Multiplication operators on non-commutative spaces

This chapter is about metric spaces, an abstract generalization of the real line that allows discussion of open and closed sets, limits, convergence, continuity, and similar properties. The usual distance function for the real line becomes an example of a metric. The other notions are defined in terms of the metric. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions. Sections 2 through 4 develop properties of open and closed sets, continuity, and convergence of sequences that are simple generalizations of known facts about $\mathbb{R}$. Section 5 shows how a subset of a metric space can be made into a metric space so that the restriction of a continuous function from the whole space to the subset remains continuous. It also shows that three natural metrics for the product of two metric spaces lead to the same open sets, continuous functions, and convergent sequences. Section 6 shows that any metric space is “Hausdorff,” “regular,” and “normal,” and it goes on to exhibit three different countability hypotheses about a metric space as equivalent. A metric space with these properties is called “separable.” Section 7 concerns compactness and completeness. A metric space is defined to be “compact” if every open cover has a finite subcover. This property is equivalent to the condition that every sequence has a convergent subsequence. The Heine–Borel Theorem says that the compact sets of $\mathbb{R}^n$ are exactly the closed bounded sets. A number of the results early in Chapter I that were proved by the Bolzano–Weierstrass Theorem in the context of the real line are seen to extend to any compact metric space. A metric space is “complete” if every Cauchy sequence is convergent. A metric space is compact if and only if it is complete and “totally bounded.” Section 8 concerns connectedness, which is an abstraction of the property of an interval of the line that accounts for the Intermediate Value Theorem. Section 9 proves a fundamental result known as the Baire Category Theorem. A sample consequence of the theorem is that the pointwise limit of a sequence of continuous complex-valued functions on a complete metric space must have points where it is continuous. Section 10 studies the spaces of real-valued and complex-valued continuous functions on a compact metric space. A generalization of Ascoli's Theorem from the setting of Chapter I provides a characterization of compact sets in either of these spaces of continuous functions. A generalization of the Weierstrass Approximation Theorem, known as the Stone–Weierstrass Theorem, gives sufficient conditions for a subalgebra of either of these spaces of continuous functions to be dense. One consequence is that these spaces of continuous functions are separable. Section 11 constructs the “completion” of a metric space out of Cauchy sequences in the given space. The result is a complete metric space and a distance-preserving map of the given metric space into the completion such that the image is dense.

We consider the spaces of ultradifferentiable functions $$\mathcal {S}_\omega $$ as introduced by Bjorck (and its dual $$\mathcal {S}'_\omega $$ ) and we use time-frequency analysis to define a suitable wave front set in this setting and obtain several applications: global regularity properties of pseudodifferential operators of infinite order and the micro-pseudolocal behaviour of partial differential operators with polynomial coefficients and of localization operators with symbols of exponential growth. Moreover, we prove that the new wave front set, defined in terms of the Gabor transform, can be described using only Gabor frames. Finally, some examples show the convenience of the use of weight functions to describe more precisely the global regularity of (ultra)distributions.

Given a Banach lattice L, the space of lattice Lipschitz operators on L has been introduced as a natural Lipschitz generalization of the linear notions of diagonal operator and multiplication operator on Banach function lattices. It is a particular space of superposition operators on Banach lattices. Motivated by certain procedures in Reinforcement Learning based on McShane–Whitney extensions of Lipschitz maps, this class has proven to be useful also in the classical context of Mathematical Analysis. In this paper, we discuss the properties of such operators when defined on spaces of continuous functions, focusing attention on the functional bounds for the pointwise Lipschitz inequalities defining the lattice Lipschitz operators, the representation theorems for these operators as vector-valued functions, and the corresponding dual spaces. Finally, and with possible applications in Artificial Intelligence in mind, we provide a McShane–Whitney extension theorem for these operators.

We study Banach envelopes for commutative symmetric sequence or function spaces, and noncommutative symmetric spaces of measurable operators. We characterize the class (HC) of quasi-normed symmetric sequence or function spaces E for which their Banach envelopes \(\widehat{E}\) are also symmetric spaces. The class of symmetric spaces satisfying (HC) contains but is not limited to order continuous spaces. Let \(\mathcal {M}\) be a non-atomic, semifinite von Neumann algebra with a faithful, normal, \(\sigma \)-finite trace \(\tau \) and E be as symmetric function space on \([0,\tau (\mathbf 1 ))\) or symmetric sequence space. We compute Banach envelope norms on \(E(\mathcal {M},\tau )\) and \(C_E\) for any quasi-normed symmetric space E. Then we show under assumption that \(E\in (HC)\) that the Banach envelope \(\widehat{E(\mathcal {M},\tau )}\) of \(E(\mathcal {M},\tau )\) is equal to \(\widehat{E}\left( \mathcal {M},\tau \right) \) isometrically. We also prove the analogous result for unitary matrix spaces \(C_E\).

Some Factorization Properties of Composition Operators

We have obtained the supersymmetric extension of a spectral triple that specifies a noncommutative geometry. We assume that the functional space |$\mathcal {H}$| consists of wave functions of matter fields and their superpartners included in the minimum supersymmetric standard model (MSSM). We introduce the internal fluctuations of the Dirac operator on the finite space as well as on the manifold by elements of the algebra |$\mathcal {A}$| in the triple. So, we obtain not only the vector supermultiplets that mediate |${\rm SU}(3)\otimes {\rm SU}(2)\otimes {\rm U}(1)_Y$| gauge degrees of freedom but also Higgs supermultiplets that appear in the MSSM from the same standpoint. According to the supersymmetric version of the spectral action principle, we calculate the square of the fluctuated total Dirac operator and verify that the Seeley–DeWitt coefficients give the correct action of the vector and Higgs supermultiplets. We also verify that the relation between the coupling constants of |${\rm SU}(3)$|⁠, |${\rm SU}(2)$|⁠, and |${\rm U}(1)_Y$| is same as that of SU(5) unification theory.

For a large class of Hilbert spaces of analytic functions in the unit disc lying between the Hardy and the Dirichlet space we prove that each element of the space is the quotient of two bounded functions in the same space. It follows that the multiplication operator on these spaces is cellular indecomposable and that each invariant subspace contains nontrivial bounded functions.

LetVbe an arbitrary system of weights on an open connected subsetGofℂN(N≥1)and letB(E)be the Banach algebra of all bounded linear operators on a Banach spaceE. LetHVb(G,E)andHV0(G,E)be the weighted locally convex spaces of vector-valued analytic functions. In this survey, we present a development of the theory of multiplication operators and composition operators from classical spaces of analytic functionsH(G)to the weighted spaces of analytic functionsHVb(G,E)andHV0(G,E).

To any periodic and full C *-dynamical system , an invertible operator s acting on the Banach space of trace functionals of the fixed point algebra is canonically associated. KMS states correspond to positive eigenvectors of s. A Perron–Frobenius type theorem asserts the existence of KMS states at inverse temperatures equals the logarithms of the inner and outer spectral radii of s (extremal KMS states). Examples arising from subshifts in symbolic dynamics, self-similar sets in fractal geometry and noncommutative metric spaces are discussed. Certain subshifts are naturally associated to the system, and criteria for the equality of their topological entropy and inverse temperatures of extremal KMS states are given. Unital completely positive maps implemented by partitions of unity {x j } of grade 1 are considered, resembling the “canonical endomorphism” of the Cuntz algebras. The relationship between the Voiculescu topological entropy of and the topological entropy of the associated subshift is studied. Examples where the equality holds are discussed among Matsumoto algebras associated to non finite type subshifts. In the general case is bounded by the sum of the entropy of the subshift and a suitable entropic quantity of the homogeneous subalgebra. Both summands are necessary. The measure-theoretic entropy of , in the sense of Connes–Narnhofer–Thirring, is compared to the classical measure-theoretic entropy of the subshift. A noncommutative analogue of the classical variational principle for the entropy is obtained for the “canonical endomorphism” of certain Matsumoto algebras. More generally, a necessary condition is discussed. In the case of Cuntz–Krieger algebras an explicit construction of the state with maximal entropy from the unique KMS state is done.

This chapter concludes the analysis of the tracial setting by showing that real interpolation extends to the context of tracial Lp and Orlicz spaces in a manner which closely mimics the classical theory. After a brief consideration of a noncommutative Marcinkiewicz interpolation theorem, we then go on to enlarge our pool of noncommutative spaces by using the K-method of interpolation to introduce a very general class of noncommutative rearrangement invariant Banach function spaces – the so-called monotone Riesz-Fischer spaces. Our approach to this by now well-known theory is slightly non-standard in that we introduce the theory by exploiting the known theory of modular interpolation spaces. As a result we are able to get to the cut of the chase a lot quicker.

We show that all Hankel operators $H$ realized as integral operators with kernels $h(t+s)$ in $L^2 ({\Bbb R}_{+}) $ can be quasi-diagonalized as $H= {\sf L}^* \Sigma {\sf L} $. Here ${\sf L}$ is the Laplace transform, $\Sigma$ is the operator of multiplication by a function (distribution) $\sigma(\lambda)$, $\lambda\in {\Bbb R}$. We find a scale of spaces of test functions where ${\sf L} $ acts as an isomorphism. Then ${\sf L}^*$ is an isomorphism of the corresponding spaces of distributions. We show that $h= {\sf L}^* \sigma$ which yields a one-to-one correspondence between kernels $h(t)$ and sigma-functions $\sigma(\lambda)$ of Hankel operators. The sigma-function of a self-adjoint Hankel operator $H$ contains substantial information about its spectral properties. Thus we show that the operators $H$ and $\Sigma$ have the same numbers of positive and negatives eigenvalues. In particular, we find necessary and sufficient conditions for sign-definiteness of Hankel operators. These results are illustrated at examples of quasi-Carleman operators generalizing the classical Carleman operator with kernel $h(t)=t^{-1}$ in various directions. The concept of the sigma-function directly leads to a criterion (equivalent of course to the classical Nehari theorem) for boundedness of Hankel operators. Our construction also shows that every Hankel operator is unitarily equivalent by the Mellin transform to a pseudo-differential operator with amplitude which is a product of functions of one variable only (of $x\in{\Bbb R}$ and of its dual variable).

The purpose of this note is to present certain aspects of the theory of spectral operators in Grothendieck spaces with the Dunford-Pettis property, briefly, GDP-spaces, thereby elaborating on the recent note [10].For example, the sum and product of commuting spectral operators in such spaces are again spectral operators (cf. Proposition 2.1) and a continuous linear operator is spectral if and only if it has finite spectrum (cf. Proposition 2.2). Accordingly, if a spectral operator is of finite type, then its spectrum consists entirely of eigenvalues. Furthermore, it turns out that there are no unbounded spectral operators in such spaces (cf. Proposition 2.4). As a simple application of these results we are able to determine which multiplication operators in certain function spaces are spectral operators.

Consider a couple of Banach function spaces X and Y over the same measure space and the space X Y of multiplication operators from X into Y. In this paper we develop the setting for characterizing certain summability properties satisfied by the elements of X Y . At this end, using the “generalized Kothe duality” for Banach function spaces, we introduce a new class of norms for spaces consisting of infinite sums of products of the type xy with \({x \in X}\) and \({y \in Y}\) .

We consider free interpolation problems, the study of which was initiated by A.F. Leontiev. We obtain new criterions for the interpolation property of sets in the space of analytic in the upper half - plane functions of finite order. We provide examples of interpolation sets in the space of analytic in the upper half - plane functions of finite order. These examples are similar to interpolation sets in the space of analytic and bounded in the upper half - plane functions. In particular, we provide examples of sets satisfying the Newman condition and uniform Frostman condition.

Dense Sets in Spaces of Integrable Functions