The study of Cowen-Douglas operators involves traditional operator-theoretic and complex-geometric tools. We study the similarity of Cowen-Douglas operators to backward shifts on analytic function spaces whose multiplier algebras equal H∞(D). Using the ratio of metrics and a model theorem, we obtain a sufficient condition for similarity. For weighted Dirichlet spaces, whose multiplier algebra is a proper subalgebra of H∞(D), we give a similarity condition via the jet bundle. The analysis relies on M\"uller’s model theory for Dirichlet shifts and the corona theorem of Kidane-Trent for weighted Dirichlet spaces.
For r between 0 and 1, let Ar={z∈C:r<|z|<1}. We study the class C1,r={T:T is invertible and ∥T∥,∥rT−1∥⩽1}. The class C1,r is closely related to the \textit{quantum annulus} given by QAr={T:T is invertible and ∥rT∥, ∥rT−1∥⩽1}. McCullough and Pascoe proved that T∈QAr dilates to an operator S satisfying (r−2+r2)I−S∗S−S−1S−∗=0. We extend this dilation result to doubly commuting tuples of operators in quantum annulus as well as in C1,r class. We also provide characterizations and decomposition results for such tuples.
We investigate the structure of the space of commuting squares around the Fourier spin model commuting square, or equivalently the structure of the space of complex n & times; n Hadamard matrices around the Fourier matrix Fn, by finding restrictions for the possible directions of tangency at Fn. As an application, we show that for n = 30 the dimension of any differentiable family of complex Hadamard matrices containing Fn is strictly less than the dimension of the enveloping tangent space at Fn (called the defect of Fn).
Let n >= 2, and let V = (V-1, . . ., V-n) be an n-tuple of isometries acting on a Hilbert space H. We say that V is an n-tuple of isometries with equal range if V-i(mi) V-j(mj) H = V-j(jm) V-i(mi) H and V-i*(mi) V-j(mj) H = V-j(mj) V-i*(mi) H for m(i), m(j) is an element of Z(+), where 1 <= i < j <= n. We prove that each n-tuple of isometries with equal range admits a unique Wold decomposition. We obtain analytic models of the above class, and as a consequence, we show that the wandering data are complete unitary invariants for n-tuples of isometries with equal range. Our results unify all prior findings on the decomposition for tuples of isometries in the existing literature.
Gonz & aacute;lez-P & eacute;rez, Parcet and Xia introduced a framework to study Lp-boundedness of certain families of idempotent multipliers on von Neumann algebras. It includes symbols m: PSL2(C) -> R arising from lifting the indicator function of a partition {L+, L+, L-} of the hyperbolic space H3 to its isometry group PSL2(C). The boundedness of Tm on Lp(LPSL2(C)) was disproved by Parcet, de la Salle and Tablate. Nevertheless, we will show that this Fourier multiplier is bounded when restricted to the arithmetic lattices PSL2(Z[root-n] ), solving a question left open by the first named authors.
. The linear action of SL2(R) on Rncorresponding to its unique irreducible representation induces an action SL2(Z) curved right arrow Tnfor every n 2 that factors through PSL2(Z) for n odd. Thus, setting Gn = SL2(Z) (respectively Gn = PSL2(Z)) for n even (respectively n odd), Gn curved right arrow Tnis free and ergodic, every ergodic sub-equivalence relation of the orbital equivalence relation is either amenable or rigid, and the fundamental group of the II1 factor Nn := L infinity(Tn) & rtimes; Gn is trivial. For n even, L infinity(Tn) & rtimes; H is a maximal Haagerup subalgebra of Nn for every suitable maximal amenable subgroup H of SL2(Z).
. Given a non-singular action F n (X, y), we define the *-algebra Cfp[F n X] of operators of finite dynamical propagation associated with this action. This assignment is completely canonical and depends only on the class of measures of y. We prove that the algebraic crossed product L infinity X & rtimes;alg F surjects onto Cfp[F n X] and that this surjection is a *-isomorphism whenever the action is essentially free. As a consequence, we canonically characterize ergodicity and strong ergodicity of the action in terms of structural properties of Cfp[F n X] and its closure. We also use these techniques to describe the Roe algebra of a warped space in terms of the Roe algebra of the (non-warped) space and the group action. We apply this result to Roe algebras of warped cones.
We classify all G-invariant von Neumann subalgebras in L(G) for G = Z(2) & rtimes; SL2(Z). This is the first result on classifying G-invariant von Neumann subalgebras in L(G) for icc groups G without the invariant von Neumann subalgebras rigidity property (ISR property for short) as introduced in Amrutam-Jiang's work. As a corollary, we show that L(Z(2) & rtimes; {+/- I-2}) is the unique maximal Haagerup G-invariant von Neumann subalgebra in L(G), where I-2 denotes the identity matrix in SL2(Z).
In this paper, we investigate the r-summing Carleson embed-dings on weighted Fock spaces F-alpha,w(p). By using duality arguments, translating techniques and block diagonal operator skills, we completely characterize the r-summability of the natural embeddings I-d : F-alpha,w(p) -> L-alpha(p)(mu) for any r >= 1 and p > 1, where w is a weight on the complex plane C that satisfies an A(p)-type condition. As applications, we establish some results on the r-summability of differentiation and integration operators, Volterra-type operators and composition operators. Especially, we completely characterize the boundedness of Volterra-type operators and composition operators on vector-valued Fock spaces for all 1 < p < infinity, which were left open before for the case 1 < p < 2.
Let 𝒢 be a Hilbert space and 𝔅(𝒢) the algebra of bounded operators, ℋ=L_2([0,∞);𝒢). An operator-valued function Q∈ L_∞, loc([0,∞);𝔅(𝒢)) determines a multiplication operator in ℋ by (Qy)(x)=Q(x)y(x), x⩾0. We say that an operator L_0 in a Hilbert space is a Schrödinger type operator, if it is unitarily equivalent to -d^2/dx^2+Q(x) on a relevant domain. The paper provides a characterization of a class of such operators. The characterization is given in terms of properties of an evolutionary dynamical system associated with L_0. It provides a way to construct a functional Schrödinger model of L_0.
In the paper we investigate the Banach space representations of Manin's quantum q-plane for |q|=/1. The Arens-Michael envelope of the quantum plane is extended up to a Fr & eacute;chet algebra presheaf over its spectrum. The obtained ringed space represents the geometry of the quantum plane as a union of two irreducible components being copies of the complex plane equipped with the q-topology and the disk topology, respectively. It turns out that the Fr & eacute;chet algebra presheaf is commutative modulo its Jacobson radical, which is decomposed into a topological direct sum. The related noncommutative functional calculus problem and the spectral mapping property are solved in terms of the noncommutative Harte spectrum.
In this paper we formulate the almost invariant subspaces theorems of backward shift operators in terms of the ranges or kernels of product of Toeplitz and Hankel operators. This approach simplifies and gives more explicit forms of these almost invariant subspaces which are derived from related nearly backward shift invariant subspaces with finite defect. Furthermore, this approach also leads to the surprising result that the almost invariant subspaces of backward shift operators are the same as the almost invariant subspaces of forward shift operators which were treated only briefly in literature.
In this paper, we study a more general version of multidimensional Bohr radii for the holomorphic functions defined on unit ball of & ell;(n)(q)(1 <= q <= infinity) spaces with values in arbitrary complex Banach spaces. More precisely, we study the multidimensional Bohr radii for bounded linear operators between complex Banach spaces, primarily motivated by the work of A. Defant, M. Maestre, and U. Schwarting (Adv. Math. 231(2012), pp. 2837-2857). We obtain the exact asymptotic estimates of multidimensional Bohr radius for both finite and infinite dimensional Banach spaces. As an application, we find the lower bound of arithmetic Bohr radius.
Let $R$ be a rational function with degree $\geq 2$ and $X$ be its Julia set, its Fatou set, or the Riemann sphere. Suppose that $X$ is not empty. We can regard $R$ as a continuous map from $X$ onto itself. Kajiwara and Watatani showed that in the case that $X$ is the Julia set, $C_0(X)$ is a maximal abelian subalgebra of $\mathcal{O}_R(X)$, where $\mathcal{O}_R(X)$ denotes the C*-algebra associated with the dynamical system $(X,R)$ introduced by them. In this paper, we develop their result and give the equivalent condition for $C_0(X)$ to be a Cartan subalgebra of $\mathcal{O}_R(X)$.
We study surjective maps between the sets of all self-adjoint elements of unital $C^*$-algebras which satisfy the multiplicatively spectrum-preserving property. We show that such maps are characterized by Jordan isomorphisms and central symmetries. This is an answer to a problem posed by Moln\'ar.
We characterize when a C*-cover admits a C*-dynamical extension of dynamics on an operator algebra in terms of the boundary ideal structure for the operator algebra in its maximal representation and show that the C*-covers that admit such an extension form a complete lattice. We study dynamical systems arising from groups acting via inner automorphisms in a C*-cover and produce an example of a C*-cover that admits no extension of dynamics on a finite-dimensional non-self-adjoint operator algebra. We construct a partial action on a class of C*-covers that recovers the crossed product of an operator algebra as a subalgebra of the partial crossed product, even when the C*-cover admits no dynamical extension.
We study the local-triviality dimensions of actions on C^*-algebras, which are invariants developed for noncommutative Borsuk-Ulam theory. While finiteness of the local-triviality dimensions is known to guarantee freeness of an action, we show that free actions need not have finite weak local-triviality dimension. Moreover, the local-triviality dimensions of a continuous field may be greater than those of its individual fibers, and the dimensions may fail to vary continuously across the fibers. However, in certain circumstances upper semicontinuity of the weak local-triviality dimension is guaranteed. We examine these results and counterexamples with a focus on noncommutative tori and noncommutative spheres, both in terms of computation and theory.
We show that the space of trace-class operators on a Hilbert module over a commutative C*-algebra, as defined and studied in earlier work of Stern and van Suijlekom (Journal of Functional Analysis, 2021), is completely isometrically isomorphic to a Haagerup tensor product of the module with its operator-theoretic adjoint. This generalises a well-known property of Hilbert spaces. In the course of proving this, we also obtain a new proof of a result of Stern-van Suijlekom concerning the equivalence between two definitions of trace-class operators on Hilbert modules.
We give a characterisation of factoriality of the groupoid von Neumann algebra $L(\mathcal{G})$ associated to a discrete measured groupoid $(\mathcal{G},\mu)$. We introduce the notion of groupoids with `infinite conjugacy classes' and show that this property together with ergodicity of the groupoid is equivalent to factoriality of $L(\mathcal{G})$.
