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

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

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

AGGRESSION AND "AUTOAGGRESSION" IN SCHIZOPHRENIA

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CLINICAL PECULIARITIES OF THE ACUTE SYNDROME OF PSYCHIC AUTOMATISM IN SCHIZOPHRENIA

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