Faithful Conditional Expectation Research Articles

A planar algebraic description of conditional expectations

Let [Formula: see text] be a unital inclusion of arbitrary von Neumann algebras. We give a 2-[Formula: see text]-categorical/planar algebraic description of normal faithful conditional expectations [Formula: see text] with finite index and their duals [Formula: see text] by means of the solutions of the conjugate equations for the inclusion morphism [Formula: see text] and its conjugate morphism [Formula: see text]. In particular, the theory of index for conditional expectations admits a 2-[Formula: see text]-categorical formulation in full generality. Moreover, we show that a pair [Formula: see text] as above can be described by a Q-system, and vice versa. These results are due to Longo in the subfactor/simple tensor unit case [R. Longo, Index of subfactors and statistics of quantum fields. II. Correspondences, braid group statistics and Jones polynomial, Comm. Math. Phys. 130(1990) 285–309; A duality for Hopf algebras and for subfactors. I, Comm. Math. Phys. 159 (1994) 133–150].

Decomposable partial actions

We define the decomposition property for partial actions of discrete groups on C⁎-algebras. Decomposable partial systems appear naturally in practice, and many commonly occurring partial actions can be decomposed into partial actions with the decomposition property. For instance, any partial action of a finite group is an iterated extension of decomposable systems.Partial actions with the decomposition property are always globalizable and amenable, regardless of the acting group, and their globalization can be explicitly described in terms of certain global sub-systems. A direct computation of their crossed products is also carried out. We show that partial actions with the decomposition property behave in many ways like global actions of finite groups (even when the acting group is infinite), which makes their study particularly accessible. For example, there exists a canonical faithful conditional expectation onto the fixed point algebra, which is moreover a corner in the crossed product in a natural way. (Both of these facts are in general false for partial actions of finite groups.) As an application, we show that freeness of a topological partial action with the decomposition property is equivalent to its fixed point algebra being Morita equivalent to its crossed product. We also show by example that this fails for general partial actions of finite groups.

Intermediate C*-algebras of Cartan embeddings

Let A A be a C ∗ ^* -algebra and let D D be a Cartan subalgebra of A A . We study the following question: if B B is a C ∗ ^* -algebra such that D ⊆ B ⊆ A D \subseteq B \subseteq A , is D D a Cartan subalgebra of B B ? We give a positive answer in two cases: the case when there is a faithful conditional expectation from A A onto B B , and the case when A A is nuclear and D D is a C ∗ ^* -diagonal of A A . In both cases there is a one-to-one correspondence between the intermediate C ∗ ^* -algebras B B , and a class of open subgroupoids of the groupoid G G , where Σ → G \Sigma \rightarrow G is the twist associated with the embedding D ⊆ A D \subseteq A .

Finite-dimensional approximations for Nica–Pimsner algebras

We give necessary and sufficient conditions for nuclearity of Cuntz–Nica–Pimsner algebras for a variety of quasi-lattice ordered groups. First we deal with the free abelian lattice case. We use this as a stepping-stone to tackle product systems over quasi-lattices that are controlled by the free abelian lattice and satisfy a minimality property. Our setting accommodates examples like the Baumslag–Solitar lattice for $n=m>0$ and the right-angled Artin groups. More generally, the class of quasi-lattices for which our results apply is closed under taking semi-direct and graph products. In the process we accomplish more. Our arguments tackle Nica–Pimsner algebras that admit a faithful conditional expectation on a small fixed point algebra and a faithful copy of the coefficient algebra. This is the case for CNP-relative quotients in-between the Toeplitz–Nica–Pimsner algebra and the Cuntz–Nica–Pimsner algebra. We complete this study with the relevant results on exactness.

Chain conditions for C*-algebras coming from Hilbert C*-modules

In this paper we define and study chain conditions for Hilbert C*-modules through their C*-algebras of compact operators and discuss their perseverance under Morita equivalence and tensor products. We show that these chain conditions are passed from the C*-algebra to its Hilbert module under certain conditions. We also study chain conditions for Hilbert modules coming from inclusion of C*-algebra with a faithful conditional expectation.

PERIODIC -GRAPH ALGEBRAS REVISITED

Let $P$ be a finitely generated cancellative abelian monoid. A $P$-graph ${\rm\Lambda}$ is a natural generalization of a $k$-graph. A pullback of ${\rm\Lambda}$ is constructed by pulling it back over a given monoid morphism to $P$, while a pushout of ${\rm\Lambda}$ is obtained by modding out its periodicity, which is deduced from a natural equivalence relation on ${\rm\Lambda}$. One of our main results in this paper shows that, for some $k$-graphs ${\rm\Lambda}$, ${\rm\Lambda}$ is isomorphic to the pullback of its pushout via a natural quotient map, and that its graph $\text{C}^{\ast }$-algebra can be embedded into the tensor product of the graph $\text{C}^{\ast }$-algebra of its pushout and $\text{C}^{\ast }(\text{Per}\,{\rm\Lambda})$. As a consequence, in this case, the cycline algebra generated by the standard generators corresponding to equivalent pairs is a maximal abelian subalgebra, and there is a faithful conditional expectation from the graph $\text{C}^{\ast }$-algebra onto it.

Formula omitted]-algebras of separated graphs

The construction of the C ⁎ -algebra associated to a directed graph E is extended to incorporate a family C consisting of partitions of the sets of edges emanating from the vertices of E. These C ⁎ -algebras C ⁎ ( E , C ) are analyzed in terms of their ideal theory and K-theory, mainly in the case of partitions by finite sets. The groups K 0 ( C ⁎ ( E , C ) ) and K 1 ( C ⁎ ( E , C ) ) are completely described via a map built from an adjacency matrix associated to ( E , C ) . One application determines the K-theory of the C ⁎ -algebras U m , n n c , confirming a conjecture of McClanahan. A reduced C ⁎ -algebra C r e d ⁎ ( E , C ) is also introduced and studied. A key tool in its construction is the existence of canonical faithful conditional expectations from the C ⁎ -algebra of any row-finite graph to the C ⁎ -subalgebra generated by its vertices. Differences between C r e d ⁎ ( E , C ) and C ⁎ ( E , C ) , such as simplicity versus non-simplicity, are exhibited in various examples, related to some algebras studied by McClanahan.

Rosenthal type inequalities for free chaos

Let $\mathscr{A}$ denote the reduced amalgamated free product of a family $\mathsf{A}_{1},\mathsf{A}_{2},\ldots,\mathsf{A}_{n}$ of von Neumann algebras over a von Neumann subalgebra $\mathscr {B}$ with respect to normal faithful conditional expectations $\mathsf {E}_{k}:\mathsf{A}_{k}\to \mathscr {B}$. We investigate the norm in $L_{p}(\mathscr {A})$ of homogeneous polynomials of a given degree d. We first generalize Voiculescu’s inequality to arbitrary degree d≥1 and indices 1≤p≤∞. This can be regarded as a free analogue of the classical Rosenthal inequality. Our second result is a length-reduction formula from which we generalize recent results of Pisier, Ricard and the authors. All constants in our estimates are independent of n so that we may consider infinitely many free factors. As applications, we study square functions of free martingales. More precisely, we show that, in contrast with the Khintchine and Rosenthal inequalities, the free analogue of the Burkholder–Gundy inequalities does not hold in $L_{\infty}(\mathscr {A})$. At the end of the paper we also consider Khintchine type inequalities for Shlyakhtenko’s generalized circular systems.

Positive Projective Space of a C*-Algebra with a Tracial Conditional Expectation

Let Open image in new window be a C*-algebra, Open image in new window a subalgebra of its center and Φ: Open image in new window → Open image in new window a tracial faithful conditional expectation. We define the positive projective space as the quotient

A-Valued Semicircular Systems

To a von Neurnann algebra A and a set of linear maps ηij:A→A, i, j∈I such that a↦(ηij)ij∈I:A→A⊗B(l2(I)) is normal and completely positive, we associate a von Neumann algebra Φ(A, η). This von Neumann algebra is generated by A and an A-valued semicircular system Xi, i∈I, associated to η. In many cases there is a faithful conditional expectation E:Φ(A, η)→A; if A is tracial, then under certain assumptions on η, Φ(A, η) also has a trace. One can think of the construction Φ(A, η) as an analogue of a crossed product construction. We show that most known algebras arising in free probability theory can be obtained from the complex field by iterating the construction Φ. Of a particular interest are free Krieger algebras, which, by analogy with crossed products and ordinary Krieger factors, are defined to be algebras of the form Φ(L∞[0, 1], η). The cores of free Araki–Woods factors are free Krieger algebras. We study the free Krieger algebras and as a result obtain several non-isomorphism results for free Araki–Woods factors. As another source of classification results for free Araki–Woods factors, we compute the τ invariant of Connes for free products of von Neumann algebras. This computation generalizes earlier work on computation of T, S, and Sd invariants for free product algebras.

Minimal upper bounds of commuting operators

Let ( x i ) (x_{i}) be a finite collection of commuting self-adjoint elements of a von Neumann algebra M \mathcal {M} . Then within the (abelian) C*-algebra they generate, these elements have a least upper bound x x . We show that within M \mathcal {M} , x x is a minimal upper bound in the sense that if y y is any self-adjoint element such that x i ≤ y ≤ x x_{i} \leq y \leq x for all i i , then y = x y = x . The corresponding assertion for infinite collections ( x i ) (x_{i}) is shown to be false in general, although it does hold in any finite von Neumann algebra. We use this sort of result to show that if N ⊂ M \mathcal {N} \subset \mathcal {M} are von Neumann algebras, Φ : M → N \Phi : \mathcal {M} \to \mathcal {N} is a faithful conditional expectation, and x ∈ M x \in \mathcal {M} is positive, then Φ ( x n ) 1 / n \Phi (x^{n})^{1/n} converges in the strong operator topology to the “spectral order majorant” of x x in N \mathcal {N} .

On the Conjugate Endomorphism in the Infinite Index Case.

We give an algebraic characterization for the conjugate endomorphism ρ of an endomorphism ρ of infinite index of a properly infinite von Neumann algebra M such that the set of normal faithful conditional expectations E(M,ρ(M)) is not empty. In the particular case of irreducible endomorphisms we obtain the same result holding in finite index case and in the representation theory of compact groups, that is if ρ is an irreducible endomorphism of an infinite factor, with E(M,ρ(M)) 6= ∅, then an irreducible endomorphism σ is conjugate to ρ iff σρ id; moreover the identity is contained only once in σρ. Some applications of the above results are also given.