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 completely isometric to the commutant of the minimal normal extension $\hat{\Cal F}$. Applications of these results are given for Toeplitz operators on strictly pseudoconvex or bounded symmetric domains.
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.
We study the structure of isometries defined on the algebra $\mathcal{A}$ of upper-triangular Toeplitz matrices. Our first result is that a continuous multiplicative isometry $\mathcal{A}\to M_n$ must be of the form either $A\mapsto UAU^*$ or $A\mapsto U\overline AU^*$, where $\overline A$ is the complex conjugation and $U$ is a unitary matrix. In our second result we use a range of ideas in operator theory and linear algebra to show that every linear isometry $\mathcal{A}\to M_n(\mathbb{C})$ is of the form $A\mapsto UAV$ where $U$ and $V$ are two unitary matrices. This implies, in particular,…
The Cuntz-Toeplitz algebra $E_{n+1}$ for $n\geq1$ is the universal C*-algebra generated by $n+1$ isometries with mutually orthogonal ranges. In this paper, we investigate the automorphism groups of the Cuntz-Toeplitz algebras and determine their homotopy groups.
The graph product of a family of groups lies somewhere between their direct and free products, with the graph determining which pairs of groups commute and which do not. We show that the graph product of quasi-lattice ordered groups is quasi-lattice ordered, and, when the underlying groups are amenable, that it satisfies Nica's amenability condition for quasi-lattice orders. As a consequence the Toeplitz algebras of these groups are universal for covariant isometric representations on Hilbert space, and their representations are faithful if the isometries satisfy a properness condition given b…
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.
The non-commutative analytic Toeplitz algebra is the weak operator topology closed algebra generated by the left regular representation of the free semigroup on $n$ generators. The structure theory of contractions in these algebras is examined. Each is shown to have an $H^\infty$ functional calculus. The isometries defined by words are shown to factor only as the words do over the unit ball of the algebra. This turns out to be false over the full algebra. The natural identification of weakly closed left ideals with invariant subspaces of the algebra is shown to hold only for a proper subcollec…
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…
In the theory of C*-algebras, interesting noncommutative structures arise as deformations of the tensor product. For instance, the rotation algebra may be seen as a scalar twist deformation of the tensor product of the functions on the circle with itself. We deform the tensor product of two Toeplitz algebras in the same way, introducing the universal C*-algebra generated by two isometries u and v such that uv=e^{it}vu and u*v=e^{-it}vu*, for a fixed real parameter t. Since the second relation implies the first one, we also consider the universal C*-algebra generated by two isometries u and v w…
There has recently been much interest in the $C^*$-algebras of directed graphs. Here we consider product systems $E$ of directed graphs over semigroups and associated $C^*$-algebras $C^*(E)$ and $\mathcal{T}C^*(E)$ which generalise the higher-rank graph algebras of Kumjian-Pask and their Toeplitz analogues. We study these algebras by constructing from $E$ a product system $X(E)$ of Hilbert bimodules, and applying recent results of Fowler about the Toeplitz algebras of such systems. Fowler's hypotheses turn out to be very interesting graph-theoretically, and indicate new relations which will ha…
We consider the C*-algebras O_2^q and A_2^q generated, respectively, by isometries s_1, s_2 satisfying the relation s_1^* s_2 = q s_2 s_1^* with |q| < 1 (the deformed Cuntz relation), and by isometries s_1, s_2 satisfying the relation s_2 s_1 = q s_1 s_2 with |q| = 1. We show that O_2^q is isomorphic to the Cuntz-Toeplitz C*-algebra O_2^0 for any |q| < 1. We further prove that A_2^{q_1} is isomorphic to A_2^{q_2} if and only if either q_1 = q_2 or q_1 = complex conjugate of q_2. In the second part of our paper, we discuss the complexity of the representation theory of A_2^q. We show that A_2^q…
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.…
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…
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 contractive $n$-tuple $A=(A_1,...,A_n)$ has a minimal joint isometric dilation $S=(S_1,...,S_n)$ where the $S_i$'s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When $A$ acts on a finite dimensional space, the \wot-closed nonself-adjoint algebra $\mathfrak{S}$ generated by $S$ is completely described in terms of the properties of $A$. This provides complete unitary invariants for the corresponding representations. In addition, we show that the algebra $\mathfrak{S}$ is always hyper-reflexive. In the last section, we describe s…
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…
We show that every biorthogonal wavelet determines a representation by operators on Hilbert space satisfying simple identities, which captures the established relationship between orthogonal wavelets and Cuntz-algebra representations in that special case. Each of these representations is shown to have tractable finite-dimensional co-invariant doubly-cyclic subspaces. Further, motivated by these representations, we introduce a general Fock-space Hilbert space construction which yields creation operators containing the Cuntz--Toeplitz isometries as a special case.
Operator algebras generated by partial isometries and their adjoints form the basis for some of the most well studied classes of C*-algebras. The primary object of this paper is the norm-closed operator algebra generated by a left invertible $T$ together with its Moore-Penrose inverse $T^\dagger$. We denote this algebra by $\mathfrak{A}_T$. In the isometric case, $T^\dagger = T^*$ and $\mathfrak{A}_T$ is a representation of the Toeplitz algebra. Of particular interest is the case when $T$ satisfies a non-degeneracy condition called analytic. We show that $T$ is analytic if and only if $T^*$ is…
Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. Relations between operations arising from the structure definitions, however, lead to restrictions which determine their possible arity shapes and lead us to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic an…
