Topological radicals, V. From algebra to spectral theory

We introduce and study procedures and constructions in the theory of the joint spectral radius that are related to the spectral theory. In particular we devlop the theory of the scattered radical. Among applications we find some sufficient conditions of continuity of the spectrum and spectral radii of various types, and prove that in GCR C*-algebras the joint spectral radius is continuous on precompact subsets and coincides with the Berger-Wang radius

Method and apparatus for producing an image from a transparent object

The contrast produced from a photographic transparency is controlled by placing the transparency between a pair of partially reflecting mirrors forming walls of an optical cavity. Mirrors trap a collimated laser beam illuminating the transparency so that at least a portion of the beam energy is passed through the transparency plural times. The distance that the light beam travels between the mirrors is controlled as a function of the wavelength of the beam energy to control the phase of light interference in the beam passing through the transparency, thereby controlling the intensity of the beam derived from the mirror downstream of the transparency

Topological radicals, II. Applications to spectral theory of multiplication operators

We develop the spectral radius technique and the theory of tensor radicals. As applications we obtain numerous results on mutiplication operators in Banach algebras and Operator bimodules

A unified framework for generalized multicategories

Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the "lax algebras" or "Kleisli monoids" relative to a "monad" on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous examples, while at the same time simplifying and clarifying much of the theory.Comment: 76 pages; final version, to appear in TA

Reduced spectral synthesis and compact operator synthesis

We introduce and study the notion of reduced spectral synthesis, which unifies the concepts of spectral synthesis and uniqueness in locally compact groups. We exhibit a number of examples and prove that every non-discrete locally compact group with an open abelian subgroup has a subset that fails reduced spectral synthesis. We introduce compact operator synthesis as an operator algebraic counterpart of this notion and link it with other exceptional sets in operator algebra theory, studied previously. We show that a closed subset $E$ of a second countable locally compact group $G$ satisfies reduced local spectral synthesis if and only if the subset $E^* = \{(s,t) : ts^{-1}\in E\}$ of $G\times G$ satisfies compact operator synthesis. We apply our results to questions about the equivalence of linear operator equations with normal commuting coefficients on Schatten $p$-classes.Comment: 43 page

Sets of multiplicity and closable multipliers on group algebras

We undertake a detailed study of the sets of multiplicity in a second countable locally compact group $G$ and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space $\mathcal{B}(L^2(G))$ of bounded linear operators on $L^2(G)$ into the von Neumann algebra $VN(G)$ of $G$ and use it to show that a closed subset $E\subseteq G$ is a set of multiplicity if and only if the set $E^* = \{(s,t)\in G\times G : ts^{-1}\in E\}$ is a set of operator multiplicity. Analogous results are established for $M_1$-sets and $M_0$-sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if $G$ satisfies a mild approximation condition, pointwise multiplication by a given measurable function $\psi : G\to \mathbb{C}$ defines a closable multiplier on the reduced C*-algebra $C_r^*(G)$ of $G$ if and only if Schur multiplication by the function $N(\psi) : G\times G\to \mathbb{C}$, given by $N(\psi)(s,t) = \psi(ts^{-1})$, is a closable operator when viewed as a densely defined linear map on the space of compact operators on $L^2(G)$. Similar results are obtained for multipliers on $VN(G)$.Comment: 51 page

Closable Multipliers

Let (X,m) and (Y,n) be standard measure spaces. A function f in $L^\infty(X\times Y,m\times n)$ is called a (measurable) Schur multiplier if the map $S_f$, defined on the space of Hilbert-Schmidt operators from $L_2(X,m)$ to $L_2(Y,n)$ by multiplying their integral kernels by f, is bounded in the operator norm. The paper studies measurable functions f for which $S_f$ is closable in the norm topology or in the weak* topology. We obtain a characterisation of w*-closable multipliers and relate the question about norm closability to the theory of operator synthesis. We also study multipliers of two special types: if f is of Toeplitz type, that is, if f(x,y)=h(x-y), x,y in G, where G is a locally compact abelian group, then the closability of f is related to the local inclusion of h in the Fourier algebra A(G) of G. If f is a divided difference, that is, a function of the form (h(x)-h(y))/(x-y), then its closability is related to the "operator smoothness" of the function h. A number of examples of non-closable, norm closable and w*-closable multipliers are presented.Comment: 35 page