Spectral scale of selfadjoint operators and trace inequalities
In a quantum system the entropy of a state can be expressed as -tr(\(\rho\) log \(\rho)\), where \(\rho\) is a density matrix and tr is the trace. This entropy function is concave in \(\rho\). J. von Neumann pointed out the reason for this concavity is that the real-valued function \(t\mapsto t \log t\) is concave and the linear function \(\rho\) \(\mapsto tr f(\rho)\) is concave for any concave function f. In this paper, motivated by the notion of the generalized singular values due to \textit{T. Fack} [J. Operator Theory 7, No.2, 307-333 (1982; Zbl 0493.46052)], the author inroduces the spectral scale of a selfadjoint element in a finite von Neumann algebra and generalizes the aforementioned result of von Neumann to C *-algebras with a tracial state. Let R be a finite von Neumann algebra with a fixed faithful normal normaized trace \(\tau\). For any selfadjoint operator A affiliated to R the spectral scale is defined as \(\lambda_ t(A)=\min [s:\tau (P_ A(s,\infty))\leq t]\), \(t\in [0,1]\), where \(\int tdP_ A(t)\) is the spectral resolution of A. It is clear that \(\lambda_ t(A)\) is continuous from the right and nonincreasing in t. The scale \(\lambda\).(A):[0,1]\(\to {\mathbb{R}}\) is also measure preserving if [0,1] is endowed with the Lebesgue measure and the Borel measure \(H\mapsto \tau (P_ A(H))\) is considered on \({\mathbb{R}}\). Thus, for any bounded Borel function f of \({\mathbb{R}}\) we have \(\tau (f(A))=\int^{1}_{0}f(\lambda_ t(A))dt\). This property together with the following minimax characterization paves the way for the two main results in this paper, which will be stated immediately after the following: \(\lambda_ t(A)=\inf_{e}\sup_{s}\{<As,s>:s\in H\), H the Hilbert space A is acting on, \(\| s\| =1\), \(es=e\}\), where inf is taken over all the projections e such that \(\tau (e^{\perp})\leq t.\) Result 1 (Theorem 2): If f:\({\mathbb{R}}\to {\mathbb{R}}\) is a monotone function then \(a\mapsto \tau (f(a))\) is monotone on \(R_{S.A.}\). Result 2 (Theorem 4): Let f be a continuous convex function on [\(\alpha\),\(\beta\) ]; then the functional \(A\mapsto \tau (f(A))\) is convex on the set \(\{A\in R_{S.A.}:Sp(A)\subset [\alpha,\beta]\}\). Furthermore, for C *- algebras \({\mathfrak U}\) with a tracial state \(\phi\), the author is able to extend the above results as: Result 3 (Theorem 5): Let \({\mathfrak U}\) be a C *-algebra with a tracial state \(\phi\) ; if f:[\(\alpha\),\(\beta\) ]\(\to {\mathbb{R}}\) is a continuous monotone [convex] function then the functional \(A\mapsto \phi (f(A))\) is monotone [convex] on the set \(\{\) \(A\in {\mathfrak U}_{S.A.}:Sp(A)\subset [\alpha,\beta]\}.\) The author is further able to generalize the result to tracial positive linear maps \(\Phi\), between two C *-algebras such that \(\Phi (AB)=\Phi (BA)\) for all A, B in the domain C *-algebra. The tracial positive linear maps were first studied by \textit{M. D. Choi} and the reviewer [Proc. Amer. Math. Soc. 87, No.1, 57-61 (1983; Zbl 0548.46045)]. Finally the author establishes an inequality for a tracial state \(\phi\) on a C *-algebra \({\mathfrak U}\), which states \(\phi (f_ 1(A)g_ 1(B))\leq \phi (f_ 2(A))+\phi (g_ 2(A))\) for all A, B in \({\mathfrak U}_{S.A.}\) with Sp(A)\(\subset [\alpha,\beta]\), Sp(B)\(\subset [\gamma,\delta]\) and continuous functions \(f_ i:[\alpha,\beta]\to {\mathbb{R}}\), \(g_ i:[\gamma,\delta]\to {\mathbb{R}}\), \(i=1,2\), such that \(f_ 1(x)g_ 1(y)\leq f_ 2(x)+g_ 2(y)\) for \(x\in [\alpha,\beta]\) and \(y\in [\gamma,\delta]\).
spectral scale of a selfadjoint element in a finite von Neumann
scientific article; zbMATH DE number 4069643
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