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We study an operator-valued generalization of the Haagerup inequality for Gromov hyperbolic groups. In 1978, U. Haagerup showed that if $f$ is a function on the free group $\mathbb{F}_r$ which is supported on the $k$-sphere $S_k=\{x\in \mathbb{F}_r:\ell(x)=k\}$, then the operator norm of its left regular representation is bounded by $(k+1)\|f\|_2$. An operator-valued generalization of it was started by U. Haagerup and G. Pisier. One of the most complete form was given by A. Buchholz, where the $\ell^2$-norm in the original inequality was replaced by $k+1$ different matrix norms associated to word decompositions (this type of inequality is also called Khintchine-type inequality). We provide a generalization of Buchholz's result for hyperbolic groups., Comment: 8 pages, comments welcome!

For a unital spectral triple $(\mathcal{A}, H,D)$, we study when its truncation converges to itself. The spectral truncation is obtained by using the spectral projection $P_{\Lambda}$ of $D$ onto $[-\Lambda,\Lambda]$ to deal with the case where only a finite range of energy levels of a physical system is available. By restricting operators in $\mathcal{A}$ and $D$ to $P_{\Lambda}H$, we obtain a sequence of operator system spectral triples $\{(P_{\Lambda}\mathcal{A}P_{\Lambda},P_{\Lambda}H,P_{\Lambda}DP_{\Lambda})\}_{\Lambda}$. We prove that if the spectral triple is the one constructed using a discrete group with polynomial growth, then the sequence of operator systems $\{P_{\Lambda}\mathcal{A}P_{\Lambda}\}_{\Lambda}$ converges to $\mathcal{A}$ in the sense of quantum Gromov-Hausdorff convergence with respect to the Lip-norm coming from high order derivatives.

We give a rigorous proof of Conjecture 3.1 by Prosen [Prosen T 2010 J. Stat. Mech. $\textbf{2010}$ P07020] on the nilpotent part of the Jordan decomposition of a quadratic fermionic Liouvillian. We also show that the number of the Jordan blocks of each size can be expressed in terms of the coefficients of a polynomial called the $q$-binomial coefficient and describe the procedure to obtain the Jordan canonical form of the nilpotent part., Comment: 12 pages, no figures; v3: some remarks added

We give a characterization of geometric property (T) for a coarse disjoint union of finite graphs with bounded degree using the idea of noncommutative real algebraic geometry. In the proof, we define a $*$-subalgebra $I_u[X]$ of real algebraic Roe algebra $\mathbb{R}_u[X]$ over a graph $X$ with bounded degree. Then we show that $I_u[X]$ contains the Laplacian $\Delta$ as an order unit with respect to the positive cone ${\sum \limits}^2I_u[X]$ which consists of sums of hermitian squares., Comment: In the proof of the Lemma 5.3, I implicitely assumed \phi(1-u)=\lim \sum \phi(1-u_m), but it was incorrect

Motivated by the idea that our access to the spacetime is limited by the resolution of our measuring device, we give a new description of $K$-homology with a finite resolution. G. Yu introduced a $C^*$-algebra called the localization algebra $C^*_L(X)$ which consists of functions from $[1,\infty)$ to the Roe algebra $C^*(X)$ whose propagations converge to $0$ and he showed that for any finite dimensional simplicial complex $X$ endowed with the spherical metric, the $K$-theory of the localization algebra is isomorphic to the $K$-homology of $X$. We give a coarse graining version of this theorem using controlled $K$-theory (also known as quantitative $K$-theory). Namely, instead of considering families of operators whose propagations converge to $0$, we prove that for each dimension $n$, there exists a threshold $r_n>0$ such that the $K$-homology of $n$-dimensional finite simplicial complex $X$ is isomorphic to a certain group of equivalence classes of operators whose propagation is less than $r_n$. This picture also enables us to represent any element in the $K$-homology group $K_*(X)$ by a finite matrix for a finite simplicial complex $X$., Comment: 20 pages