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Register a new userConnes Integration Formula without singular traces
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A version of Connes Integration Formula which provides concrete asymptotics of the eigenvalues is given. This radically extending the class of quantum-integrable functions on compact Riemannian manifolds.
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We shall develop a notion of amenability for dual Banach algebras, namely weak Connes amenability, which will play the role that weak amenability does for usual Banach algebras
Let $f$ and $g$ be weakly holomorphic modular functions on $Gamma_0(N)$ with the trivial character. For an integer $d$, let $Tr_d(f)$ denote the modular trace of $f$ of index $d$. Let $r$ be a rational number equivalent to $iinfty$ under the action of $Gamma_0(4N)$. In this paper, we prove that, when $z$ goes radially to $r$, the limit $Q_{hat{H}(f)}(r)$ of the sum $H(f)(z) = sum_{d>0}Tr_d(f)e^{2pi idz}$ is a special value of a regularized twisted $L$-function defined by $Tr_d(f)$ for $dleq0$. It is proved that the regularized $L$-function is meromorphic on $mathbb{C}$ and satisfies a certain functional equation. Finally, under the assumption that $N$ is square free, we prove that if $Q_{hat{H}(f)}(r)=Q_{hat{H}(g)}(r)$ for all $r$ equivalent to $i infty$ under the action of $Gamma_0(4N)$, then $Tr_d(f)=Tr_d(g)$ for all integers $d$.
Let $A$ be a positive semidefinite $mtimes m$ block matrix with each block $n$-square, then the following determinantal inequality for partial traces holds [ (mathrm{tr} A)^{mn} - det(mathrm{tr}_2 A)^n ge bigl| det A - det(mathrm{tr}_1 A)^m bigr|, ] where $mathrm{tr}_1$ and $mathrm{tr}_2$ stand for the first and second partial trace, respectively. This result improves a recent result of Lin [14].
This article - a part of a multipaper project investigating arithmetic mean ideals - investigates the codimension of commutator spaces [I, B(H)] of operator ideals on a separable Hilbert space, i.e., ``How many traces can an ideal support? We conjecture that the codimension can be only zero, one, or infinity. Using the arithmetic mean (am) operations on ideals introduced by Dykema, Figiel, Weiss, and Wodzicki, and the analogous am operations at infinity that we develop in this article, the conjecture is proven for all ideals not contained in the largest am-infinity stable ideal and not containing the smallest am-stable ideal. It is also proven for all soft-edged ideals (i.e., I= IK(H)) and all soft-complemented ideals (i.e., I= I/K(H)), which include many classical operator ideals. In the process, we prove that an ideal of trace class operators supports a unique trace (up to scalar multiples) if and only if it is am-infinity stable and that, for a principal ideal, am-infinity stability is equivalent to regularity at infinity of the sequence of s-numbers of the generator. Furthermore, we apply trace extension methods to two problems on elementary operators studied by V. Shulman and to Fuglede-Putnam type problems of the second author.
Let $A$ be an $mtimes m$ positive semidefinite block matrix with each block being $n$-square. We write $mathrm{tr}_1$ and $mathrm{tr}_2$ for the first and second partial trace, respectively. In this note, we prove the following inequality [ (mathrm{tr} A)I_{mn} - (mathrm{tr}_2 A) otimes I_n ge pm bigl( I_motimes (mathrm{tr}_1 A) -Abigr). ] This inequality is not only a generalization of Andos result [1], but it also could be regarded as a complement of a recent result of Choi [8]. Additionally, some new partial traces inequalities for positive semidefinite block matrices are also included.
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