In this version we explicitly exhibit enough (irreducible) boundary repns to faithfully represent the C*-envelope of the tensor algebra of the system. We also show that piecewise conjugacy is not a complete invariant for complete isomorphisms between semicrossed products. The paper has been reorganized
Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $τ_i:X \to X$ for $1 \le i \le n$. To this we associate two topological conjugacy algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra $\A(X, τ)$ and the semicrossed product $\rC_0(X)\times_τ\Fn$.
We introduce a concept of conjugacy for multidimensional systems, which we coin piecewise conjugacy. We prove that the piecewise conjugacy class of the system can be recovered from either the algebraic structure of $\A(X, τ)$ or $\rC_0(X)\times_τ\Fn$. Various classification results follow as a consequence. For example, for $n=2,3$, the tensor algebras are (algebraically or even completely isometrically) isomorphic if and only if the systems are piecewise topologically conjugate. In order to establish these results we make use of analytic varieties as well as homotopy theory for Lie groups
We define a generalized notion of wandering sets and recurrence. Using this, it is shown that $\A(X, τ)$ or $\rC_0(X)\times_τ\Fn$ is semisimple if and only if there are no generalized wandering sets. In the metrizable case, this is equivalent to each $τ_i$ being surjective and $v$-recurrent points being dense for each $v \in \Fn$.
We resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis relating to work of Arveson from the sixties, and extends related work of Kakariadis and Katsoulis.
We examine the completely isometric automorphisms of a free product of noncommutative disc algebras. It will be established that such an automorphism is given simply by a completely isometric automorphism of each component of the free product and a permutation of the components. This mirrors a similar fact in topology concerning biholomorphic automorphisms of product spaces with nice boundaries due to Rudin, Ligocka and Tsyganov. This paper is also a study of multivariable dynamical systems by their semicrossed product algebras. A new form of dynamical system conjugacy is introduced and is sho…
We show that if (A,a) and (B,b) are automorphic multivariable C*-dynamical systems with isometrically isomorphic tensor algebras (or semi crossed products), then the systems are piecewise conjugate over their Jacobson spectrum. This answers a question of Kakariadis and the author.
Unitary invariants in multivariable operator theory
本文中考虑的问题是我们开发Sz自由类似物计划的自然延续。Nagy-Foias理论，用于行收缩。本文件的结构如下：n元组算子的酉不变量1.接头数值半径2.欧几里德算子半径3.联合数值域和频谱4. $ρ$ -操作员半径第二部分。联合算子半径、不等式和应用5. von Neumann不等式6.约束冯诺依曼不等式7.多变量Haagerup-de la Harpe不等式8.多变量Fejer不等式参考文献
On Bilateral Weighted Shifts in Noncommutative Multivariable Operator Theory
我们提出了一个推广的双边加权移位算子的非交换多变量设置。我们发现了一个周期性的概念，这些变化，这有一个有吸引力的数学解释的无限树结构与底层的希尔伯特空间。这些移位通过Cuntz C $^*$ -代数$O_n$的某些表示的加权形式自然产生。在研究这些算子的联合约化子空间结构时，考虑这些算子生成的弱算子拓扑闭代数是方便且等价的。我们证明了这些代数有非平凡的约化子空间时，正是双周期的移位，也就是说，移位的权重有周期性的行为，和相应的表示$O_n$有一定的空间周期性。这推广了单变量情况下的Nikolskii定理。
To a given multivariable C*-dynamical system $(A, \al)$ consisting of *-automorphisms, we associate a family of operator algebras $\alg(A, \al)$, which includes as specific examples the tensor algebra and the semicrossed product. It is shown that if two such operator algebras $\alg(A, \al)$ and $\alg(B, \be)$ are isometrically isomorphic, then the induced dynamical systems $(\hat{A}, \hat{\al})$ and $(\hat{B}, \hat{\be})$ on the Fell spectra are piecewise conjugate, in the sense of Davidson and Katsoulis. In the course of proving the above theorem we obtain several results of independent inter…
本文得到了如下算子值多元Bohr型不等式：一类非交换全纯函数，推广了开单位圆盘上的解析函数;非交换圆盘代数和非交换解析Toeplitz代数;一类非交换自伴调和函数，推广了开单位圆盘上的实值调和函数;具有n个生成元的自由群的约化（或全）群C* 代数; C^n的开单位球上的一类解析函数。证明了经典的Bohr不等式是正三角多项式系数的Fejer不等式和幂零算子幂零算子的Haagerup-de la Harpe不等式的一个推论.此外，我们提供了一个不等式，它是开放的单位圆盘上的解析多项式，比玻尔的不等式更尖锐。
A commuting $n$-tuple $(T_1, \ldots, T_n)$ of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h,\]where $p \in \mathbb{C}[z_1, \ldots, z_n]$ and $h \in \mathcal{H}$. A companion survey provides an introduction to the theory of Hilbert modules and some (Hilbert) module point of view to multivariable operator theory. The purpose of this survey is to emphasize algebraic and g…
Let $\{T_1, \ldots, T_n\}$ be a set of $n$ commuting bounded linear operators on a Hilbert space $\mathcal{H}$. Then the $n$-tuple $(T_1, \ldots, T_n)$ turns $\mathcal{H}$ into a module over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \raro \clh, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h,\]where $p \in \mathbb{C}[z_1, \ldots, z_n]$ and $h \in \mathcal{H}$. The above module is usually called the Hilbert module over $\mathbb{C}[z_1, \ldots, z_n]$. Hilbert modules over $\mathbb{C}[z_1, \ldots, z_n]$ (or natural function algebras) …
Multivariable operator theory is used to provide Bohr inequalities for free holomorphic functions with operator coefficients on the regular polyball. In addition, we obtain analogues of Caratheodory, Fejer, and Egervary-Szazs inequalities for free holomorhic functions with operator coefficients and positive real parts on the polyball. These results are used to provide multivariable analogues of Landau's inequality and Bohr's inequality when the norm is replaced by the numerical radius of an operator.
We examine the theory of the KMS states on Pimsner algebras arising from multivariable unital C*-dynamical systems. As an application we show that Pimsner algebras of piecewise conjugate classical systems attain the same KMS states, even though it is still an open problem whether or not they are $*$-isomorphic.
In this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital $C^*$-algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the $C^*$-algebra of operators on a Hilbert space, even to general unital $C^*$-algebras, and fina…
