Annals of Functional Analysis最新文献

For a (C^*)-algebra ({mathcal {A}},) its scattered radical ({mathcal {R}}_s({mathcal {A}})) is the largest scattered ideal of ({mathcal {A}};) an ideal is scattered if its elements all have countable spectrum. We say that ({mathcal {A}}) is scattered if ({mathcal {R}}_s({mathcal {A}})={mathcal {A}}.) In this paper, we first show that any scattered von Neumann algebra is finite dimensional and then obtain a complete characterization of scattered radical of von Neumann algebras. Furthermore, we give a topological characterization of ({mathcal {R}}_s(C(M)),) that is, ({mathcal {R}}_s(C(M))={fin C(M): f(P(M))=0},) where M is a Hausdorff compact space and P(M) is the largest perfect subset of M. Finally, we show that ({mathcal {R}}_s({mathcal {A}}otimes _{min } {mathcal {B}})={mathcal {R}}_s({mathcal {A}})otimes _{min } {mathcal {R}}_s({mathcal {B}})) if ({mathcal {A}},{mathcal {B}},) satisfying one of the following conditions: (i) ({mathcal {A}},{mathcal {B}}) are (C^*)-algebras and ({mathcal {A}},{mathcal {B}}) are exact; (ii) ({mathcal {A}},{mathcal {B}}) are (C^*)-algebras and ({mathcal {A}}) or ({mathcal {B}}) is nuclear.

Let (mathcal {E}) be a finite dimensional Hilbert space. This note finds all factorizations of the right shift semigroup ({mathcal {S}}^mathcal {E}=(S_t^mathcal {E})_{tge 0}) on (L^2(mathbb {R}_+,mathcal {E})) into the product of n commuting contractive semigroups, i.e., characterizes all n-tuples of commuting semigroups (({mathcal {V}}_1,{mathcal {V}}_2,ldots ,{mathcal {V}}_n)) where ({mathcal {V}}_i=(V_{i,t})_{tge 0}) for (i=1,2,ldots ,n) are semigroups of contractions satisfying (V_{i,t}V_{j,t}=V_{j,t}V_{i,t}) for all i and j and (S_t^mathcal {E}=V_{1,t}V_{2,t}cdots V_{n,t}) for all (tge 0.) The factorizations are characterized by tuples of self-adjoint operators (underline{A}=(A_1,A_2,ldots ,A_n)) and tuples of positive contractions (underline{B}=(B_1,B_2,ldots ,B_n)) on (mathcal {E}) satisfying certain conditions which are stated in Theorem 4.10. One of the tools of our analysis is a convexity argument using the extreme points of the Herglotz class of functions

A representation of a right-angled Artin monoid is determined by a family of operators whose commutativity is dictated by a graph. We introduce the notion of the weak Brehmer’s condition and prove that the Cauchy transform for a representation of a right-angled Artin monoid is bounded under such conditions. As a result, we obtain the Poisson transform on right-angled Artin monoids, which generalizes Popescu’s notion of Cauchy and Poisson transforms for commuting families of row contractions. Finally, we prove that having (*)-regular dilation is equivalent to the weak Brehmer’s condition plus the property (P), thereby establishing their equivalence to the generalized Brehmer’s condition.

Let A be a unital (C^*)-algebra and U(A) be the unitary group of A. For (xin A), the unitary orbit of x is the set ({u^*xu: uin U(A)}), we denote by ({mathcal {U}}(x)) the closure of the unitary orbit of x. In this paper, we show that if A is a unital simple (C^*)-algebra of stable rank one and real rank zero, and (V_1,V_2in U(A)) with ([V_1]_1=[V_2]_1=0) in (K_1(A)), then (D_c(V_1,V_2)=textrm{dist}({mathcal {U}}(V_1),{mathcal {U}}(V_2))), where (D_c(V_1,V_2)) is a spectral distance function introduced by Hu and Lin and (textrm{dist}({mathcal {U}}(V_1),{mathcal {U}}(V_2))) is the distance between ({mathcal {U}}(V_1)) and ({mathcal {U}}(V_2)). Furthermore, we show that if A is a unital simple (C^*)-algebra of tracial rank zero, and (V_1,V_2in U(A)) with ([lambda -V_1]_1=[lambda -V_2]_1) for all (lambda notin {mathbb {T}}) in (K_1(A)), then (D_c(V_1,V_2)=textrm{dist}({mathcal {U}}(V_1),{mathcal {U}}(V_2))). Thus, we generalize the results by Bhatia and Davis for distance between unitary orbits of unitary matrices.

In this article, we investigate the ({mathbb {Z}}^d)-action (Phi) on the Banach space such that each generator of (Phi) consists of a linear part and a perturbed part. By adding certain conditions for the linear and perturbed parts of the generator, the notions Lipschitz hyperbolic ({mathbb {Z}}^d)-action and the strong partially hyperbolic ({mathbb {Z}}^d)-action are introduced. We show that (Phi) has the shadowing (quasi-stability) property when (Phi) is a Lipschitz hyperbolic (strong partially hyperbolic) ({mathbb {Z}}^d)-action.

Equivalence of Furuta inequality and Ando–Hiai inequality had shown in our previous paper. Furuta inequality has several advanced forms; satellite one, grand one and chaotic one. Corresponding to these, we will change the costume of Ando–Hiai inequality. We discuss quantum relative entropies as applications of Furuta inequality and grand Furuta inequality.

Let ({mathcal{H}}) and ({mathcal{K}}) be complex infinite-dimensional separable Hilbert spaces and ({mathcal{B}}({mathcal{K}},{mathcal{H}})) be the algebra of all bounded linear operators from ({mathcal{K}}) into ({mathcal{H}}). Given (Ain {mathcal{B}}({mathcal{H}})), (Bin {mathcal{B}}({mathcal{K}})) and (Cin {mathcal{B}}({mathcal{K}},{mathcal{H}})), we denote by (M_{C}=left( begin{array}{cc} A & C 0 & B end{array} right)) the upper triangular operator matrix acting on ({mathcal{H}}oplus {mathcal{K}}). In this paper, we give the characterization on the existence of (Cin {mathcal{B}}({mathcal{K}},{mathcal{H}})) such that (M_C) to be upper semi-Fredholm with fixed nullity and to be Fredholm with fixed index, respectively. Besides, we also show that the existence of invertible (C_0in {mathcal{B}}({mathcal{K}},{mathcal{H}})) such that (M_{C_0}) is a CI operator(resp. CW operator) is equivalent with (M_0) is a CI operator (resp. CW operator).

In this paper, we establish that a norm bounded set A in a Banach lattice E is an order bounded subset of (E^a) if and only if every disjoint sequence in its solid hull is ru-convergent to zero. Based on this result, we define a quantitative measure (delta (cdot )) and show that for every norm bounded set A in E, A is order bounded in (E^a) if and only if (delta (A)=0). As applications, we investigate the order boundedness of sets in atomic order continuous Banach lattices, and provide several necessary and sufficient conditions for an order continuous normed Riesz space to be a Banach lattice. In addition, we also obtain several sufficient conditions for a set to be b-order bounded in Dedekind complete Riesz spaces.

Let (mathcal {B}) be a separable simple stable purely infinite C*-algebra, and let (mathcal {M}(mathcal {B})) be the multiplier algebra of (mathcal {B}). We find a multiplier algebra analog of a result of Brown, Pearcy and Salinas, proving that for all (X in mathcal {M}(mathcal {B})), there exists a nilpotent operator (N in mathcal {M}(mathcal {B})) for which (X + N) is invertible in (mathcal {M}(mathcal {B})) if and only if (X notin mathcal {B}). Related to the above, we also have multiplier analogs of results of Dyer–Porcelli–Rosenfeld and Aiken, as well as results in the simple C*-algebra context.

This paper investigates the properties of trajectories in harmonic oscillator systems equipped with a point, absolutely continuous, or singular measure. Infinite-dimensional linear flows of countable oscillator systems exhibit a new class of trajectory behavior. Specifically, these trajectories are non-periodic, and their projections onto any four-dimensional symplectic subspace fail to be dense in the corresponding projection of the invariant torus. Such trajectories do not arise in finite-dimensional systems, are non-generic for countable oscillator systems, but become generic in the continual case. It is proved that for a countable harmonic oscillators system, every point on a nondegenerate invariant torus is a non-wandering point of the flow. Conversely, for a continuous system with an absolutely continuous measure, all points on such a torus are wandering. Furthermore, for continuous systems with singular measure, sufficient conditions on the measure and the torus are established, excluding the existence of both transitive trajectories and non-wandering points. As an application, a class of singular Bernoulli measures satisfying these conditions is presented.