arXiv subjects

Dilian Yang

Publications and source records attributed to Dilian Yang.

32 records · Page 2Linked to original sources

arXiv2014

Periodic higher rank graphs revisited

Let $P$ be a finitely generated cancellative abelian monoid. A $P$-graph $Λ$ is a natural generalization of a higher rank graph. A pullback of $Λ$ is constructed by pulling it back over a given monoid morphism to $P$, while a pushout of $Λ$ is obtained by modding out its periodicity $\PerΛ$, which is deduced from a natural equivalence relation on $Λ$. One of our main results in this paper shows that, for a class of higher rank graphs $Λ$, $Λ$ is isomorphic to the pullback of its pushout via a natural quotient map, and that its graph C*-algebra can be embedded into the tensor product of the graph C*-algebra of its pushout and $\ca(\PerΛ)$. As a consequence, its cycline C*-algebra generated by the standard generators with equivalent pairs is an abelian core (particularly a MASA). Along the way, we give an in-depth study on periodicity of $P$-graphs.

math.OA↗

arXiv2013

The Hopf structure of some dual operator algebras

We study the Hopf structure of a class of dual operator algebras corresponding to certain semigroups. This class of algebras arises in dilation theory, and includes the noncommutative analytic Toeplitz algebra and the multiplier algebra of the Drury-Arveson space, which correspond to the free semigroup and the free commutative semigroup respectively. The preduals of the algebras in this class naturally form Hopf (convolution) algebras. The original algebras and their preduals form (non-self-adjoint) dual Hopf algebras in the sense of Effros and Ruan. We study these algebras from this perspective, and obtain a number of results about their structure.

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arXiv2013

Orthogonally Additive Mappings on Hilbert Modules

In this paper, we study the representation of orthogonally additive mappings acting on Hilbert $C^*$-modules and Hilbert $H^*$-modules. One of our main results shows that every continuous orthogonally additive mapping $f$ from a Hilbert module $W$ over $\K(\H)$ or $\H§(\H)$ to a complex normed space is of the form $f(x)=T(x)+Φ(< x, x >)$ for all $x\in W$, where $T$ is a continuous additive mapping, and $Φ$ is a continuous linear mapping.

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arXiv2013

A non-self-adjoint Lebesgue decomposition

We study the structure of bounded linear functionals on a class of non-self-adjoint operator algebras that includes the multiplier algebra of every complete Nevanlinna-Pick space, and in particular the multiplier algebra of the Drury-Arveson space. Our main result is a Lebesgue decomposition expressing every linear functional as the sum of an absolutely continuous (i.e. weak-* continuous) linear functional, and a singular linear functional that is far from being absolutely continuous. This is a non-self-adjoint analogue of Takesaki's decomposition theorem for linear functionals on von Neumann algebras. We apply our decomposition theorem to prove that the predual of every algebra in this class is (strongly) unique.

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arXiv2013

Nonself-adjoint 2-graph algebras

We study the structure of weakly-closed nonself-adjoint algebras arising from representations of single vertex 2-graphs. These are the algebras generated by 2 isometric tuples which satisfy a certain commutation relation. We show that these algebras have a lower-triangular $3\times 3$ form. The left-hand side of this matrix decomposition is a slice of the enveloping von Neumann algebra generated by the 2-graph algebra. We further give necessary and sufficient conditions for these algebras themselves to be von Neumann algebras. The paper concludes with further study of atomic representations.

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arXiv2012

Analytic Free Semigroup Algebras and Hopf Algebras

Let $\fS$ be an analytic free semigroup algebra. In this paper, we explore richer structures of $\fS$ and its predual $\fS_*$. We prove that $\fS$ and $\fS_*$ both are Hopf algebras. Moreover, the structures of $\fS$ and $\fS_*$ are closely connected with each other: There is a bijection between the set of completely bounded representations of $\fS_*$ and the set of corepresentations of $\fS$ on one hand, and $\fS$ can be recovered from the coefficient operators of completely bounded representations of $\fS_*$ on the other hand. As an amusing application of our results, the (Gelfand) spectrum of $\fS_*$ is identified. Surprisingly, the main results of this paper seem new even in the classical case.

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arXiv2011

Type III von Neumann Algebras associated with $Ø_θ$

Let $\Fth$ be a 2 graph generated by $m$ blue edges and $n$ red edges, and $ω$ be the distinguished faithful state associated with its graph C*-algebra $Ø_θ$. In this paper, we characterize the factorness of the von Neumann algebra $π_ω(Ø_θ)"$ induced from the GNS representation of $ω$ under a certain condition. Moreover, when $π_ω(Ø_θ)"$ is a factor, then it is of type III$_{m^{-\frac{1}{b}}}$ (or III$_{n^{-\frac{1}{a}}}$) if $\frac{\ln m}{\ln n}\in\bQ$, where $a,b\in\bN$ with $\gcd(a,b)=1$ satisfy $m^a=n^b$, and of type III$_1$ if $\frac{\ln m}{\ln n}\not\in\bQ$. In the case of $θ$ being the identity permutation, our condition turns out to be redundant. On the way to our main results, we also obtain the structure of the fixed point algebra $Ø_θ^σ$ of the modular action $σ$ from $ω$. This could be useful in proving the redundancy of our extra condition.

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arXiv2010

Functional Equations and Fourier Analysis

By exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations -- the d'Alembert equation, the Wilson equation, and the d'Alembert long equation, on compact groups.

math.FA↗

arXiv2009

Endomorphisms and Modular Theory of 2-Graph C*-Algebras

In this paper, we initiate the study of endomorphisms and modular theory of the graph C*-algebras $Ø_θ$of a 2-graph $\Fth$ on a single vertex. We prove that there is a semigroup isomorphism between unital endomorphisms of $Ø_θ$ and its unitary pairs with a \textit{twisted property}. We characterize when endomorphisms preserve the fixed point algebra $\fF$ of the gauge automorphisms and its canonical masa $\fD$. Some other properties of endomorphisms are also investigated. As far as the modular theory of $Ø_θ$ is concerned, we show that the algebraic *-algebra generated by the generators of $Ø_θ$ with the inner product induced from a distinguished state $ω$ is a modular Hilbert algebra. Consequently, we obtain that the von Neumann algebra $π(Ø_θ)"$ generated by the GNS representation of $ω$ is an AFD factor of type III$_1$, provided $\frac{\ln m}{\ln n}\not\in\bQ$. Here $m,n$ are the numbers of generators of $\Fth$ of degree $(1,0)$ and $(0,1)$, respectively. This work is a continuation of \cite{DPY1, DPY2} by Davidson-Power-Yang and \cite{DY} by Davidson-Yang.

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arXiv2008

Atomic Representations of Rank 2 Graph Algebras

We provide a detailed analysis of atomic *-representations of rank 2 graphs on a single vertex. They are completely classified up to unitary equivalence, and decomposed into a direct sum or direct integral of irreducible atomic representations. The building blocks are described as the minimal *-dilations of defect free representations modelled on finite groups of rank 2.

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arXiv2008

Nonabelian harmonic analysis and functional equations on compact groups

Making use of nonabelian harmonic analysis and representation theory, we solve the functional equation $$f_1(xy)+f_2(yx)+f_3(xy^{-1})+f_4(y^{-1}x)=f_5(x)f_6(y)$$ on arbitrary compact groups. The structure of its general solution is completely described. Consequently, several special cases of the above equation, in particular, the Wilson equation and the d'Alembert long equation, are solved on compact groups.

math.FA↗

arXiv2008

Representations of Higher Rank Graph Algebras

Let $\Fth$ be a $\Bk$-graph on a single vertex. We show that every irreducible atomic $*$-representation is the minimal $*$-dilation of a group construction representation. It follows that every atomic representation decomposes as a direct sum or integral of such representations. We characterize periodicity of $\Fth$ and identify a symmetry subgroup $H_θ$ of $\bZ^\Bk$. If this has rank $s$, then $\ca(\Fth) \cong \rC(\bT^s) \otimes \fA$ for some simple C*-algebra $\fA$.

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arXiv2007

Dilation Theory for Rank 2 Graph Algebras

An analysis is given of $*$-representations of rank 2 single vertex graphs. We develop dilation theory for the non-selfadjoint algebras $\A_θ$ and $\A_u$ which are associated with the commutation relation permutation $θ$ of a 2 graph and, more generally, with commutation relations determined by a unitary matrix $u$ in $M_m(\bC) \otimes M_n(\bC)$. We show that a defect free row contractive representation has a unique minimal dilation to a $*$-representation and we provide a new simpler proof of Solel's row isometric dilation of two $u$-commuting row contractions. Furthermore it is shown that the C*-envelope of $\A_u$ is the generalised Cuntz algebra $Ø_{X_u}$ for the product system $X_u$ of $u$; that for $m\geq 2 $ and $n \geq 2 $ contractive representations of $\Ath$ need not be completely contractive; and that the universal tensor algebra $\T_+(X_u)$ need not be isometrically isomorphic to $\A_u$.

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arXiv2007

Periodicity in Rank 2 Graph Algebras

Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of $\ca(\Fth)$. The periodic C*-algebras are characterized, and it is shown that $\ca(\Fth) \simeq \rC(\bT) \otimes \fA$ where $\fA$ is a simple C*-algebra.

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