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Poly-infix operators and operator families are introduced as an alternative for working modulo associativity and the corresponding bracket deletion convention. Poly-infix operators represent the basic intuition of repetitively connecting an ordered sequence of entities with the same connecting primitive.
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We give another proof for [ sum_{n=1}^{infty}frac{1}{n^2}=frac{pi^2}{6} ] that basically follows from the theory of difference equations.
Following [44], we introduce the notion of families of projective operators on fibrations equipped with an Azumaya bundle $mathcal{A}$. We define and compute the index of such families using the cohomological index formula from [7]. More precisely, a family of projective operators $A$ can be pulled back in a family $tilde{A}$ of $SU(N)$-transversally elliptic operators on the $PU(N)$-principal bundle of trivialisations of $mathcal{A}$. Through the distributional index of $tilde{A}$, we can define an index for the family $A$ of projective operators and using the cohomological index formula from [7], we obtain an explicit cohomological index formula. Let $1 to Gamma to tilde{G} to G to 1$ be a central extension by an abelian finite group. As a preliminary result, we compute the index of families of $tilde{G}$-transversally elliptic operators on a $G$-principal bundle $P$.
Let $mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $H$ equipped with a semifinite faithful normal trace $tau$. Let $E(mathcal{M},tau) $ be a symmetric operator space affiliated with $ mathcal{M} $, whose norm is order continuous and is not proportional to the Hilbertian norm $left|cdotright|_2$ on $L_2(mathcal{M},tau)$. We obtain general description of all bounded hermitian operators on $E(mathcal{M},tau)$. This is the first time that the description of hermitian operators on asymmetric operator space (even for a noncommutative $L_p$-space) is obtained in the setting of general (non-hyperfinite) von Neumann algebras. As an application, we resolve a long-standing open problem concerning the description of isometries raised in the 1980s, which generalizes and unifies numerous earlier results.
We introduce a Banach Lie group $G$ of unitary operators subject to a natural trace condition. We compute the homotopy groups of $G$, describe its cohomology and construct an $S^1$-central extension. We show that the central extension determines a non-trivial gerbe on the action Lie groupoid $Gltimes mathfrak{k}$, where $mathfrak{k}$ denotes the Hilbert space of self-adjoint Hilbert-Schmidt operators. With an eye towards constructing elements in twisted K-theory, we prove the existence of a cubic Dirac operator $mathbb{D}$ in a suitable completion of the quantum Weil algebra $mathcal{U}(mathfrak{g}) otimes Cl(mathfrak{k})$, which is subsequently extended to a projective family of self-adjoint operators $mathbb{D}_A$ on $Gltimes frak{k}$. While the kernel of $mathbb{D}_A$ is infinite-dimensional, we show that there is still a notion of finite reducibility at every point, which suggests a generalized definition of twisted K-theory for action Lie groupoids.
On the unit ball B^n we consider the weighted Bergman spaces H_lambda and their Toeplitz operators with bounded symbols. It is known from our previous work that if a closed subgroup H of widetilde{SU(n,1)} has a multiplicity-free restriction for the holomorphic discrete series of $widetilde{SU(n,1)}$, then the family of Toeplitz operators with H-invariant symbols pairwise commute. In this work we consider the case of maximal abelian subgroups of widetilde{SU(n,1)} and provide a detailed proof of the pairwise commutativity of the corresponding Toeplitz operators. To achieve this we explicitly develop the restriction principle for each (conjugacy class of) maximal abelian subgroup and obtain the corresponding Segal-Bargmann transform. In particular, we obtain a multiplicity one result for the restriction of the holomorphic discrete series to all maximal abelian subgroups. We also observe that the Segal-Bargman transform is (up to a unitary transformation) a convolution operator against a function that we write down explicitly for each case. This can be used to obtain the explicit simultaneous diagonalization of Toeplitz operators whose symbols are invariant by one of these maximal abelian subgroups.
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