Family Of Partial Isometries Research Articles

Unitary Representations of the Baumslag–Solitar Group on the Cantor Set

The Cantor Set supports a Borel probability measure known as the Hutchinson measure which satisfies a well known fixed point relationship (Hutchinson in Indiana University Math J 30(5):713–747, 1981). Previously it has been shown by Jorgensen and Dutkay that the Cantor set can be extended to an inflated Cantor set, $${\mathcal {R}}$$, on a subset of the real line, which supports an extended Hutchinson measure $${\bar{\mu }}$$ (Dutkay and Jorgensen in Rev. Mat. Iberoamericana 22(1):131–180, 2006). Unitary dilation and translation operators can be defined on $$L^2({\mathcal {R}}, {\bar{\mu }})$$ which satisfy the Baumslag–Solitar group relation, and give rise to a filtration of the Hilbert space $$L^2({\mathcal {R}}, {\bar{\mu }})$$ called a multi-resolution analysis (Dutkay and Jorgensen 2006). The low pass filter function corresponding to this construction can be used to produce a measure, m, on a compact topological group called the 3-solenoid, denoted $${\mathcal {S}}_3$$ (Dutkay in Trans Am Math Soc 358(12):5271–5291, 2006). The Hilbert space $$L^2({\mathcal {S}}_3, m)$$ also admits a unitary representation of the Baumslag–Solitar group, and there exists a generalized Fourier transform between $$L^2({\mathcal {R}}, {\bar{\mu }})$$ and $$L^2({\mathcal {S}}_3,m)$$ (Dutkay 2006). In this paper, we build off of Jorgensen and Dutkay’s work to show that the unitary operators on $$L^2({\mathcal {S}}_3,m)$$ mentioned above are related to each other via a family of partial isometries, which satisfy properties resembling the Cuntz relations.

Algebra Associated with a Map Inducing an Inverse Semigroup

The algebra under study belongs to the class of operator algebras generated by a family of partial isometries, satisfying some relations on the initial and final projections. In turn, this family is uniquely determined by a self-mapping of a countable set. In the paper we consider a situation when isometry family generates an inverse semigroup. It is shown that in this (and only in this) case the corresponding C*-algebra has a nontrivial commutative AF-subalgebra, generated by a semi-lattice of projections of inverse semigroup. All invariant subspaces of the mentioned C*-algebra and its irreducible representations are described.

C*-Algebras Generated by Mappings. Classification of Invariant Subspaces

We continue the study of an operator algebra associated with a self-mapping ϕ on a countable setX which can be represented as a directed graph. This C*-algebra belongs to a class of operator algebras, generated by a family of partial isometries satisfying some relations on their source and range projections. Earlier we have formulated the irreducibility criterion of such algebras, which give us a possibility to examine the structure of the corresponding Hilbert space. We will show that for reducible algebras the underlying Hilbert space can be represented either as an infinite sum of invariant subspaces or as a tensor product of a finite-dimensional Hilbert space with l2(Z). In the first case we present a conditions under which the studied algebra has an irreducible representation into a C*-algebra generated by a weighted shift operator. In the second case, the algebra has the irreducible finite-dimensional representations indexed by the unit circle.

C*-Algebras Generated by Mappings. Criterion of Irreducibility

We study the operator algebra associated with a self-mapping ϕ on a countable set X which can be represented as a directed graph. The algebra is generated by the family of partial isometries acting on the corresponding l2(X). We study the structure of involutive semigroup multiplicatively generated by the family of partial isometries. We formulate the criterion when the algebra is irreducible on the Hilbert space. We consider the concrete examples of operator algebras. In particular, we give the examples of nonisomorphic C*-algebras, which are the extensions by compact operators of the algebra of continuous functions on the unit circle.

On a Class of Operator Algebras Generated by a Family of Partial Isometries

The paper provides a short overview of a series of articles devoted to C*-algebras generated by a self-mapping on a countable set. Such an algebra can be seen as a representation of the universal C*-algebra generated by a family of partial isometries satisfying a set of conditions. These conditions are determined by the initial mapping.

On the ideals of a C*-algebra generated by a family of partial isometries and multipliers

The C*-subalgebra of the algebra of all bounded operators on the Hilbert space l2 generated by the multiplier algebra and a family of partial isometries satisfying certain relations is considered. The ideals of the algebra under examination and of its quotient by the algebra of compact operators are studied. It is shown that the quotient algebra can be represented as a direct sum of two principal ideals and has nontrivial center.

Noncommutative integration

We will show that if $\mathcal{M}$ is a factor, then for any pair $\varphi ,p\in\mathcal{M}_*^+$ of normal positive linear functionals on $\mathcal{M}$, the inequality: $$ \|\varphi\|\leq \|\psi\| $$ is equivalent to the fact that there exist a countable family $\{\varphi_i : i\in I\}\subset \mathcal{M}_*^+$ in $\mathcal{M}_*^+$ and a family $\{u_i : i\in I\}\subset\mathcal{M}$ of partial isometries in $\mathcal{M}$ such that $$ \varphi=\sum_{ i\in I} \varphi_i,\quad \sum_{ i\in I} u_i{\varphi_i}u_i^*\leq \psi, \quad \text{and} \quad u_i^*u_i=s(\varphi_i), i\in I, $$ where $s(\omega), \omega\in \mathcal{M}_*^+$, means the support projection of $\omega$. Furthermore, if $\|\varphi\|=\|\psi\|$, then the equality replaces the inequality in the second statement. In the case that $\mathcal{M}$ is not of type III$_1$, the family of partial isometries can be replaced by a family of unitaries in $\mathcal{M}$. One cannot expect to have this result in the usual integration theory. To have a similar result, one needs to bring in some kind of non-commutativity. Let $\{X, \mu\}$ be a $\sigma$-finite semifinite measure space and $G$ be an ergodic group of automorphisms of $L^\infty(X,\mu)$, then for a pair $f$ and $g$ of $\mu$-integrable positive functions on $X$, the inequality: $$ \int_X f(x)\text{d} \mu(x)\leq \int_X g(x)\text{d} \mu(x) $$ is equivalent to the existence of a countable families $\{f_i: i\in I\}\subset L^1(X, \mu)$ of positive integrable functions and $\{\gamma_i: i\in I\}$ in $G$ such that $$ f=\sum_{ i\in I} f_i\quad\text{and}\quad \sum_{ i\in I} \gamma_i(f_i)\leq g, $$ where the summation and inequality are all taken in the ordered Banach space $L^1(X, \mu)$ and the action of $G$ on $L^1(X, \mu)$ is defined through the duality between $L^\infty(X, \mu)$ and $L^1(X, \mu)$, i.e., $$ (\gamma(f))(x)=f(\gamma^{-1} x)\frac{d\mu\circ \gamma^{-1}}{d\mu}(x), \quad f\in L^1(X, \mu). $$

One class of C*-algebras generated by a family of partial isometries and multiplicators

We consider a C*-subalgebra of the algebra of all bounded operators on the Hilbert space of square-summable functions defined on some countable set. The algebra under consideration is generated by a family of partial isometries and the multiplier algebra isomorphic to the algebra of all bounded functions defined on the mentioned set. The partial isometry operators satisfy correlations defined by a prescribed map on the set. We show that the considered algebra is ℤ-graduated. After that we construct the conditional expectation from the latter onto the subalgebra responding to zero. Using this conditional expectation, we prove that the algebra under consideration is nuclear.

On Unitary Extensions of Multiplicative Families of Partial Isometries with a Generating Subspace

We consider a two parameter family of operators on a Hilbert space given by a multiplicative family of partial isometries with a generating subspace that commutes with an unitary group. The first parameter runs on a generalized interval of an abelian ordered group and the second on an abelian group. We show that it can be extended to a two parameter group of unitary operators on a larger Hilbert space. Applications to the problem of extending generalized semigroups of isometries and locally defined positive definite functions are given.

Hyper-reflexivity of free semigroupoid algebras

As a generalization of the free semigroup algebras considered by Davidson and Pitts, and others, the second author and D.W. Kribs initiated a study of reflexive algebras associated with directed graphs. A free semigroupoid algebra L G \mathcal {L}_G is generated by a family of partial isometries, and initial projections, which act on a generalized Fock space spawned by the directed graph G G . We show that if the graph is finite, then L G \mathcal {L}_G is hyper-reflexive.

Graph C*-Algebras with Real Rank Zero

Given a row-finite directed graph E, a universal C*-algebra C*(E) generated by a family of partial isometries and projections subject to the relations determined by E is associated to the graph E. The Cuntz–Krieger algebras are those graph C*-algebras associated to some finite graphs. We prove that a graph C*-algebra C*(E) has real rank zero in the sense that the set of invertible self-adjoint elements is dense in the set of all self-adjoint elements in C*(E) (or in the unitization of C*(E) if C*(E) is nonunital) if and only if E satisfies a loop condition (K) that is analogous to the condition for a finite {0, 1} matrix A under which Cuntz analyzed the ideal structure of the Cuntz–Krieger algebra OA.

Nonclassical states of light and canonical transformations

Representations of nonlinear nonbijective canonical transformations in quantum mechanics are discussed. Due to the nonbijectivity the classical phase space has a Riemann sheet structure, and a family of partial isometries translating this structure into quantum mechanics is constructed. If a unitary representation is required, a new variable-the ambiguity spin-has to be introduced in order to recover bijectivity following the approach of Moshinsky (1981) and co-workers. The new degree of freedom is analysed in terms of multiboson operators. The application of this formalism to some non-classical states of light is discussed.