A. Yu. Pirkovskii
We introduce a family of dense subalgebras of the Toeplitz algebra and give conditions under which our algebras are quasi-free. As a corollary, we show that the smooth Toeplitz algebra introduced by Cuntz is quasi-free.
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We formulate and establish a noncommutative version of the well known Helson-Szego theorem about the angle between past and future for subdiagonal subalgebras. We then proceed to use this theorem to characterise the symbols of invertible Toeplitz operators on the noncommutative Hardy spaces associated to subdiagonal subalgebras.
In this paper, we study Toeplitz algebras generated by certain class of Toeplitz operators on the $p$-Fock space and the $p$-Bergman space with $1<p<\infty$. Let BUC($\mathbb C^n$) and BUC($\mathbb B_n$) denote the collections of bounded uniformly continuous functions on $\mathbb C^n$ and $\mathbb B_n$ (the unit ball in $\mathbb C^n$), respectively. On the $p$-Fock space, we show that the Toeplitz algebra which has a translation invariant closed subalgebra of BUC($\mathbb C^n$) as its set of symbols is linearly generated by Toeplitz operators with the same space of symbols. This answers a ques…
We prove several results concerning the theory of Toeplitz algebras over $p$-Fock spaces using a correspondence theory of translation invariant symbol and operator spaces. The most notable results are: The full Toeplitz algebra is the norm closure of all Toeplitz operators with bounded uniformly continuous symbols. This generalizes a result obtained by J. Xia (J. Funct. Anal. 269:781-814, 2015) in the case $p = 2$, which was proven by different methods. Further, we prove that every Toeplitz algebra which has a translation invariant $C^\ast$ subalgebra of the bounded uniformly continuous functi…
Let L^\star be a filtered algebra of abstract pseudodifferential operators equipped with a notion of ellipticity, and T^\star be a subalgebra of operators of the form P_1AP_0, where P_0 and P_1 are two projections. The elements of L^\star act as linear continuous operators in certain scales of abstract Sobolev spaces, the elements of the subalgebra in the corresponding subspaces determined by the projections. We study how the ellipticity in L^\star descends to T^\star, focusing on parametrix construction, Fredholm property, and homogeneous principal symbols. Applications concern SG-pseudodiffe…
Commutative $G$-invariant Toeplitz C$^\ast$ algebras on the Fock space and their Gelfand theory through Quantum Harmonic Analysis
本文基于量子调和分析中的谱综合概念，研究了具有特定不变符号的 Toeplitz 代数子代数及其交换子，发现了一类新的 Fock 空间上的交换 Toeplitz C$^\ast$ 代数，并探讨了其 Gelfand 理论。
We discuss various aspects of noncommutative geometry of a smooth subalgebra of the Toeplitz algebra. In particular, we study the structure of derivations on this subalgebra.
Density of Toeplitz operators in rotation-invariant Toeplitz algebras
本文运用 Werner 的量子调和分析技术，证明了仿射酉群子群下不变及准径向 Toeplitz 算子在相应代数中的稠密性，并给出有界算子空间中 SOT 稠密性的构造性证明。
We show how to reconstruct a finite directed graph E from its Toeplitz algebra, its gauge action, and the canonical finite-dimensional abelian subalgebra generated by the vertex projections. We also show that if E has no sinks, then we can recover E from its Toeplitz algebra and the generalised gauge action that has, for each vertex, an independent copy of the circle acting on the generators corresponding to edges emanating from that vertex. We show by example that it is not possible to recover E from its Toeplitz algebra and gauge action alone.
On the density of Toeplitz operators in the Toeplitz algebra over the Bergman space of the unit ball
该研究采用量子调和分析与表示论方法，给出单位球Bergman空间上Toeplitz代数中Toeplitz算子按范数稠密的夏氏定理的新证明。
We consider C*-algebras generated by a single Hilbert bimodule (Pimsner-Toeplitz algebras) and by a product systems of Hilbert bimodules. We give a new proof of a theorem of Pimsner, which states that any representation of the generating bimodule gives rise to a representation of the Pimsner-Toeplitz algebra. Our proof does not make use of the conditional expectation onto the subalgebra invariant under the dual action of the circle group. We then prove the analogous statement for the case of product systems, generalizing a theorem of Arveson from the case of product systems of Hilbert spaces.
It is a well-known fact that endomorphisms of $B(H)$ are intimately connected with families of mutually orthogonal isometries, i.e. with representations of the so-called Toeplitz $C^*$-algebras. In this paper we consider a natural generalization of this connection between the representation theory of certain $C^*$-algebras associated to graphs and endomorphisms of certain von Neumann subalgebras of $B(H)$. Our primary results give criteria by which it may be determined if two representations give rise to equal or conjugate endomorphisms.
Topological full groups arising from Cuntz and Cuntz-Toeplitz algebras and their crossed products
论文研究 Cuntz 与 Cuntz-Toeplitz 代数及其 Cartan 交叉积所诱导的拓扑满群，分析其正规子群和交换化，并完全刻画相关约化交叉积在规范作用下的 KMS 态。
We prove six-term exact sequences of Pimsner-Voiculescu type for certain subalgebras of the Cuntz-Pimsner algebras. This sequence may, in particular, be applied to smooth subalgebras of the Quantum Heisenberg Manifolds in order to compute the generators of their cyclic cohomology. Further, our results include the known results for smooth crossed products. Our proof is based on a combination of arguments from the setting of (Cuntz-)Pimsner algebras and the Toeplitz proof of Bott-periodicity.
We construct dense Banach subalgebras $A$ of the null sequence algebra $c_0$ which are spectral invariant, but do not satisfy the $D_1$-condition $\|ab \|_A \leq K(\|a\|_{\infty} \|b\|_A + \|a \|_A \|b \|_{\infty})$, for all $a, b \in A$. The sequences in $A$ vanish in a skewed manner with respect to an unbounded function $σ\colon {\mathbb N} \rightarrow [1, \infty)$.
We construct a unital pre-C*-algebra $A_0$ which is stably finite, in the sense that every left invertible square matrix over $A_0$ is right invertible, while the C*-completion of $A_0$ contains a non-unitary isometry, and so it is infinite.
On Spectral Invariant Dense Subalgebras of Uniform Roe Algebras with Subexponential Growth
针对满足性质 P 的次指数增长群，构造 R^∞(G) 子代数，并利用容许权证明其在一致 Roe 代数 C_u^*(G) 中的谱不变性，从而把相关结果推广到多项式增长之外。
The Laplacian of an operator and the radial Toeplitz algebra
本文利用量子调和分析证明算子拉普拉斯算子的定义域在 Fock 空间上的 Toeplitz 代数中稠密，并据此简化了径向 Toeplitz 代数的 Gelfand 理论。
In this paper we show that the C*-algebra generated by radial Toeplitz operators with $L_{\infty}$-symbols acting on the Fock space is isometrically isomorphic to the C*-algebra of bounded sequences uniformly continuous with respect to the square-root-metric $ρ(j,k)=|\sqrt{\vphantom{jk}j}-\sqrt{\vphantom{jk}k}\,|$. More precisely, we prove that the sequences of eigenvalues of radial Toeplitz operators form a dense subset of the latter C*-algebra of sequences.
We give an example of a dense, simple, unital Banach subalgebra $A$ of the irrational rotation C*-algebra $B$, such that $A$ is not a spectral subalgebra of $B$. This answers a question posed in T.W. Palmer's paper [1].
Toeplitz algebra of bounded symmetric domains: A quantum harmonic analysis approach via localization
本文利用表示论、量子调和分析与弱局部化算子，证明了有界对称域加权 Bergman 空间上 Toeplitz 算子在 Toeplitz 代数中范数稠密，并刻画了 α-弱局部化算子自伴代数的范数闭包。
Consider the pair $(G,P)$ consisting of an abelian lattice-ordered discrete group $G$ and its positive cone $P$. Let $α$ be an action of $P$ by extendible endomorphisms of a $C^*$-algebra $A$. We show that the Nica-Toeplitz algebra $\mathcal{T}_{\textrm{cov}}(A\times_α P)$ is a full corner of a group crossed product $\mathcal{B}\rtimes_βG$, where $\mathcal{B}$ is a subalgebra of $\ell^{\infty}(G,A)$ generated by a collection of faithful copies of $A$, and the action $β$ on $\mathcal{B}$ is given by the shift on $\ell^{\infty}(G,A)$. By using this realization, we show that the ideal $\mathcal{I…
We define and study dense Frechet subalgebras of compact quantum groups consisting of elements rapidly decreasing with respect to an unbounded sequence of real numbers. Further, this sequence can be viewed as the eigenvalues of a Dirac-like operator and we characterize the boundedness of its commutators in terms of the eigenvalues. Grotendieck's theory of topological tensor products immediately yields a Schwartz kernel theorem for linear operators on compact quantum groups and allows us to introduce a natural class of pseudo-differential operators on compact quantum groups. As a by-product, we…
In this paper we construct a compact quantum semigroup structure on the Toeplitz algebra $\mathcal{T}$. The existence of a subalgebra, isomorphic to the algebra of regular Borel's measures on a circle with convolution product, in the dual algebra $\mathcal{T}^*$ is shown. The existence of Haar functionals in the dual algebra and in the above-mentioned subalgebra is proved. Also we show the connection between $\mathcal{T}$ and the structure of weak Hopf algebra.
A simple Steinberg algebra associated to an ample Hausdorff groupoid $G$ is algebraically purely infinite if and only if the characteristic functions of compact open subsets of the unit space are infinite idempotents. If a simple Steinberg algebra is algebraically purely infinite, then the reduced groupoid $C^*$-algebra $C^*_r(G)$ is simple and purely infinite. But the Steinberg algebra seems to small for the converse to hold. For this purpose we introduce an intermediate $*$-algebra $B(G)$ constructed using corners $1_U C^*_r(G) 1_U$ for all compact open subsets $U$ of the unit space of the g…
On dense subalgebras of the singular ideal in groupoid C*-algebras
研究群oid C*-代数中奇异理想稠密子代数的确定性，通过同态纤维性质和有限- nilpotent同态群的C*-代数密度特性进行分析。
We define the notion of strong spectral invariance for a dense Frechet subalgebra A of a Banach algebra B. We show that if A is strongly spectral invariant in a C*-algebra B, and G is a compactly generated polynomial growth Type R Lie group, not necessarily connected, then the smooth crossed product G\rtimes A is spectral invariant in the C*-crossed product G\rtimes B. Examples of such groups are given by finitely generated polynomial growth discrete groups, compact or connected nilpotent Lie groups, the group of Euclidean motions on the plane, the Mautner group, or any closed subgroup of one …
Let A be a dense Frechet *-subalgebra of a C*-algebra B. (We do not require Frechet algebras to be m-convex.) Let G be a Lie group, not necessarily con- nected, which acts on both $A$ and B by *-automorphisms, and let \s be a sub- multiplicative function from G to the nonnegative real numbers. If \s and the action of G on A satisfy certain simple properties, we define a dense Frechet *-subalgebra G\rtimes^{\s} A of the crossed product L^{1}(G, B). Our algebra consists of differentiable A-valued functions on G, rapidly vanishing in \s. We give conditions on the action of G on A which imply the …
To a graph $Γ$ one can associate a C^*-algebra $C^*(Γ)$ generated by isometries. Such $C^*$-algebras were studied recently by Crisp and Laca. They are a special case of the Toeplitz C^*-algebras $\mathcal{T}(G, P)$ associated to quasi-latice ordered groups (G, P) introduced by Nica. Crisp and Laca proved that the so called "boundary quotients" $C^*_q(Γ)$ of $C^*(Γ)$ are simple and purely infinite. For a certain class of finite graphs $Γ$ we show that $C^*_q(Γ)$ can be represented as a full corner of a crossed product of an appropriate C^*-subalgebra of $C^*_q(Γ)$ built by using $C^*(Γ')$, wher…
We construct a representation of each finitely aligned aperiodic k-graph Λ on the Hilbert space H^{ap} with basis indexed by aperiodic boundary paths in Λ. We show that the canonical expectation on B(H^{ap}) restricts to an expectation of the image of this representation onto the subalgebra spanned by the final projections of the generating partial isometries. We then show that every quotient of the Toeplitz algebra of the k-graph admits an expectation compatible with this one. Using this, we prove that the image of our representation, which is canonically isomorphic to the Cuntz-Krieger algeb…
We construct dense, unconditional subalgebras of the reduced group $C^*$-algebra of a word-hyperbolic group, which are closed under holomorphic functional calculus and possess many bounded traces. Applications to the cyclic cohomology of group $C^*$-algebras and to delocalized $L^2$-invariants of negatively curved manifolds are given.
We consider an action of the circle group, T on a von Neumann algebra, M. Similarly to the case when the algebra of essentially bounded functions on T is acted upon by translations, we define the generalized Hardy subspace of H,where H is the Hilbert space of a standard representation of M and the subalgebra of analytic elements of M with respect to the action. We prove that the subalgebra of analytic elements is a reflexive algebra of operators if the Arveson spectrum is finite or, if the spectrum is infinite, the spectral subspace corresponding to the least positive element contains an unita…
Relative topological principality and the ideal intersection property for groupoid C*-algebras
本文引入相对拓扑主性概念，证明群胚C*-代数表示忠实性可由子代数限制忠实性判定，推广了Toeplitz代数唯一性定理。
Analytic subalgebras of Beurling-Fourier algebras and complexification of Lie groups
本文通过引入解析子代数，将加权Fourier代数的Gelfand谱元素解释为李群复化中特定点上的取值，并分别给出一般连通李群的局部解、连通单连通幂零李群的全局解，以及$ax+b$-群的全局解实例。
The theory of Toeplitz quantization presented in our previous paper is extended and further developed to include diverse and interesting non-commutative realizations of the classical Euclidean plane. This is done using Hilbert spaces of entire functions, where polynomials in one complex variable form a dense subspace. The complex coordinate naturally acts as an unbounded multiplication operator generating, together with its adjoint, a highly non-commutative *-algebra of operators. The Toeplitz operators are then geometrically constructed as special elements from this algebra; they are associat…
An extension of Haagerup's reduction theorem with applications to subdiagonal subalgebras of general von Neumann algebras
本文将 Haagerup 约化定理推广到带任意忠实正规半有限权的一般 von Neumann 代数，并据此扩展 $H^p$ 空间应用、提出近似次对角子代数概念。
A quantum metric on the Cantor Space
本研究针对康托空间上已有量子度量，通过推导稠密子代数上的Lip-范数公式、对比其在Hamel基及元素族上的取值，证明该量子度量与经典度量诱导的量子度量存在差异。
Let $\M$ be a finite von Neumann algebra acting on a Hilbert space $\H$ and $Å$ be a transitive algebra containing $\M'$. In this paper we prove that if $Å$ is 2-fold transitive, then $Å$ is strongly dense in $\B(\H)$. This implies that if a transitive algebra containing a standard finite von Neumann algebra (in the sense of [Ha1]) is 2-fold transitive, then $Å$ is strongly dense in $\B(\H)$. Non-selfadjoint algebras related to free products of finite von Neumann algebras, e.g., $Ł{\mathbb{F}_n}$ and $(M_2(\cc), {1/2}Tr)*(M_2(\cc), {1/2}Tr)$, are studied. Brown measures of certain operators in…
Subproduct systems with quantum group symmetry
本文引入一类纤维由Temperley-Lieb代数Jones-Wenzl投影定义的子积系统，研究其量子对称性为自由正交量子群的子类，证明对应Toeplitz代数的性质及规范不变子代数的结构，同时得到等变子积系统的相关一般性结果。
Let $G$ be a countable discrete amenable group, ${\cal M}$ a McDuff factor von Neumann algebra, and $A$ a separable nuclear weakly dense C$^*$-subalgebra of ${\cal M}$. We show that if two centrally free actions of $G$ on ${\cal M}$ differ up to approximately inner automorphisms then they are outer conjugate by an approximately inner automorphism, in the operator norm topology, which makes $A$ invariant. In addition, when $A$ is unital, simple, and with a unique tracial state and $α$ is an automorphism of $A$ we also show that the aperiodicity of $α$ on the von Neumann algebra is equivalent to…
In this paper we use the description of free group factors as the von Neumann algebras of Berezin's deformation of the upper half-plane, modulo PSL$(2,{\Bbb Z})$. The derivative, in the deformation parameter, of the product in the corresponding algebras, is a positive Hochschild 2-cocycle, defined on a dense subalgebra. By analyzing the structure of the cocycle we prove that there is a generator $\cal L$ for a quantum dynamical semigroup that implements the cocycle on a strongly dense subalgebra. For $x$ in the dense subalgebra, ${\cal L}(x)$ is the (diffusion) operator $$ {\cal L}(x)=Λ(x)-(1/…
The pure infiniteness transfer problem
本文针对C*-代数A的稠密子代数A₀的纯无限性转移问题，在A具实秩零、A₀为hfc-闭的假设下解决了该问题，还证明了相关阻碍，并以Cuntz代数的光滑子代数为例验证假设成立。
To each monoid $P$ that embeds in a group we associate a universal Toeplitz C*-algebra $T_u(P)$ defined via generators and relations; $T_u(P)$ is a quotient of Li's semigroup C*-algebra $C^*(P)$ and they are isomorphic if and only if $P$ satisfies independence. We give a partial crossed product realization of $T_u(P)$ and show that several key results known for $C^*(P)$ when $P$ satisfies independence are also valid for $T_u(P)$ when independence fails. At the level of the reduced semigroup C*-algebra $T_r(P)$, we show that nontrivial ideals have nontrivial intersection with the reduced crosse…
The inverse-closed subalgebra of $C^{*}(G,A)$
本文研究系数为非交换 $C^*$-代数的 Roe 代数 $C^*(G,A)$ 的逆闭子代数，通过 Sobolev 型代数取交集构造 $W_a^{\infty}(G,A)$，证明其为谱不变稠密子代数，充分条件为群作用具有多项式增长。
We consider Hilsum's notion of bordism as an equivalence relation on unbounded $KK$-cycles and study the equivalence classes. Upon fixing two $C^*$-algebras, and a $*$-subalgebra dense in the first $C^*$-algebra, a $\mathbb{Z}/2\mathbb{Z}$-graded abelian group is obtained; it maps to the Kasparov $KK$-group of the two $C^*$-algebras via the bounded transform. We study properties of this map both in general and in specific examples. In particular, it is an isomorphism if the first $C^*$-algebra is the complex numbers (i.e., for $K$-theory) and is a split surjection if the first $C^*$-algebra is…
Spaces of UCP maps and subalgebras of von Neumann algebras
研究保态幺正完全正映射空间拓扑，分析保态条件期望子空间拓扑，应用于子代数结构类，证明相关性质，还研究了特定态下非保态条件期望像的子代数空间，得出部分代数此类子代数构成开且稠密子集的结论。
The reduced $C^*$-algebra of the interior of the isotropy in any Hausdorff étale groupoid $G$ embeds as a $C^*$-subalgebra $M$ of the reduced $C^*$-algebra of $G$. We prove that the set of pure states of $M$ with unique extension is dense, and deduce that any representation of the reduced $C^*$-algebra of $G$ that is injective on $M$ is faithful. We prove that there is a conditional expectation from the reduced $C^*$-algebra of $G$ onto $M$ if and only if the interior of the isotropy in $G$ is closed. Using this, we prove that when the interior of the isotropy is abelian and closed, $M$ is a C…
Let $\A$ and $\B$ be unital Banach algebras and $ϕ\colon\A\to\B$ be a unital continuous homomorphism. We prove that if $ϕ$ is relatively spectral (i.e., there is a dense subalgebra $X$ of $\A$ such that $\sp_\B(ϕ(a))=\sp_\A(a)$ for every $a\in X$) and has dense range, then $ϕ$ induces monomorphisms from $K_i(\A)$ to $K_i(\B)$, $i=0,1$.
We give sufficient conditions, in terms of the existence of unbounded derivations satisfying certain properties, which ensure that a II$_1$ factor $M$ is prime or has at most one Cartan subalgebra. For instance, we prove that if there exists a real closable unbounded densely defined derivation $δ:M\rightarrow L^2(M)\bar{\otimes}L^2(M)$ whose domain contains a non-amenability set, then $M$ is prime. If $δ$ is moreover "algebraic" (i.e. its domain $M_0$ is finitely generated, $δ(M_0)\subset M_0\otimes M_0$ and $δ^*(1\otimes 1)\in M_0$), then we show that $M$ has no Cartan subalgebra. We also giv…
A subalgebra $\mathcal{A}$ of a $C^*$-algebra $\mathcal{M}$ is logmodular (resp. has factorization) if the set $\{a^*a; a\text{ is invertible with }a,a^{-1}\in\mathcal{A}\}$ is dense in (resp. equal to) the set of all positive and invertible elements of $\mathcal{M}$. There are large classes of well studied algebras, both in commutative and non-commutative settings, which are known to be logmodular. In this paper, we show that the lattice of projections in a von Neumann algebra $\mathcal{M}$ whose ranges are invariant under a logmodular algebra in $\mathcal{M}$, is a commutative subspace latti…
A tiling with infinite rotational symmetry, such as the Conway-Radin Pinwheel Tiling, gives rise to a topological dynamical system to which an étale equivalence relation is associated. A groupoid C*-algebra for a tiling is produced and a separating dense set is exhibited in the C*-algebra which encodes the structure of the topological dynamical system. In the case of a substitution tiling, natural subsets of this separating dense set are used to define an AT-subalgebra of the C*-algebra. Finally our results are applied to the Pinwheel Tiling.
The author's recent results on spectral invariant dense subalgebras of C*-algebras associated with dynamical systems are summarized. If G is a compactly generated polynomial growth Type R Lie group, and the action of G on S(M) (Schwartz functions on a locally compact G-space M) is tempered in a certain sense, then there is a natural smooth crossed product S(G X M) which is dense and spectral invariant in the C*-crossed product C*(G X M).
It is shown how to construct *-homomorphic quantum stochastic Feller cocycles for certain unbounded generators, and so obtain dilations of strongly continuous quantum dynamical semigroups on C* algebras; this generalises the construction of a classical Feller process and semigroup from a given generator. The construction is possible provided the generator satisfies an invariance property for some dense subalgebra A_0 of the C* algebra A and obeys the necessary structure relations; the iterates of the generator, when applied to a generating set for A_0, must satisfy a growth condition. Furtherm…
For each 1<s<\infty, a Popa algebra A_s is constructed that embeds as a weakly dense C*-subalgebra of the interpolated free group factor L(F_s). Certain approximation properties for A_s are shown. It follows that L(F_s) has the weak expectation property of Lance with respect to A_s. In the course of the demonstration, it is proved that under certain conditions, full amalgamated free products of matrix algebras are residually finite dimensional.
Partial actions of groups on C*-algebras and the closely related actions and coactions of Hopf algebras received much attention over the last decades. They arise naturally as restrictions of their global counterparts to non-invariant subalgebras, and the ambient eveloping global (co)actions have proven useful for the study of associated crossed products. In this article, we introduce the partial coactions of C*-bialgebras, focussing on C*-quantum, and prove existence of an enveloping global coaction under mild technical assumptions. The construction of the latter provides a left adjoint to the…
Let M be a tracial von Neumann algebra and A be a weakly dense unital C*-subalgebra of M. We say that a set X is a W*-generating set for M if the von Neumann algebra generated by X is M and that X is a C*-generating set for A if the unital C*-algebra generated by X is A. For any finite W*-generating set X for M we show that $δ_0(X) \leq sup {δ_0(Y): Y is a finite C*-generating set for A}$ where $δ_0$ denotes the microstates free entropy dimension. It follows that if $sup {δ_0(Y): Y is a finite C*-generating set for C*_{red}(\mathbb F_2)} < \infty$, then the free group factors are all nonisomor…
We show that certain dense and spectral invariant subalgebras of a $C^*$-algebra have the same bilateral Bass stable rank. This is a partial answer for (a version of) an open problem raised by R.G. Swan. Then, for certain Banach algebras, we indicate when the homotopy groups $π_{i}(GL_{n}(A))$ stabilize for large $n$. This is an improvement of a result due to G. Corach and A. Larotonda. Using some results due to M. Karoubi, we show the isomorphism of the Witt group of a symmetric Banach algebra with the $K_0$-group of its enveloping $C^*$-algebra. The question if this is true for all involutiv…
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