Todd Kemp
We introduce holomorphic algebras $H_q$ in the context of the q-Gaussian algebra $Γ_q$ of Bozejko, Kümmerer, and Speicher, and give a q-Segal-Bargmann transform for them. We then prove a strong hypercontractivity theorem, generalizing Janson's strong (holomorphic) hypercontractivity, from $L^2(H_q) \to L^r(H_q)$ for r an even integer.
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We prove that the twisted property RD fails to hold for all non Kac type, non amenable orthogonal free quantum groups. In the Kac case we revisit property RD, proving an analogue of the $L_p-L_2$ non-commutative Khintchine inequality for free groups. As an application, we give new and improved hypercontractivity and ultracontractivity estimates for the generalized heat semigroups on free orthogonal quantum groups, both in the Kac and non Kac cases.
We give a non-commutative Positivstellensatz for CP^n: The (commutative) *-algebra of polynomials on the real algebraic set CP^n with the pointwise product can be realized by phase space reduction as the U(1)-invariant polynomials on C^{1+n}, restricted to the real (2n+1)-sphere inside C^{1+n}, and Schmüdgen's Positivstellensatz gives an algebraic description of the real-valued U(1)-invariant polynomials on CP^n that are strictly pointwise positive on the sphere. In analogy to this commutative case, we consider a non-commutative *-algebra of polynomials on C^{1+n}, the Weyl algebra, and give a…
We generalize the Serre-Swan theorem to non-commutative C$^{*}$-algebras. For a Hilbert C$^{*}$-module $X$ over a C$^{*}$-algebra ${\cal A}$, we introduce a hermitian vector bundle $\exx$ associated to $X$. We show that there is a linear subspace $Γ_{X}$ of the space of all holomorphic sections of ${\cal E}_{X}$ and a flat connection $D$ on ${\cal E}_{X}$ with the following properties: (i) $Γ_{X}$ is a Hilbert ${\cal A}$-module with the action of ${\cal A}$ defined by $D$, (ii) the C$^{*}$-inner product of $Γ_{X}$ is induced by the hermitian metric of ${\cal E}_{X}$, (iii) ${\cal E}_{X}$ is is…
For any given bounded symmetric domain, we prove the existence of commutative $C^*$-algebras generated by Toeplitz operators acting on any weighted Bergman space. The symbols of the Toeplitz operators that generate such algebras are defined by essentially bounded functions invariant under suitable subgroups of the group of biholomorphisms of the domain. These subgroups include the maximal compact groups of biholomorphisms. We prove the commutativity of the Toeplitz operators by considering the Bergman spaces as the underlying space of the holomorphic discrete series and then applying known mul…
Representations of $C^*$-algebras are realized on section spaces of holomorphic homogeneous vector bundles. The corresponding section spaces are investigated by means of a new notion of reproducing kernel, suitable for dealing with involutive diffeomorphisms defined on the base spaces of the bundles. Applications of this technique to dilation theory of completely positive maps are explored and the critical role of complexified homogeneous spaces in connection with the Stinespring dilations is pointed out. The general results are further illustrated by a discussion of several specific topics, i…
We introduce and study holomorphically finitely generated (HFG) Fréchet algebras, which are analytic counterparts of affine (i.e., finitely generated) $\mathbb C$-algebras. Using a theorem of O. Forster, we prove that the category of commutative HFG algebras is anti-equivalent to the category of Stein spaces of finite embedding dimension. We also show that the class of HFG algebras is stable under some standard constructions. This enables us to give a series of concrete examples of HFG algebras, including Arens-Michael envelopes of affine algebras (such as the algebras of holomorphic functions…
The Smirnov class for the classical Hardy space is the set of ratios of bounded analytic functions on the open complex unit disk with outer denominators. This definition extends naturally to the commutative and non-commutative multi-variable settings of the Drury-Arveson space and the full Fock space over $\mathbb C ^d$. Identifying the Fock space with the free multi-variable Hardy space of non-commutative or free holomorphic functions on the non-commutative open unit ball, we prove that any closed, densely-defined operator affiliated to the right free multiplier algebra of the full Fock space…
Divided Differences and Multivariate Holomorphic Calculus
本文回顾交换巴拿赫代数元组的多元全纯函数演算，扩展至一般巴拿赫代数交换元组，提出朴素演算框架并应用于单变量函数均差生成的多元函数，证明相关引理、推导展开式与定理，提供组合公式。
Let $A,B$ be C*-algebras, $B_A(0;r)$ the open ball in $A$ centered at $0$ with radius $r>0$, and $H:B_A(0;r)\to B$ an orthogonally additive holomorphic map. If $H$ is zero product preserving on positive elements in $B_A(0;r)$, we show, in the commutative case when $A=C_0(X)$ and $B=C_0(Y)$, that there exist weight functions $h_n$'s and a symbol map $φ: Y\to X$ such that $$ H(f)=\sum_{n\geq1} h_n (f\circφ)^n, \quad\forall f\in B_{C_0(X)}(0;r). $$ In the general case, we show that if $H$ is also conformal then there exist central multipliers $h_n$'s of $B$ and a surjective Jordan isomorphism $J:…
The following topics are presented in these notes: Elements of Banach algebras, Banach algebras of the form $L^1(G)$, where $G$ is a locally compact group, spectrum of elements of Banach algebras, the spectral theory of compact operators on Banach spaces, the holomorphic functional calculus in Banach algebras, the Gelfand transform on commutative Banach algebras and C*-algebras, the continuous functional calculus, the Gelfand duality between commutative C*-algebras and locally compact and Hausdorff topological spaces, positivity in C*-algebras, approximate units, ideals of C*-algebras, heredit…
We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, H_d denotes the subspace of the von Neumann algebra of a free group F_I spanned by the words of length d in the generators (but not their inverses), then we provide in this paper an explicit upper bound on the norm on M_n(H_d), which improves and generalizes previous results by Kemp-Speicher (in the scalar case) and Buchholz and Parcet-Pisier (in the non-holomorphic setting). Namely the norm of an element of the form $\sum_{i=(i_1,..., i_d)} a_i \otimes λ(g_{i_1} ... g_{i_d})$ is less than $4^5 \sqrt e (\|M_…
On the Formality of Configuration Spaces of $\mathbb{R}^{n'} \times \mathbb{C}^{n}$
本文针对$\boldsymbol{\reals^{n'} \times \boldsymbol{\reals^{n}}}$的构形空间，通过定义可构造de Rham-Dolbeault上同调理论，分类其形式化性质并给出对应CDGA模型，还将其应用于拓扑-全纯场论的局部算子代数研究。
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