Extensions of C*-algebras by partial isometries

Complex symmetric partial isometries

An operator $T \in B(\h)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:\h\to\h$ so that $T = CT^*C$. We provide a concrete description of all complex symmetric partial isometries. In particular, we prove that any partial isometry on a Hilbert space of dimension $\leq 4$ is complex symmetric.

On a joint $(m, (q_1, ..., q_d))$-partial isometries and a joint $m$-invertible $d$-tuple of operators on a Hilbert space

The aim of the present paper is, firstly we study the concepts of (m, (q_1, ..., q_d))- partial isometries on a Hilbert space, secondly, we introduce the notion of m- invertibility of tuples of operators as a natural generalization of the m-invertibility in single variable operators.

Open partial isometries and positivity in operator spaces

We study positivity in C*-modules and operator spaces using open tripotents, and an ordered version of the `noncommutative Shilov boundary'. Because of their independent interest, we also systematically study open tripotents and their properties.

On The Extension of the Toeplitz Algebra by Isometries

We introduce the notion of π-extension of the semigroup \mathbb{Z}_+ and study the extensions of the Toeplitz algebras by isometric operators. We show that when the action of the Toeplitz algebra is irreducible all such extensions generate the same algebra, i.e. there are no non-trivial extensions of the Toeplitz algebra. Also we provide the examples of the non-trivial extensions of the Toeplitz algebra in case its representation is reducible.

Partial dynamical systems and C*-algebras generated by partial isometries

A collection of partial isometries whose range and initial projections satisfy a specified set of conditions often gives rise to a partial representation of a group. The C*-algebra generated by the partial isometries is thus a quotient of the universal C*-algebra for partial representations of the group, from which it inherits a crossed product structure, of an abelian C*-algebra by a partial action of the group. Questions of faithfulness of representations, simplicity, and ideal structure of these C*-algebras can then be addressed in a unified manner from within the theory of partial actions.…

希尔伯特模之间的部分等距及其复合

Partial Isometries Between Hilbert Modules and Their Compositions

受Accardi和Lu的问题启发，研究希尔伯特空间中两部分等距乘积仍为部分等距的准则，定义新复合运算，并证明该运算无法直接推广到希尔伯特模，需转向部分定义等距类。

On the circular numerical range of 5-by-5 partial isometries

We prove, in some cases in term of kippenhahn curve, that if 5-by-5 partial isometry whose numerical range is a circular disc then its center is must be the origin. This gives a partial affirmative answer of the Conjecture 5.1. of [H. l. Gau et al., Linear and Multilinear Algebra, 64 (1) 2016, 14--35.], for the five dimensional case.

Semigroups of Partial Isometries

We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of self-adjoint semigroups of partial isometries. We obtain a general structure result showing that every self-adjoint semigroup of partial isometries consists of "generalized weighted composition" operators on a space of square-integrable Hilbert-space valued functions. If the semigroup is irreducible and contains a compact operator then the underlying measure space is p…

On selfadjoint extensions of semigroups of partial isometries

Let $\mathcal S$ be a semigroup of partial isometries acting on a complex, infinite-dimensional, separable Hilbert space. In this paper we seek criteria which will guarantee that the selfadjoint semigroup $\mathcal T$ generated by $\mathcal S$ consists of partial isometries as well. Amongst other things, we show that this is the case when the set of final projections of elements of $\mathcal S$ generates an abelian von Neumann algebra of uniform finite multiplicity.

Symbolic Dynamics, Partial Dynamical Systems, Boolean Algebras and C*-algebras Generated by Partial Isometries

We associate to each discrete partial dynamical system a universal C*-algebra generated by partial isometries satisfying relations given by a Boolean algebra connected to the discrete partial dynamical system in question. We show that for symbolic dynamical systems like one-sided and two-sided shift spaces and topological Markov chains with an arbitrary state space the C*-algebras usually associated to them, can be obtained in this way. As a consequence of this, we will be able to show that for two-sided shift spaces having a certain property, the crossed product of the two-sided shift space i…

The algebraic structure of quantum partial isometries

The partial isometries of $\mathbb R^N,\mathbb C^N$ form compact semigroups $\widetilde{O}_N,\widetilde{U}_N$. We discuss here the liberation question for these semigroups, and for their discrete versions $\widetilde{H}_N,\widetilde{K}_N$. Our main results concern the construction of half-liberations $\widetilde{H}_N^\times,\widetilde{K}_N^\times,\widetilde{O}_N^\times,\widetilde{U}_N^\times$ and of liberations $\widetilde{H}_N^+,\widetilde{K}_N^+,\widetilde{O}_N^+,\widetilde{U}_N^+$. We include a detailed algebraic and probabilistic study of all these objects, justifying our "half-liberation"…

幂部分等距算子

Power partial isometries

本文完整刻画了幂部分等距算子完全非酉部分的约化、不变与超不变子空间，并特别给出了截断移位算子三类子空间的精确刻画。

On the extension of isometries between the unit spheres of a C*-algebra and B(H)

Given two complex Hilbert spaces $H$ and $K$, let $S(B(H))$ and $S(B(K))$ denote the unit spheres of the C$^*$-algebras $B(H)$ and $B(K)$ of all bounded linear operators on $H$ and $K$, respectively. We prove that every surjective isometry $f: S(B(K)) \to S(B(H))$ admits an extension to a surjective complex linear or conjugate linear isometry $T: B(K)\to B(H)$. This provides a positive answer to Tingley's problem in the setting of $B(H)$ spaces.

关于圆状Kippenhahn曲线及Gau-Wang-Wu关于幂等偏置运算的猜想

On circular Kippenhahn curves and the Gau-Wang-Wu conjecture about nilpotent partial isometries

本文研究具有圆状Kippenhahn曲线的线性算子，探讨其圆性性质，并通过构造更广的算子类推翻Gau-Wang-Wu猜想，提出数值验证旋转不变性的方法。

关于core EP Drazin矩阵与部分等距算子的若干结果

Some results on core EP Drazin matrices and partial isometries

本文基于core EP逆与Drazin逆引入CEPD矩阵类，研究其代数性质，推导部分等距算子相关广义逆的结果并给出其为CEPD矩阵的条件，辅以数值示例验证。

On the extension of surjective isometries whose domain is the unit sphere of a space of compact operators

We prove that every surjective isometry from the unit sphere of the space $K(H),$ of all compact operators on an arbitrary complex Hilbert space $H$, onto the unit sphere of an arbitrary real Banach space $Y$ can be extended to a surjective real linear isometry from $K(H)$ onto $Y$. This is probably the first example of an infinite dimensional non-commutative C$^*$-algebra containing no unitaries and satisfying the Mazur--Ulam property. We also prove that all compact C$^*$-algebras and all weakly compact JB$^*$-triples satisfy the Mazur--Ulam property.

On the extension of isometries between the unit spheres of a JBW$^*$-triple and a Banach space

We prove that every JBW$^*$-triple $M$ with rank one or rank bigger than or equal to three satisfies the Mazur--Ulam property, that is, every surjective isometry from the unit sphere of $M$ onto the unit sphere of another Banach space $Y$ extends to a surjective real linear isometry from $M$ onto $Y$. We also show that the same conclusion holds if $M$ is not a JBW$^*$-triple factor, or more generally, if the atomic part of $M^{**}$ is not a rank two Cartan factor.

Extension of isometries from the unit sphere of a rank-2 Cartan factor

We prove that every surjective isometry from the unit sphere of a rank-2 Cartan factor $C$ onto the unit sphere of a real Banach space $Y$, admits an extension to a surjective real linear isometry from $C$ onto $Y$. The conclusion also covers the case in which $C$ is a spin factor. This result closes an open problem and, combined with the conclusion in a previous paper, allows us to establish that every JBW$^*$-triple $M$ satisfies the Mazur--Ulam property, that is, every surjective isometry from its unit sphere onto the unit sphere of a arbitrary real Banach space $Y$ admits an extension to a…

The structure of twisted power partial isometries

Let $n>1$ and let $\{U_{ij}\}_{1\leq i<j\leq n}$ be $n\choose 2$ commuting unitaries on a Hilbert space $\mathcal{H}$. Suppose $U_{ji}:=U^*_{ij}$, $1\leq i<j\leq n$. An n-tuple of power partial isometries $(V_1,...,V_n)$ on Hilbert space $\mathcal{H}$ is called $\mathcal{U}_n$-twisted power partial isometry with respect to $\{U_{ij}\}_{i<j}$ (or simply $\mathcal{U}_n$-twisted power partial isometry if $\{U_{ij}\}_{i<j}$ is clear from the context) if $V_i^*V_j=U_{ij}V_jV^*_i, ~~ V_iV_j=U_{ji}V_jV_i ~~\text{and}~~ V_kU_{ij}=U_{ij}V_k~~(i,j,k=1,2,...,n,~\text{and}~i\neq j).$ We prove that each $\…

Partial Dynamical Systems, Fell Bundles and Applications

This is a book about Partial Actions and Fell Bundles with applications to C*-algebras generated by partial isometries. Here is the table of contents: 1-Introduction, 2-Partial actions, 3-Restriction and globalization, 4-Inverse semigroups, 5-Topological partial dynamical systems, 6-Algebraic partial dynamical systems, 7-Multipliers, 8-Crossed products, 9-Partial group representations, 10-Partial group algebras, 11-C*-algebraic partial dynamical systems, 12-Partial isometries, 13-Covariant representations of C*-algebraic dynamical systems, 14-Partial representations subject to relations, 15-Hi…

A classification for 2-isometries of noncommutative Lp-spaces

In this paper we extend previous results of Banach, Lamperti and Yeadon on isometries of Lp-spaces to the non-tracial case first introduced by Haagerup. Specifically, we use operator space techniques and an extrapolation argument to prove that every 2-isometry T : Lp(M) to Lp(N) between arbitrary noncommutative Lp-spaces can always be written in the form T(phi^{1/p}) = w (phi circ pi^{-1} circ E)^{1/p}, for phi in M_*^+. Here pi is a normal *-isomorphism from M onto the von Neumann subalgebra pi(M) of N, w is a partial isometry in N, and E is a normal conditional expectation from N onto pi(M).…

Some Exact Sequences for Toeplitz Algebras of Spherical Isometries

A family $\{T_j\}_{j\in J}$ of commuting Hilbert space operators is said to be a spherical isometry if $\sum_{j\in J}T^*_jT_j=1$ in the weak operator topology. We show that every commuting family $\Cal F$ of spherical isometries has a commuting normal extension $\hat{\Cal F}$. Moreover, if $\hat{\Cal F}$ is minimal, then there exists a natural short exact sequence $0\to\Cal C\to C^*(\Cal F)\to C^*(\hat{\Cal F})\to 0$ with a completely isometric cross-section, where $\Cal C$ is the commutator ideal in $C^*(\Cal F)$. We also show that the space of Toeplitz operators associated to $\Cal F$ is com…

乘积部分等距、可管理性及C*-代数量子群体

Multiplicative partial isometries, manageability, and C*-algebraic quantum groupoids

本文提出乘积部分等距的概念，并通过可管理性条件构造C*-代数量子群体结构。

On the structure of isometries between noncommutative Lp spaces

We prove some structure results for isometries between noncommutative Lp spaces associated to von Neumann algebras. We find that an isometry T: Lp(M_1) to Lp(M_2) (1 le p < infty, p not 2) can be canonically expressed in a certain simple form whenever M_1 has variants of Watanabe's extension property. Conversely, this form always defines an isometry provided that M_1 is "approximately semifinite" (defined below). Although neither of these properties is fully understood, we show that they are enjoyed by all semifinite algebras and hyperfinite algebras (with no summand of type I_2), plus others.…

Haagerup–Schultz代数的等距映射

Isometries of Haagerup--Schultz algebras

该研究在Haagerup–Schultz代数框架下，推广了Kadison和Yeadon的经典等距映射结果，证明这类代数上的非满射等距映射由部分等距映射与保迹Jordan *-单射实现。

A classification theorem for direct limits of extensions of circle algebras by purely infinite C*-algebras

We give a classification theorem for a class of C*-algebras which are direct limits of extensions of circle algebras by purely infinite C*-algebras. The invariant consists of the following: (1) the set of Murray-von Neumann equivalence classes of projections; (2) the set of homotopy classes of hyponormal partial isometries; (3) a map d; and (4) total K-theory.

Co-universal C*-algebras associated to aperiodic k-graphs

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…

拓扑满群、可逆等距与广群代数的自同构

Topological full groups, invertible isometries, and automorphisms of groupoid algebras

本文研究紧单位空间Hausdorff丰沛广群的拓扑满群与对应伪函数代数可逆等距同伦类群的关系，并给出有效广群下伪函数代数自同构群的分裂扩张结构。

Partially Isometric Dilations of Noncommuting $N$-tuples of Operators

Given a row contraction of operators on Hilbert space and a family of projections on the space which stabilize the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries which satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in our analysis is a connection with directed graphs. We use a Wold Decomposition for partial isometries to describe the models for these dilations, and discuss how the basic properties of a dilation depend…

Subalgebras of Graph C*-Algebras

We prove a spectral theorem for bimodules in the context of graph C*-algebras. A bimodule over a suitable abelian algebra is determined by its spectrum (i.e., its groupoid partial order) iff it is generated by the Cuntz-Krieger partial isometries which it contains iff it is invariant under the gauge automorphisms. We study 1-cocycles on the Cuntz-Krieger groupoid associated with a graph C*-algebra, obtaining results on when integer valued or bounded cocycles on the natural AF subgroupoid extend. To a finite graph with a total order, we associate a nest subalgebra of the graph C*-algebra and th…

由满足扭曲交换关系的等距算子生成的$C^*$-代数的酉不变量

A unitary invariant for $C^*$-algebras generated by isometries with a twisted commutation relation

本文研究满足$V_1V_2=qV_2V_1$（$q$为无理旋转）的等距算子对生成的$C^*$-代数的酉分类，通过函数模型和半Fredholm指标构造酉不变量，扩展了Berger-Coburn-Lebow定理至非交换情形。

Partial-isometric crossed products of dynamical systems by left LCM semigroups

Let P be a left LCM semigroup, and $α$ an action of $P$ by endomorphisms of a $C^{*}$-algebra $A$. We study a semigroup crossed product $C^{*}$-algebra in which the action $α$ is implemented by partial isometries. This crossed product gives a model for the Nica-Teoplitz algebras of product systems of Hilbert bimodules (associated with semigroup dynamical systems) studied first by Fowler, for which we provide a structure theorem as it behaves well under short exact sequences and tensor products.

Extension of projection mappings

We show that a map between projection lattices of semi-finite von Neumann algebras can be extended to a Jordan $*$-homomorphism between the von Neumann algebras if this map is defined in terms of the support projections of images (under the linear map) of projections and the images of orthogonal projections have orthogonal support projections. This has numerous fundamental applications in the study of isometries and composition operators on quantum symmetric spaces and is of independent interest, since it provides a partial generalization of Dye's Theorem without the requirement that the initi…

Partial-isometric crossed products by semigroups of endomorphisms

Let $Γ^+$ be the positive cone in a totally ordered abelian group $Γ$, and let $α$ be an action of $Γ^+$ by endomorphisms of a $C^*$-algebra $A$. We consider a new kind of crossed-product $C^*$-algebra $A\times_αΓ^+$, which is generated by a faithful copy of $A$ and a representation of $Γ^+$ as partial isometries. We claim that these crossed products provide a rich and tractable family of Toeplitz algebras for product systems of Hilbert bimodules, as recently studied by Fowler, and we illustrate this by proving detailed structure theorems for actions by forward and backward shifts.

Linear isometries between real JB*-triples and C*-algebras

Let $T: A\to B$ be a (not necessarily surjective) linear isometry between two real JB$^*$-triples. Then for each $a\in A$ there exists a tripotent $u_a$ in the bidual, $B'',$ of $B$ such that \begin{enumerate}[$(a)$] \item $\{u_a,T(\{f,g,h\}),u_a\}=\{u_a,\{T(f),T(g),T(h)\},u_a\}$, for all $f,g,h$ in the real JB$^*$-subtriple, $A_a,$ generated by $a$; \item The mapping $\{u_a,T(\cdot),u_a\} :A_a\rightarrow B''$ is a linear isometry. \end{enumerate} Furthermore, when $B$ is a real C$^*$-algebra, the projection $p=p_a= u_a^* u_a$ satisfies that $T(\cdot)p :A_a\rightarrow B''$ is an isometric trip…

图C*-代数是单生成的

Graph C*-algebras are singly generated

本文证明了可数有向图的C*-代数是单生成的，并展示了由满足Cuntz-Krieger关系的投影和部分等距生成的C*-代数也是单生成的。

Surjective isometries between unitary sets of unital JB$^*$-algebras

This paper is, in a first stage, devoted to establish a topological--algebraic characterization of the principal component, $\mathcal{U}^0 (M)$, of the set of unitary elements, $\mathcal{U} (M)$, in a unital JB$^*$-algebra $M$. We arrive to the conclusion that, as in the case of unital C$^*$-algebras, $$\begin{aligned}\mathcal{U}^0(M) &= M^{-1}_{\textbf{1}}\cap\mathcal{U} (M) =\left\lbrace U_{e^{i h_n}}\cdots U_{e^{i h_1}}(\textbf{1}) \colon \begin{array}{c} n\in \mathbb{N}, \ h_j\in M_{sa} \forall\ 1\leq j \leq n \end{array} \right\rbrace \end{aligned}$$ is analytically arcwise connected. Our…

顺从迹与联合数值半径

Amenable traces and the joint numerical radius

本文利用单位元、等距元和部分等距元的联合自由数值半径，给出了C*-代数上存在顺从迹的充要刻画，并应用于获得各种提升性质的新障碍。

Geometric characterizations of some classes of operators in C*-algebras and von Neumann algebras

We present geometric characterizations of the partial isometries, unitaries, and invertible operators in C*-algebras and von Neumann algebras.

Cuntz-Krieger algebras and a generalization of Catalan numbers

We first observe that the relations of the canonical generating isometries of the Cuntz algebra ${\cal O}_N$ are naturally related to the $N$-colored Catalan numbers. For a directed graph $G$, we generalize the Catalan numbers by using the canonical generating partial isometries of the Cuntz-Krieger algebra ${\cal O}_{A^G}$ for the transition matrix $A^G$ of $G$. The generalized Catalan numbers $c_n^G, n=0,1,2,...$ enumerate the number of Dyck paths and oriented rooted trees for the graph $G$. Its generating functions will be studied.

Open projections in operator algebras I: Comparison theory

We begin a program of generalizing basic elements of the theory of comparison, equivalence, and subequivalence, of elements in C*-algebras, to the setting of more general algebras. In particular, we follow the recent lead of Lin, Ortega, Rordam, and Thiel of studying these equivalences, etc., in terms of open projections or module isomorphisms. We also define and characterize a new class of inner ideals in operator algebras, and develop a matching theory of open partial isometries in operator ideals which simultaneously generalize the open projections in operator algebras (in the sense of the …

Certain free products of graph operator algebras

We develop a notion of a generalized Cuntz-Krieger family of projections and partial isometries where the range of the partial isometries need not have trivial intersection. We associate to these generalized Cuntz-Krieger families a directed graph, with a coloring function on the edge set. We call such a directed graph an edge-colored directed graph. We then study the $C^*$-algebras and the non-selfadjoint operator algebras associated to edge-colored directed graphs. These algebras arise as free products of directed graph algebras with amalgamation. We then determine the $C^*$-envelopes for a …

关于巢代数的一类左理想

On a class of left ideals of nest algebras

本文研究由全序偏等距族确定的巢代数左理想，刻画了其为子代数的充要条件，并证明其可分解性、有限秩算子强稠密性及方程可解条件。

A survey on Tingley's problem for operator algebras

We survey the most recent results on extension of isometries between special subsets of the unit spheres of C$^*$-algebras, von Neumann algebras, trace class operators, preduals of von Neumann algebras, and $p$-Schatten-von Neumann spaces, with special interest on Tingley's problem.

The group of L^2 isometries on H^1_0

Let U be an open subset of R^n. Let L^2=L^2(U,dx) and H^1_0=H^1_0(U) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group G of invertible operators on H^1_0 which preserve the L^2-inner product. When U is bounded and the border $\partial U$ is smooth, this group acts as the intertwiner of the H^1_0 solutions of the non-homogeneous Helmholtz equation $u-Δu=f$, $u|_{\partial U}=0$. We show that G is a real Banach-Lie group, whose Lie algebra is (i times) the space of symmetrizable operators. We discuss the spectrum of operators belo…

Quantum isometry groups of noncommutative manifolds associated to group C*-algebras

Let G be a finitely generated discrete group. The standard spectral triple on the group C*-algebra C*(G) is shown to admit the quantum group of orientation preserving isometries. This leads to new examples of compact quantum groups. In particular the quantum isometry group of the C*-algebra of the free group on n-generators is computed and turns out to be a quantum group extension of the quantum permutation group A_{2n} of Wang. The quantum groups of orientation and real structure preserving isometries are also considered and construction of the Laplacian for the standard spectral triple on C*…