Partial Isometries Research Articles

Ordered *-Semigroups and a C<sup>*</sup>-Correspondence for a Partial Isometry

Certain $*$-semigroups are associated with the universal $C^*$-algebra generated by a partial isometry, which is itself the universal $C^*$-algebra of a $*$-semigroup. A fundamental role for a $*$-structure on a semigroup is emphasized, and ordered and matricially ordered $*$-semigroups are introduced, along with their universal $C^*$-algebras. The universal $C^*$-algebra generated by a partial isometry is isomorphic to a relative Cuntz-Pimsner $C^*$-algebra of a $C^*$-correspondence over the $C^*$-algebra of a matricially ordered $*$-semigroup. One may view the $C^*$-algebra of a partial isometry as the crossed product algebra associated with a dynamical system defined by a complete order map modelled by a partial isometry acting on a matricially ordered $*$-semigroup.

Partial isometries, unitary operators, and complementary operators in polarization optics.

We show that for nonnormal singular operators corresponding to the nonorthogonal polarizers, the unitary polar component is constituted by two partial isometries, one of them "active" and the other "hidden" or "mute." For each such operator there exists a complementary one, corresponding also to a nonorthogonal polarizer, which has the same unitary polar component and whose partial isometries reverse their roles.

Tight frames, partial isometries, and signal reconstruction

This article gives a procedure to convert a frame which is not a tight frame into a Parseval frame for the same space, with the requirement that each element in the resulting Parseval frame can be explicitly written as a linear combination of the elements in the original frame. Several examples are considered, such as a Fourier frame on a spiral. The procedure can be applied to the construction of Parseval frames for the space of square integrable functions whose domain is the ball of radius When a finite number of measurements is used to reconstruct a signal in error estimates arising from such approximation are discussed.

Skew symmetry of a class of operators

An operator $T$ on a complex Hilbert space $\mathcal{H}$ is said to be skew symmetric if there exists a conjugate-linear, isometric involution $C:\mathcal{H}\rightarrow\mathcal{H}$ such that $CTC=-T^*$. In this paper, using an interpolation theorem related to conjugations, we give a geometric characterization for a class of operators to be skew symmetric. As an application, we get a description of skew symmetric partial isometries.

Structures and numerical ranges of power partial isometries

We derive a matrix model, under unitary similarity, of an n-by-n matrix A such that A,A2,…,Ak (k⩾1) are all partial isometries, which generalizes the known fact that if A is a partial isometry, then it is unitarily similar to a matrix of the form [0B0C] with B⁎B+C⁎C=I. Using this model, we show that if A has ascent k and A,A2,…,Ak−1 are partial isometries, then the numerical range W(A) of A is a circular disc centered at the origin if and only if A is unitarily similar to a direct sum of Jordan blocks whose largest size is k. As an application, this yields that, for any Sn-matrix A, W(A) (resp., W(A⊗A)) is a circular disc centered at the origin if and only if A is unitarily similar to the Jordan block Jn. Finally, examples are given to show that, for a general matrix A, the conditions that W(A) and W(A⊗A) are circular discs at 0 are independent of each other.

Fuglede-Kadison determinants for operators in the von Neumann algebra of an equivalence relation

We calculate the Fuglede-Kadison determinant for operators of the form ∑n i=1 MfiLgi where Lgi are unitaries or partial isometries coming from Borel (partial) isomorphisms gi on a probability space which generate an ergodic equivalence relation, and Mfi are multiplication operators. We obtain formulas for the cases when the relation is treeable or the fi’s and gi’s satisfy some restrictions.

Inverse Semigroups and Sheu's Groupoid for Odd Dimensional Quantum Spheres

Abstract.In this paper, we give a different proof of the fact that the odd dimensional quantum spheres are groupoid C*-algebras. We show that the C*-algebra is generated by an inverse semigroup T of partial isometries. We show that the groupoid 𝓖tight associated with the inverse semigroup T by Exel is exactly the same as the groupoid considered by Sheu.

Erratum to: Semigroups of partial isometries

Proof For each i, let Li be the lattice of subspaces of H generated by the ranges and the kernels of T k i , k = 0,1,2, . . . . Let us show that Li = Lj for all i and j . It is enough to establish that ranT k i = ranT k j and kerT k i = kerT k j , k = 0,1,2, . . . . If k = 0, the statement is trivial. Assume by induction that the statement is proved for some k. Pick x ∈ ranT k+1 i . There is y ∈ H such that x = T k+1 i y = TiT k i y. Since ranT k i = ranT k j , there is z ∈ H such that T k i y = T k j z. Then x = TiT k j z. By commutativity, x = T k j Tiz. Since ranTi = ranTj by the assumptions of the theorem, there is v ∈ H such that x = T k j Tj v = T k+1 j v. So, ranT k+1 i = ranT k+1 j . Similarly,

CO-UNIVERSAL C*-ALGEBRAS ASSOCIATED TO APERIODIC k-GRAPHS

AbstractWe construct a representation of each finitely aligned aperiodic k-graph Λ on the Hilbert space $\mathcal{H}^{\rm ap}$ with basis indexed by aperiodic boundary paths in Λ. We show that the canonical expectation on $\mathcal{B}(\mathcal{H}^{\rm ap})$ 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 algebra, is co-universal for Toeplitz–Cuntz–Krieger families consisting of non-zero partial isometries.

LINEAR ISOMETRIES BETWEEN REAL JB*-TRIPLES AND C*-ALGEBRAS

Let T : A ! B be a (not necessarily surjective) linear isometry between two real JB ∗ -triples. Then for each a 2 A there exists a tripotent ua in the bidual, B '' , of B such that (a) {ua,T({f,g,h}),ua} = {ua,{T(f),T(g),T(h)},ua}, for all f,g,h in the real JB ∗ -subtriple, Aa, generated by a; (b) The mapping {ua,T(·),ua} : Aa ! B '' is a linear isometry. Furthermore, when B is a real C ∗ -algebra, the projection p = pa = u ∗ua satisfies that T(·)p : Aa ! B '' is an isometric triple homomorphism. When A and B are real C ∗ -algebras and A is abelian of real type, then there exists a partial isometry u 2 B '' such that the mapping T(·)u ∗ u : A ! B '' is an isometric triple homomorphism. These results generalise, to the real setting, some previous contributions due to C.-H. Chu and N.-C. Wong, and C.-H. Chu and M. Mackey in 2004 and 2005. We give an example of a non-surjective real linear isometry which cannot be complexified to a complex isometry, showing that the results in the real setting can not be derived by a mere complexification argument.

A-partial isometries and generalized inverses

In this work we study the relationship between A-partial isometries and generalized inverses.

On finite-dimensional representations of compact inverse semigroups

W.D. Munn proved that a finite-dimensional representation of an inverse semigroup is equivalent to a ∗-representation (by partial isometries) if and only if it is bounded. This paper gives a new analytic proof that every finite-dimensional representation of a compact inverse semigroup is equivalent to a ∗-representation. This will be the main result of this paper.

CUNTZ–KRIEGER ALGEBRAS AND A GENERALIZATION OF CATALAN NUMBERS

For a directed graph G, we generalize the Catalan numbers by using the canonical generating partial isometries of the Cuntz–Krieger algebra [Formula: see text] for the transition matrix AGof the directed edges of G. The generalized Catalan numbers [Formula: see text] enumerate the number of Dyck paths for the graph G. Its generating functions will be studied.

Groupoid normalisers of tensor products: infinite von Neumann algebras

The groupoid normalisers of a unital inclusion $B\subseteq M$ of von Neumann algebras consist of the set $\mathcal{GN}_M(B)$ of partial isometries $v\in M$ with $vBv^*\subseteq B$ and $v^*Bv\subseteq B$. Given two unital inclusions $B_i\subseteq M_i$ of von Neumann algebras, we examine groupoid normalisers for the tensor product inclusion $B_1\ \overline{\otimes}\ B_2\subseteq M_1\ \overline{\otimes}\ M_2$ establishing the formula $$ \mathcal{GN}_{M_1\,\overline{\otimes}\,M_2}(B_1\ \overline{\otimes}\ B_2)''=\mathcal{GN}_{M_1}(B_1)''\ \overline{\otimes}\ \mathcal{GN}_{M_2}(B_2)'' $$ when one inclusion has a discrete relative commutant $B_1'\cap M_1$ equal to the centre of $B_1$ (no assumption is made on the second inclusion). This result also holds when one inclusion is a generator masa in a free group factor. We also examine when a unitary $u\in M_1\ \overline{\otimes}\ M_2$ normalising a tensor product $B_1\ \overline{\otimes}\ B_2$ of irreducible subfactors factorises as $w(v_1\otimes v_2)$ (for some unitary $w\in B_1\ \overline{\otimes}\ B_2$ and normalisers $v_i\in\mathcal{N}_{M_i}(B_i)$). We obtain a positive result when one of the $M_i$ is finite or both of the $B_i$ are infinite. For the remaining case, we characterise the II$_1$ factors $B_1$ for which such factorisations always occur (for all $M_1, B_2$ and $M_2$) as those with a trivial fundamental group.

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 finitely generated then the underlying measure space is purely atomic, so that the semigroup is represented as “zero-unitary” matrices. The same is true if the semigroup contains a compact operator, in which case it is not even required that the semigroup be self-adjoint.

On matrix semigroups bounded above and below

An irreducible norm closed semigroup of complex matrices is simultaneously similar to a semigroup of partial isometries if and only if (a) the norms of all nonzero members of it are uniformly bounded above and below, and (b) its idempotents commute. This is a generalization of the well-known result on bounded groups.

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., $$ (\gamma(f))(x)=f(\gamma^{-1} x)\frac{d\mu\circ \gamma^{-1}}{d\mu}(x), \quad f\in L^1(X, \mu). $$

On the decomposition of families of quasinormal operators

The canonical injective decomposition of a jointly quasinormal family of operators is given. Relations between the decomposition of a quasinormal operator T and the decompo- sition of a partial isometry in the polar decomposition of T are described. The decomposition of pairs of commuting quasinormal partial isometries and its applications to pairs of com- muting quasinormal operators is shown. Examples are given.

The commuting graph of bounded linear operators on a Hilbert space

An operator T on the separable infinite-dimensional Hilbert space is constructed so that the commutant of every operator which is not a scalar multiple of the identity operator and commutes with T coincides with the commutant of T. On the other hand, it is shown that for several classes of operators it is possible to construct a finite sequence of operators, starting at a given operator from the class and ending in a rank-one projection such that each operator in the sequence commutes with its predecessor. The classes which we study are: finite-rank operators, normal operators, partial isometries, and C0 contractions. It is also shown that for any given set of yes/no conditions between points in some finite set, there always exist operators on a finite-dimensional Hilbert space such that their commutativity relations exactly satisfy those conditions.

GENERIC QUANTUM MARKOV SEMIGROUPS, CYCLE DECOMPOSITION AND DEVIATION FROM EQUILIBRIUM

A generic quantum Markov semigroup [Formula: see text] of a d-level quantum open system with a faithful normal invariant state ρ admits a dual semigroup [Formula: see text] with respect to the scalar product induced by ρ. We show that the difference of the generators [Formula: see text] can be written as the sum of a derivation 2i[H, ⋅] and a weighted difference of automorphisms [Formula: see text] where [Formula: see text] is a family of cycles on the d levels of the system, wc are positive weights and [Formula: see text] are unitaries. This formula allows us to represent the deviation from equilibrium (in a "small" time interval) as the superposition of cycles of the system where the difference between the forward and backward evolution is written as the difference of a reversible evolution and its time reversal. Moreover, it generalises cycle decomposition of Markov jump processes. We also find a similar formula with partial isometries instead of unitaries.