Search

Finiteness conditions for $C^*$-algebras like AF-embeddability, quasidiagonality, stable finiteness have been studied by many authors and shown to be equivalent for certain classes of $C^*$-algebras. For example, Schfhauser proves that these conditions are all equivalent for $C^*$-algebras of compact topological graphs, and similar results were established by Clark, an Huef, and Sims for $k$-graph algebras. If $C^*(E,\mathcal L)$ is a labeled graph $C^*$-algebra over finite alphabet, it can be viewed as a $C^*$-algebra of a compact topological graph. For these labeled graph $C^*$-algebras, we provide conditions on labeled paths and show that they are equivalent to AF-embeddability of $C^*(E,\mathcal L)$.

Stratum corneum (SC) pH regulates skin barrier functions and elevated SC pH is an important factor in various inflammatory skin diseases. Acidic topical formulas have emerged as treatments for impaired skin barriers. Sodium proton exchanger 1 (NHE1) is an important factor in SC acidification. We investigated whether topical applications containing an NHE1 activator could improve skin barrier functions. We screened plant extracts to identify NHE1 activators in vitro and found Melissa officinalis leaf extract. Rosmarinic acid, a component of Melissa officinalis leaf extract, significantly increased NHE1 mRNA expression levels and NHE1 production. Immunofluorescence staining of NHE1 in 3D-cultured skin revealed greater upregulation of NHE1 expression by NHE1 activator cream, compared to vehicle cream. Epidermal lipid analysis revealed that the ceramide level was significantly higher upon application of the NHE1 activator cream on 3D-cultured skin, compared to application of a vehicle cream. In a clinical study of 50–60-year-old adult females (n = 21), application of the NHE1 activator-containing cream significantly improved skin barrier functions by reducing skin surface pH and transepidermal water loss and increasing skin hydration, compared to patients who applied vehicle cream and those receiving no treatment. Thus, creams containing NHE1 activators, such as rosmarinic acid, could help maintain or recover skin barrier functions.

By a labeled graph $C^*$-algebra we mean a $C^*$-algebra associated to a labeled space $(E,\mathcal L,\mathcal E)$ consisting of a labeled graph $(E,\mathcal L)$ and the smallest normal accommodating set $\mathcal E$ of vertex subsets. Every graph $C^*$-algebra $C^*(E)$ is a labeled graph $C^*$-algebra and it is well known that $C^*(E)$ is simple if and only if the graph $E$ is cofinal and satisfies Condition (L). Bates and Pask extend these conditions of graphs $E$ to labeled spaces, and show that if a set-finite and receiver set-finite labeled space $(E,\mathcal L, \mathcal E)$ is cofinal and disagreeable, then its $C^*$-algebra $C^*(E,\mathcal L, \mathcal E)$ is simple. In this paper, we show that the converse is also true., arXiv admin note: text overlap with arXiv:1703.01583

In this paper, we consider pure infiniteness of generalized Cuntz–Krieger algebras associated to labeled spaces $(E,{\mathcal{L}},{\mathcal{E}})$. It is shown that a $C^{\ast }$-algebra $C^{\ast }(E,{\mathcal{L}},{\mathcal{E}})$ is purely infinite in the sense that every non-zero hereditary subalgebra contains an infinite projection (we call this property (IH)) if $(E,{\mathcal{L}},{\mathcal{E}})$ is disagreeable and every vertex connects to a loop. We also prove that under the condition analogous to (K) for usual graphs, $C^{\ast }(E,{\mathcal{L}},{\mathcal{E}})=C^{\ast }(p_{A},s_{a})$ is purely infinite in the sense of Kirchberg and Rørdam if and only if every generating projection $p_{A}$, $A\in {\mathcal{E}}$, is properly infinite, and also if and only if every quotient of $C^{\ast }(E,{\mathcal{L}},{\mathcal{E}})$ has property (IH).

If a finite group action α on a unital C*-algebra M is saturated, the canonical conditional expectation E : M → Mα onto the fixed point algebra is known to be of index finite type with Index (E) = |G| in the sense of Watatani. More generally, if a finite-dimensional Hopf *-algebra A acts on M and the action is saturated, the same is true with Index (E) = dim (A). In this paper, we prove that the converse is true. Especially in case M is a commutative C*-algebra C(X) and α is a finite group action, we give an equivalent condition in order that the expectation E : C(X) → C(X)α is of index finite type, from which we obtain that α is saturated if and only if G acts freely on X. Actions by compact groups are also considered to show that the gauge action γ on a graph C*-algebra C*(E) associated with a locally finite directed graph E is saturated.

Let E be a row finite directed graph with no sinks and (XE, σE) the one-sided edge shift space. Then the graph C*-algebra C*(E) contains the commutative algebra C0(XE). Moreover if E is locally finite so that the canonical completely positive map ϕE on C*(E) is well-defined, ϕE|C0(XE) coincides with the *-homomorphism [Formula: see text]. In this paper we first show that if two edge shift spaces (XE, σE) and (XF, σF) are topologically conjugate, there is an isomorphism of C*(E) onto C*(F), and if the graphs are locally finite the isomorphism transforms ϕE|C0(XE) onto ϕF|C0(XF), which has been known for Cuntz–Krieger algebras. Let ht(ϕE) be Voiculescu–Brown topological entropy of ϕE. In case E is finite, it is well-known that the values ht(ϕE), [Formula: see text], hl(E) and hb(E) all coincide, where [Formula: see text] is the AF core of C*(E) and hl(E), hb(E) are the loop, block entropies of E respectively. If E is irreducible and infinite, [Formula: see text] has been known recently, and here we show that [Formula: see text], where Et is the transposed graph of E. Also some dynamical systems related with AF subalgebras [Formula: see text] of [Formula: see text] are examined to prove that [Formula: see text] for each vertex v.

Let A E \mathcal A_E be the canonical AF subalgebra of a graph C ∗ C^* -algebra C ∗ ( E ) C^*(E) associated with a locally finite directed graph E E . For Brown and Voiculescu’s topological entropy h t ( Φ E ) ht(\Phi _E) of the canonical completely positive map Φ E \Phi _E on C ∗ ( E ) C^*(E) , h t ( Φ E ) = h t ( Φ E | A E ) = h l ( E ) = h b ( E ) ht(\Phi _E)=ht(\Phi _E|_{\mathcal A_E})=h_l(E)=h_b(E) is known to hold for a finite graph E E , where h l ( E ) h_l(E) is the loop entropy of Gurevic and h b ( E ) h_b(E) is the block entropy of Salama. For an irreducible infinite graph E E , the inequality h l ( E ) ≤ h t ( Φ E | A E ) h_l(E)\leq ht(\Phi _E|_{\mathcal A_E}) has recently been known. It is shown in this paper that \[ h t ( Φ E | A E ) ≤ max { h b ( E ) , h b ( t E ) } , ht(\Phi _E|_{\mathcal A_E})\leq \max \{ h_b(E), h_b(\,{}^{t}E)\}, \] where t E {}^tE is the graph E E with the direction of the edges reversed. Some irreducible infinite graphs E p ( p > 1 ) E_p\ (p>1) with h t ( Φ E | A E p ) = log ⁡ p ht(\Phi _E|_{\mathcal A_{E_p}})=\log p are also examined.

Let C*(E) = C*(se, pv) be the graph C*-algebra of a directed graph E = (E0, E1) with the vertices E0 and the edges E1. We prove that if E is a finite graph (possibly with sinks) and φE: C*(E) → C*(E) is the canonical completely positive map defined by then Voiculescu's topological entropy ht(φE) of φE is log r(AE), where r(AE) is the spectral radius of the edge matrix AE of E. This extends the same result known for finite graphs with no sinks. We also consider the map φE when E is a locally finite irreducible infinite graph and prove that , where the supremum is taken over the set of all finite subgraphs of E.

It is now well known that a simple graph $C^*$-algebra $C^*(E)$ of a directed graph $E$ is either AF or purely infinite. In this paper, we address the question of whether this is the case for labeled graph $C^*$-algebras recently introduced by Bates and Pask as one of the generalizations of graph $C^*$-algebras, and show that there exists a family of simple unital labeled graph $C^*$-algebras which are neither AF nor purely infinite. Actually these algebras are shown to be isomorphic to crossed products $C(X)\times_T \mathbb Z$ where the dynamical systems $(X,T)$ are Cantor minimal subshifts. Then it is an immediate consequence of well known results about this type of crossed products that each labeled graph $C^*$-algebra in the family obtained here is an $A\mathbb T$ algebra with real rank zero and has $\mathbb Z$ as its $K_1$-group., Comment: 18 pages

In this paper, we consider the gauge-invariant ideal structure of a $C^*$-algebra $C^*(E,\mathcal{L},\mathcal{B})$ associated to a set-finite, receiver set-finite and weakly left-resolving labelled space $(E,\mathcal{L},\mathcal{B})$, where $\mathcal{L}$ is a labelling map assigning an alphabet to each edge of the directed graph $E$ with no sinks. Under the assumption that an accommodating set $\mathcal{B}$ is closed under taking relative complement, it is obtained that there is a one to one correspondence between the set of all hereditary saturated subsets of $\mathcal{B}$ and the gauge-invariant ideals of $C^*(E,\mathcal{L},\mathcal{B})$. For this, we introduce a quotient labelled space $(E,\mathcal{L},[\mathcal{B}]_R)$ arising from an equivalence relation $\sim_R$ on $\mathcal{B}$ and show the existence of the $C^*$-algebra $C^*(E,\mathcal{L},[\mathcal{B}]_R)$ generated by a universal representation of $(E,\mathcal{L},[\mathcal{B}]_R)$. Also the gauge-invariant uniqueness theorem for $C^*(E,\mathcal{L},[\mathcal{B}]_R)$ is obtained. For simple labelled graph $C^*$-algebras $C^*(E,\mathcal{L},\bar{\mathcal{E}})$, where $\bar{\mathcal{E}}$ is the smallest accommodating set containing all the generalized vertices, it is observed that if for each vertex $v$ of $E$, a generalized vertex $[v]_l$ is finite for some $l$, then $C^*(E,\mathcal{L},\bar{\mathcal{E}})$ is simple if and only if $(E,\mathcal{L},\bar{\mathcal{E}})$ is strongly cofinal and disagreeable. This is done by examining the merged labelled graph $(F,\mathcal{L}_F)$ of $(E,\mathcal{L})$ and the common properties that $C^*(E,\mathcal{L},\bar{\mathcal{E}})$ and $C^*(F,\mathcal{L},\bar{\mathcal{F}})$ share.

Let C ∗ ( E ) be the C ∗ -algebra associated with a locally finite directed graph E and A E be the AF core of C ∗ ( E ) . For the topological entropy ht ( Φ E ) (in the sense of Brown–Voiculescu) of the canonical completely positive map Φ E on the graph C ∗ -algebra, it is known that if E is finite ht ( Φ E ) = ht ( Φ E | A E ) = h b ( E ) = h l ( E ) , where h b ( E ) (respectively, h l ( E ) ) is the block (respectively, the loop) entropy of E. In case E is irreducible and infinite, h l ( E ) ⩽ ht ( Φ E | A E ) ⩽ h b ( E t ) is known recently, where E t is the graph E with the edges directed reversely. Then by monotonicity of entropy, h l ( E ) ⩽ ht ( Φ E ) is clear. In this paper we show that ht ( Φ E ) ⩽ h b ( E t ) holds for locally finite infinite graphs E. The AF core A E is known to be stably isomorphic to the graph C ∗ -algebra C ∗ ( E × c Z ) of certain skew product E × c Z and we also show that ht ( Φ E × c Z ) = ht ( Φ E | A E ) . Examples E p ( p > 1 ) of irreducible graphs with ht ( Φ E p ) = log p are discussed.

Given a row-finite directed graph E, a universal C*-algebra C*(E) generated by a family of partial isometries and projections subject to the relations determined by E is associated to the graph E. The Cuntz–Krieger algebras are those graph C*-algebras associated to some finite graphs. We prove that a graph C*-algebra C*(E) has real rank zero in the sense that the set of invertible self-adjoint elements is dense in the set of all self-adjoint elements in C*(E) (or in the unitization of C*(E) if C*(E) is nonunital) if and only if E satisfies a loop condition (K) that is analogous to the condition for a finite {0, 1} matrix A under which Cuntz analyzed the ideal structure of the Cuntz–Krieger algebra OA.