本文刻画了 C*-代数上有限指标条件期望的等价条件，证明其与完全正性条件等价，并推广了 Pimsner、Popa、Watatani 等人的结果，同时描述了 Jones 塔存在的情形及正规条件期望与 W*-型分解的交换性。
摘要：
对于（含单位元）C*-代数 A 上的条件期望 E，存在实数 K≥1 使得映射 (K·E−id_A) 为正，当且仅当存在实数 L≥1 使得映射 (L·E−id_A) 为完全正，此外还有其他等价条件。估计式 min(K) ≤ min(L) ≤ min(K)·[min(K)] 成立，其中 [·] 表示实数的整数部分。由此，“有限指标条件期望”的概念被等同于该类条件期望，这推广并完善了 M. Pimsner、S. Popa、M. Baillet、Y. Denizeau、J.-F. Havet、Y. Watatani 等人的结果。在一般框架下描述了指标值和 Jones 塔存在的情形。特别地，在 W*-情形下 Jones 塔总是存在；在 C*-情形下，当 Ind(E) 属于 E(A) 时 Jones 塔也存在。此外，有限指标的正规条件期望与一般 W*-投影到其有限、无限、离散和连续的 I 型、II_1 型、II_∞ 型和 III 型部分可交换，即初始 W*-代数中心与像 W*-代数中心中的相应投影重合。我们用非交换拓扑的语言解释了上述结果，并给出了一些维数估计公式和一个不等式。
英文摘要：
For a conditional expectation E on a (unital) C*-algebra A there exists a real number K>=1 such that the mapping (K.E-id_A) is positive if and only if there exists a real number L>=1 such that the mapping (L.E-id_A) is completely positive, among other equivalent conditions. The estimate min(K) <= min(L) <= min(K).[min(K)] is valid, where [.] denotes the integer part of a real number. As a consequence the notion of a 'conditional expectation of finite index' is identified with that class of conditional expectations, which extends and completes results of M. Pimsner, S. Popa; M. Baillet, Y. Denizeau, J.-F. Havet; Y. Watatani, and others. Situations for which the index value and the Jones' tower exist are described in the general setting. In particular, the Jones' tower always exists in the W*-case and for Ind(E) in E(A) in the C*-case. Furthermore, normal conditional expectations of finite index commute with the general W*-projections to their finite, infinite, discrete and continuous type I, type II_1, type II_\infty and type III parts, i.e. the respective projections in the centers of the initial and the image W*-algebra coincide. We give an interpretation of our result in terms of non-commutative topology and indicate some dimension estimation formulae and an inequality.
Nevanlinna showed that Cauchy transforms of probability measures parametrize all functions from the upper half plane into itself satisfying a certain asymptotic condition at infinity. We show that the correspondence fails in general for the unbounded case for somewhat trivial reasons; however, we show that in a setting of "homomorphic" operator valued free probability that Cauchy transforms of homomorphic conditional expectations parametrize free Pick functions.
We determine when there is a unique conditional expectation from a semifinite von Neumann algebra onto a singly-generated maximal abelian *-subalgebra. Our work extends the results of Kadison and Singer via new methods, notably the observation that a unique conditional expectation onto a singly-generated maximal abelian *-subalgebra must be normal.
In this paper we investigate the conditional expectation on the non-commutative $H^{(r,s)}_{p}(\mathcal A;\ell_{\infty})$ and $H_{p}(\mathcal A;\ell_{1})$ spaces associated with semifinite subdiagonal algebra, and prove the contractibility of the underlying conditional expectation on these spaces.
Anzai skew-products are shown to be uniquely ergodic with respect to the fixed-point subalgebra if and only if there is a unique conditional expectation onto such a subalgebra which is invariant under the dynamics. For the particular case of skew-products, this solves a question raised by B. Abadie and K. Dykema in the wider context of $C^*$-dynamical systems.
In this paper we prove the existence of conditional expectations in the noncommutative $L_p(M,Φ)$ spaces associated with center-valued traces. Moreover, their description is also provided. As an application of the obtained results, we establish the norm convergence of the martingales in noncommutative $L_p(M,Φ)$ spaces.
We study conditional expectations generated by an abelian $ C^* $-subalgebra in the centralizer of a positive functional. We formulate and prove Jensen's inequality for functions of several variables with respect to this type of conditional expectations, and we obtain as a corollary Jensen's inequality for expectation values.
Let $D \subseteq A$ be an inclusion of unital abelian $C^*$-algebras. In this note we characterize (in topological terms) when there is a unique conditional expectation $E:A \to D$, at least when $A$ is separable. As an application, we provide the first example of an inclusion with a unique conditional expectation, but multiple pseudo-expectations (in the sense of Pitts).
We shall introduce the notions of the strong Morita equivalence for unital inclusions of unital $C^*$-algebras and conditional expectations from an equivalence bimodule onto its closed subspace with respect to conditional expectations from unital $C^*$-algebras onto their unital $C^*$-subalgebras. Also, we shall study their basic properties.
Let $A$ be a $C^*$-algebra and $E\colon A \to A$ a conditional expectation. The Kadison-Schwarz inequality for completely positive maps, $$E(x)^*E(x) \leq E(x^* x),$$ implies that $$ \|E(x)\|^2 \leq \|E(x^* x)\|.$$ In this note we show that $E$ is a homomorphism if and only if $$\|E(x)\|^2 = \|E(x^*x)\|,$$ for every $x$ in $A$. We also prove that a homomorphic conditional expectation on a commutative $C^*$-algebra $C_0(X)$ is given by composition with a continuous retraction of $X$. One may therefore consider homomorphic conditional expectations as noncommutative retractions.
We use here a recent idea of studying functions of free random variables using Boolean cumulants. We develop idea of explicit calculations of conditional expectation using Boolean cumulants. We demonstrate Boolean cumulants approach allows to calculate explicitly some conditional expectations of functions in free random variables. We present how Boolean cumulants together with subordination simplify proofs of some results which are known in research literature.
We initiate the effective study of conditional expectations onto subfactors. Our main result is that if $M$ is an existentially closed II$_1$ factor with a w-spectral gap subfactor $N$, then the conditional expectation function onto $N$ can be computed from a (Turing) oracle that computes a presentation of $M$, the inclusion of $N$ into $M$, and a spectral gap function for the pair $(M,N)$.
It is shown that if T is a ternary ring of operators (TRO), X is a nondegenerate sub-TRO of T and there exists a contractive idempotent surjective map P:T-->X, then P has a unique, explicitly described extension to a conditional expectation between the associated linking algebras. A version of the result for W*-TROs is also presented and some applications mentioned.
Private quantum channels are the quantum analogue of the classical one-time pad. Conditional expectations and trace vectors are notions that have been part of operator algebra theory for several decades. We show that the theory of conditional expectations and trace vectors is intimately related to that of private quantum channels. Specifically we give a new geometric characterization of single qubit private quantum channels that relies on trace vectors. We further show that trace vectors completely describe the private states for quantum channels that are themselves conditional expectations. W…
Let A be a von Neumann algebra with a finite trace $τ$, represented in $H=L^2(A,τ)$, and let $B_t\subset A$ be sub-algebras, for $t$ in an interval $I$. Let $E_t:A\to B_t$ be the unique $τ$-preserving conditional expectation. We say that the path $t\mapsto E_t$ is smooth if for every $a\in A$ and $v \in H$, the map $$ I\ni t\mapsto E_t(a)v\in H $$ is continuously differentiable. This condition implies the existence of the derivative operator $$ dE_t(a):H\to H, \ dE_t(a)v=\frac{d}{dt}E_t(a)v. $$ If this operator verifies the additional boundedness condition, $$ \int_J \|dE_t(a)\|_2^2 d t\le C_J…
Most of this article is an expanded version of our conference talk. It is essentially a survey, but some part, like most of the lengthy Section 5, is comprised of new results whose proofs are unpublished elsewhere. We begin by reviewing the theory of real positivity of operator algebras initiated by the author and Charles Read. Then we present several new general results (mostly joint work with Matthew Neal) about real positive maps. The key point is that real positivity is often the right replacement in a general algebra $A$ for positivity in C*-algebras. We then apply this to studying contra…
Let N \subseteq M be von Neumann algebras and E:M\to N a faithful normal conditional expectation. In this work it is shown that the similarity orbit S(E) of E by the natural action of the invertible group of G_M of M has a natural complex analytic structure and the map given by this action: G_M\to S(E) is a smooth principal bundle. It is also shown that if N is finite then S(E) admits a reductive structure. These results were known previously under the conditions of finite index and N'\cap M \subseteq N, which are removed in this work. Conversely, if the orbit S(E) has an homogeneous reductive…
Let $(X_1,\dots,X_m)$ be self-adjoint non-commutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential $V$, and let $(S_1,\dots,S_m)$ be a free semicircular family. We show that conditional expectations and conditional non-microstates free entropy given $X_1$, \dots, $X_k$ arise as the large $N$ limit of the corresponding conditional expectations and entropy for the random matrix models associated to $V$. Then by studying conditional transport of measure for the matrix models, we construct an isomorphism $\mathrm{W}^…
Normal conditional expectations E: M --> N in M of finite index on von Neumann algebras M with discrete center are investigated to find an estimate for the minimal number of generators of M as a Hilbert N-module. Analyzing the case of M being finite type I with discrete center we obtain that these von Neumann algebras M are always finitely generated Hilbert N-modules with a minimal generator set consisting of at most [K(E)]^2 generators, where [.] denotes the integer part of a real number and K(E) = {K: K.E-id_M >= 0}. This result contrasts remarkable examples by P. Jolissaint and S. Popa show…
We prove that all the compact metric spaces are in the closure of the class of full matrix algebras for the quantum Gromov-Hausdorff propinquity. We also show that given an action of a compact metrizable group G on a quasi-Leibniz compact quantum metric space (A,Lip), the function associating any closed subgroup of G group to its fixed point C*-subalgebra in A is continuous from the topology of the Hausdorff distance to the topology induced by the propinquity. Our techniques are inspired from our work on AF algebras as quantum metric spaces, as they are based on the use of various types of con…
We establish several deep existence criteria for conditional expectations on von Neumann algebras, and then apply this theory to develop a noncommutative theory of representing measures of characters of a function algebra. Our main cycle of results describes what may be understood as a `noncommutative Hoffman-Rossi theorem' giving the existence of weak* continuous `noncommutative representing measures' for so-called D-characters. These results may also be viewed as `module' Hahn-Banach extension theorems for weak* continuous `characters' into possibly noninjective von Neumann algebras. In clos…
Let $\mathcal{N}\subset\mathcal{M}$ be a unital inclusion of arbitrary von Neumann algebras. We give a 2-{$C^*$}-categorical/planar algebraic description of normal faithful conditional expectations $E:\mathcal{M}\to\mathcal{N}\subset\mathcal{M}$ with finite index and their duals $E':\mathcal{N}'\to\mathcal{M}'\subset\mathcal{N}'$ by means of the solutions of the conjugate equations for the inclusion morphism $ι:\mathcal{N}\to\mathcal{M}$ and its conjugate morphism $\overlineι:\mathcal{M}\to\mathcal{N}$. In particular, the theory of index for conditional expectations admits a 2-{$C^*$}-categori…
We develop a new operator algebraic formulation of the Nakajima-Mori-Zwanzig (NMZ) method of projections. The new theory is built upon rigorous mathematical foundations, and it can be applied to both classical and quantum systems. We show that a duality principle between the NMZ formulation in the space of observables and in the state space can be established, analogous to the Heisenberg and Schrödinger pictures in quantum mechanics. Based on this duality we prove that, under natural assumptions, the projection operators appearing in the NMZ equation must be conditional expectations. The propo…
We consider two inclusions of $C^*$-algebras whose small $C^*$-algebras have approximate units of the large $C^*$-algebras and their two spaces of all bounded bimodule linear maps. We suppose that the two inclusions of $C^*$-algebras are strongly Morita equivalent. In this paper, we shall show that there exists an isometric isomorphism from one of the spaces of all bounded bimodule linear maps to the other space and we shall study on basic properties about the isometric isomorphism. And, using this isometric isomorphism, we define the Picard group for a bimodule linear map and discuss on the P…
We study the holographic map in AdS/CFT, as modeled by a quantum error correcting code with exact complementary recovery. We show that the map is determined by local conditional expectations acting on the operator algebras of the boundary/physical Hilbert space. Several existing results in the literature follow easily from this perspective. The Black Hole area law, and more generally the Ryu-Takayanagi area operator, arises from a central sum of entropies on the relative commutant. These entropies are determined in a state independent way by the conditional expectation. The conditional expecta…
Quantum steering is equivalent to state-preserving conditional expectations
Perimeter Institute for Theoretical Physics（佩里尔理论物理研究所）；Institute for Quantum Computing（量子计算研究所）；Unité de Mathématiques Pures et Appliquées, CNRS - ENS Lyon（纯粹与应用数学单位，法国国家科学研究中心-里昂高等师范学院）；Leibniz Universität Hannover, Institut für Theoretische Physik（汉诺威莱布尼茨大学理论物理研究所）
