exists a continuum of pairwise non (weakly) cocycle conjugate, strongly outer actions of $Λ$ on $D$. The proof consists in assigning, to any second countable abelian pro-$p$ group $G$, a strongly outer action of $Λ$ on $D$ whose (weak) cocycle conjugacy class completely remembers the group $G$. The group $G$ is reconstructed from the action through its (weak) 1-cohomology set endowed with a canonical pairing function. Our construction also shows the following stronger statement:…
We prove several results of the following general form: automorphisms of (or actions of ${\mathbb{Z}}^d$ on) certain kinds of simple separable unital C*-algebras $A$ which have a suitable version of the Rokhlin property are generic among all automorphisms (or actions), or in a suitable class of automorphisms. That is, the ones with the version of the Rokhlin property contain a dense $G_δ$-subset of the set of all such automorphisms (or actions). Specifically, we prove the following. If $A$ is stable under tensoring with the Jiang-Su algebra $Z,$ and has tracial rank zero, then automorphisms wi…
We give examples of actions of Z/2Z on AF algebras and AT algebras which demonstrate the differences between the (strict) Rokhlin property and the tracial Rokhlin property, and between (strict) approximate representability and tracial approximate representability. Specific results include the following. We determine exactly when a product type action of Z/2Z on a UHF algebra has the tracial Rokhlin property; in particular, unlike for the strict Rokhlin property, every UHF algebra admits such an action. We prove that Blackadar's action of Z/2Z on the 2^{\infty} UHF algebra, whose crossed produc…
We initiate a detailed and systematic study of automorphisms of the Cuntz algebras $Ø_n$ which preserve both the diagonal and the core $UHF$-subalgebra. A general criterion of invertibility of endomorphisms yielding such automorphisms is given. Combinatorial investigations of endomorphisms related to permutation matrices are presented. Key objects entering this analysis are labeled rooted trees equipped with additional data. Our analysis provides insight into the structure of ${\rm Aut}(Ø_n)$ and leads to numerous new examples. In particular, we completely classify all such automorphisms of ${…
Let G be a finite group acting on {1,...,n}. For any C*-algebra A, this defines an action of αof G on A^{\otimes n}. We show that if A tensorially absorbs a UHF algebra of infinite type, the Jiang-Su algebra, or is approximately divisible, then A \times_α G has the corresponding property as well.
We prove that every unital stably finite simple amenable $C^*$-algebra $A$ with finite nuclear dimension and with UCT such that every trace is quasi-diagonal has the property that $A\otimes Q$ has generalized tracial rank at most one, where $Q$ is the universal UHF-algebra. Consequently, $A$ is classifiable in the sense of Elliott.
Using the natural duality between linear functionals on tensor products of C*-algebras with the trace class operators on a Hilbert space H and linear maps of the C*-algebra into B(H), we give two characterizations of separability, one relating it to abelianness of the definite set of the map, and one on tensor products of nuclear and UHF C*-algebras
It is shown that a C*-algebra of the form C(X,U), where U is a UHF algebra, is not an inductive limit of subhomogeneous C*-algebras of topological dimension less than that of X. This is in sharp contrast to dimension-reduction phenomenon in (i) simple inductive limits of such algebras, where classification implies low-dimensional approximations, and (ii) when dimension is measured using decomposition rank, as the author and Winter proved that dr(C(X,U)) $\leq$ 2.
In this paper we exploit the fact that a Cuntz C$^*$-algebra is a groupoid C$^*$-algebra to facilitate the study of non-self-adjoint subalgebras of $O_n$. The Cuntz groupoid is not principal and the spectral theorem for bimodules does not apply in full generality. We characterize the bimodules (over a natural masa) which are determined by their spectra in the Cuntz groupoid; these are exactly the ones which are invariant under the guage automorphisms and exactly the ones which are generated by the Cuntz partial isometries which they contain. We investigate analytic subalgebras of $O_n, n$ fini…
The paper is devoted to 2-local automorphisms on $AW^\ast$-algebras. Using the technique of matrix algebras over a unital Banach algebra we prove that any 2-local automorphism on an arbitrary $AW^\ast$-algebra without finite type~I direct summands is a global automorphism.
We obtain two characterizations of the bi-inner Hopf *-automorphisms of a finite-dimensional Hopf C*-algebra, by means of an analysis of the structure of convolution products in this class of Hopf C*-algebra.
For projectionless C*-algebras absorbing the Jiang-Su algebra tensorially, we study a kind of the Rohlin property for autmorphisms. We show that the crossed products obtained by automorphisms with this Rohlin property also absorb the Jiang-Su algebra tensorially under a mild technical condition on the C*-algebras. In particular, for the Jiang-Su algebra we show the uniqueness up to outer conjugacy of the automorphism with this Rohlin property.
The Cuntz-Toeplitz algebra $E_{n+1}$ for $n\geq1$ is the universal C*-algebra generated by $n+1$ isometries with mutually orthogonal ranges. In this paper, we investigate the automorphism groups of the Cuntz-Toeplitz algebras and determine their homotopy groups.
Certain classes of automorphisms of recued amalgamated free products of C*-algebras are shown to have Brown-Voiculescu topological entropy zero. Also, for automorphisms of exact C*-algebras, the Connes-Narnhofer-Thirring entropy is shown to be bounded above by the Brown-Voiculescu entropy. These facts are applied to generalize Stormer's result about entropy of automorphisms of the II_1-factor of a free group.
