Commutative subalgebras of the algebra of smooth operators

Approximate Double Commutants and Distance Formulas

We extend work of the first author concering relative double commutants and approximate double commutants of unital subalgebras of unital C*-algebras, including metric versions involving distance estimates. We prove metric results for AH subalgebras of von Neumann algebras or AF subalgebras of primitive C*-algebras. We prove other general results, including some for nonselfadjoint commutative subalgebras, using C*-algebraic versions of the Stone-Weierstrass and Bishop-Stone-Weierstrass theorems.

Inverse Closed Ultradifferential Subalgebras

In previous work we have shown that classical approximation theory provides methods for the systematic construction of inverse-closed smooth subalgebras. Now we extend this work to treat inverse-closed subalgebras of ultradifferentiable elements. In particular, Carleman classes and Dales-Davie algebras are treated. As an application the result of Demko, Smith and Moss and Jaffard on the inverse of a matrix with exponential decay is obtained within the framework of a general theory of smoothness.

Heat-smoothing for holomorphic subalgebras of free group von Neumann algebras

The heat semigroup on discrete hypercubes is well-known to be contractive over $L_p$-spaces for $1<p<\infty$. A question of Mendel and Naor \cite{MN14} concerns a stronger contraction property in the tail spaces, which is known as the heat-smoothing conjecture. Eskenazis and Ivanisvili \cite{EI20} considered a Gaussian analog of this conjecture and resolved some special cases. In particular, they proved that heat-smoothing type conjecture holds for holomorphic functions in the Gaussian spaces with sharp constants. In this paper, we prove analogous sharp inequalities for holomorphic subalgebras…

Invariant function algebras on compact commutative homogeneous spaces

Let $M$ be a commutative homogeneous space of a compact Lie group $G$ and $A$ be a closed $G$-invariant subalgebra of the Banach algebra $C(M)$. A function algebra is called antisymmetric if it does not contain nonconstant real functions. By the main result of this paper, $A$ is antisymmetric if and only if the invariant probability measure on $M$ is multiplicative on $A$. This implies, for example, the following theorem: if $G^{\mathbb C}$ acts transitively on a Stein manifold $\cal M$, $v\in{\cal M}$, and the compact orbit $M=Gv$ is a commutative homogeneous space, then $M$ is a real form of…

Lattices of logmodular algebras

A subalgebra $\mathcal{A}$ of a $C^*$-algebra $\mathcal{M}$ is logmodular (resp. has factorization) if the set $\{a^*a; a\text{ is invertible with }a,a^{-1}\in\mathcal{A}\}$ is dense in (resp. equal to) the set of all positive and invertible elements of $\mathcal{M}$. There are large classes of well studied algebras, both in commutative and non-commutative settings, which are known to be logmodular. In this paper, we show that the lattice of projections in a von Neumann algebra $\mathcal{M}$ whose ranges are invariant under a logmodular algebra in $\mathcal{M}$, is a commutative subspace latti…

Smooth dense subalgebras and Fourier multipliers on compact quantum groups

We define and study dense Frechet subalgebras of compact quantum groups consisting of elements rapidly decreasing with respect to an unbounded sequence of real numbers. Further, this sequence can be viewed as the eigenvalues of a Dirac-like operator and we characterize the boundedness of its commutators in terms of the eigenvalues. Grotendieck's theory of topological tensor products immediately yields a Schwartz kernel theorem for linear operators on compact quantum groups and allows us to introduce a natural class of pseudo-differential operators on compact quantum groups. As a by-product, we…

交换 $G$-不变 Toeplitz C$^\ast$ 代数在 Fock 空间上及其通过量子调和分析的 Gelfand 理论

Commutative $G$-invariant Toeplitz C$^\ast$ algebras on the Fock space and their Gelfand theory through Quantum Harmonic Analysis

本文基于量子调和分析中的谱综合概念，研究了具有特定不变符号的 Toeplitz 代数子代数及其交换子，发现了一类新的 Fock 空间上的交换 Toeplitz C$^\ast$ 代数，并探讨了其 Gelfand 理论。

On Galois Correspondence and Non-Commutative Martingales

The subject of this thesis is Galois correspondence for von Neumann algebras and its interplay with non-commutative probability theory. After a brief introduction to representation theory for compact groups, in particular to Peter-Weyl theorem, and to operator algebras, including von Neumann algebras, automorphism groups, crossed products and decomposition theory, we formulate first steps of a non-commutative version of probability theory and introduce non-abelian analogues of stochastic processes and martingales. The central objects are a von Neumann algebra $\Ma$ and a compact group $\Gr$ ac…

Dense quasi-free subalgebras of the Toeplitz algebra

We introduce a family of dense subalgebras of the Toeplitz algebra and give conditions under which our algebras are quasi-free. As a corollary, we show that the smooth Toeplitz algebra introduced by Cuntz is quasi-free.