On unbounded, non-trivial Hochschild cohomology in finite von Neumann algebras and higher order Berezin's quantization

We introduce a class of densely defined, unbounded, 2-Hochschild cocycles ([PT]) on finite von Neumann algebras $M$. Our cocycles admit a coboundary, determined by an unbounded operator on the standard Hilbert space associated to the von Neumann algebra $M$. For the cocycles associated to the $\Gamma$-equivariant deformation ([Ra]) of the upper halfplane $(\Gamma=PSL_2(\mathbb Z))$, the "imaginary" part of the coboundary operator is a cohomological obstruction - in the sense that it can not be removed by a "large class" of closable derivations, with non-trivial real part, that have a joint core domain, with the given coboundary.Comment: This is the print versio

Endomorphisms of spaces of virtual vectors fixed by a discrete group

Consider a unitary representation $\pi$ of a discrete group $G$, which, when restricted to an almost normal subgroup $\Gamma\subseteq G$, is of type II. We analyze the associated unitary representation $\overline{\pi}^{\rm{p}}$ of $G$ on the Hilbert space of "virtual" $\Gamma_0$-invariant vectors, where $\Gamma_0$ runs over a suitable class of finite index subgroups of $\Gamma$. The unitary representation $\overline{\pi}^{\rm{p}}$ of $G$ is uniquely determined by the requirement that the Hecke operators, for all $\Gamma_0$, are the "block matrix coefficients" of $\overline{\pi}^{\rm{p}}$. If $\pi|_\Gamma$ is an integer multiple of the regular representation, there exists a subspace $L$ of the Hilbert space of the representation $\pi$, acting as a fundamental domain for $\Gamma$. In this case, the space of $\Gamma$-invariant vectors is identified with $L$. When $\pi|_\Gamma$ is not an integer multiple of the regular representation, (e.g. if $G=PGL(2,\mathbb Z[\frac{1}{p}])$, $\Gamma$ is the modular group, $\pi$ belongs to the discrete series of representations of $PSL(2,\mathbb R)$, and the $\Gamma$-invariant vectors are the cusp forms) we assume that $\pi$ is the restriction to a subspace $H_0$ of a larger unitary representation having a subspace $L$ as above. The operator angle between the projection $P_L$ onto $L$ (typically the characteristic function of the fundamental domain) and the projection $P_0$ onto the subspace $H_0$ (typically a Bergman projection onto a space of analytic functions), is the analogue of the space of $\Gamma$- invariant vectors. We prove that the character of the unitary representation $\overline{\pi}^{\rm{p}}$ is uniquely determined by the character of the representation $\pi$.Comment: The exposition has been improved and a normalization constant has been addressed. The result allows a direct computation for the characters of the unitary representation on spaces of invariant vectors (for example automorphic forms) in terms of the characters of the representation to which the fixed vectors are associated (e.g discrete series of PSL(2, R) for automorphic forms

Token-Reflexivity and Repetition

The classical rule of Repetition says that if you take any sentence as a premise, and repeat it as a conclusion, you have a valid argument. It's a very basic rule of logic, and many other rules depend on the guarantee that repeating a sentence, or really, any expression, guarantees sameness of referent, or semantic value. However, Repetition fails for token-reflexive expressions. In this paper, I offer three ways that one might replace Repetition, and still keep an interesting notion of validity. Each is a fine way to go for certain purposes, but I argue that one in particular is to be preferred by the semanticist who thinks that there are token-reflexive expressions in natural languages

Synonymy between Token-Reflexive Expressions

Synonymy, at its most basic, is sameness of meaning. A token-reflexive expression is an expression whose meaning assigns a referent to its tokens by relating each particular token of that particular expression to its referent. In doing so, the formulation of its meaning mentions the particular expression whose meaning it is. This seems to entail that no two token-reflexive expressions are synonymous, which would constitute a strong objection against token-reflexive semantics. In this paper, I propose and defend a notion of synonymy for token-reflexive expressions that allows such expressions to be synonymous, while being a fairly conservative extension of the customary notion of synonymy