The structure of gauge-invariant ideals of labelled graph C∗C^*-algebras

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

Topological entropy and the AF core of a graph C∗-algebra

AbstractLet C∗(E) be the C∗-algebra associated with a locally finite directed graph E and AE be the AF core of C∗(E). For the topological entropy ht(ΦE) (in the sense of Brown–Voiculescu) of the canonical completely positive map ΦE on the graph C∗-algebra, it is known that if E is finiteht(ΦE)=ht(ΦE|AE)=hb(E)=hl(E), where hb(E) (respectively, hl(E)) is the block (respectively, the loop) entropy of E. In case E is irreducible and infinite, hl(E)⩽ht(ΦE|AE)⩽hb(Et) is known recently, where Et is the graph E with the edges directed reversely. Then by monotonicity of entropy, hl(E)⩽ht(ΦE) is clear. In this paper we show that ht(ΦE)⩽hb(Et) holds for locally finite infinite graphs E. The AF core AE is known to be stably isomorphic to the graph C∗-algebra C∗(E×cZ) of certain skew product E×cZ and we also show that ht(ΦE×cZ)=ht(ΦE|AE). Examples Ep (p>1) of irreducible graphs with ht(ΦEp)=logp are discussed

Three-way Translocation of MLL/MLLT3, t(1;9;11)(p34.2;p22;q23), in a Pediatric Case of Acute Myeloid Leukemia

The chromosome band 11q23 is a common target region of chromosomal translocation in different types of leukemia, including infantile leukemia and therapy-related leukemia. The target gene at 11q23, MLL, is disrupted by the translocation and becomes fused to various translocation partners. We report a case of AML with a rare 3-way translocation involving chromosomes 1, 9, and 11: t(1;9;11)(p34.2;p22;q23). A 3-yr-old Korean girl presented with a 5-day history of fever. A diagnosis of AML was made on the basis of the morphological evaluation and immunophenotyping of bone marrow specimens. Flow cytometric immunophenotyping showed blasts positive for myeloid lineage markers and aberrant CD19 expression. Karyotypic analysis showed 46,XX,t(1;9;11)(p34.2;p22;q23) in 19 of the 20 cells analyzed. This abnormality was involved in MLL/MLLT3 rearrangement, which was confirmed by qualitative multiplex reverse transcription-PCR and interphase FISH. She achieved morphological and cytogenetic remission after 1 month of chemotherapy and remained event-free for 6 months. Four cases of t(1;9;11)(v;p22;q23) have been reported previously in a series that included cases with other 11q23 abnormalities, making it difficult to determine the distinctive clinical features associated with this abnormality. To our knowledge, this is the first description of t(1;9;11) with clinical and laboratory data, including the data for the involved genes, MLL/MLLT3