We consider some conditions similar to Ozawa's condition (AO), and prove that if a non-injective factor satisfies such a condition and has the W*CBAP, then it has no Cartan subalgebras. As a corollary, we prove that $\rm II_1$ factors of universal orthogonal and unitary discrete quantum groups have no Cartan subalgebras. We also prove that continuous cores of type $\rm III_1$ factors with such a condition are semisolid as a $\rm II_\infty$ factor.

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Theory of Operator Algebras II

Examples of factors which have no Cartan subalgebras

We prove the quantum version of an ergodic result of H. Furstenberg relative to noninvariant measures. The natural setting will be the case of the "quantum diagonal measure" relative to the product measure. Even if in all the interesting situations such diagonal measures are neither invariant nor normal with respect to the corresponding product ones, we still provide an ergodic theorem for them, generalizing the classical case. As a natural application, we are able to prove the entangled ergodic theorem in some interesting situations out of the known ones, that is when the unitary is not almost periodic, or when the involved operators are not compact.

An ergodic theorem for quantum diagonal measures

For states in infinite dimensional Hilbert spaces entanglement quantities like the entanglement of distillation can become infinite. This leads naturally to the question, whether one system in such an infinitely entangled state can serve as a resource for tasks like the teleportation of arbitrarily many qubits. We show that appropriate states cannot be obtained by density operators in an infinite dimensional Hilbert space. However, using techniques for the description of infinitely many degrees of freedom from field theory and statistical mechanics, such states can nevertheless be constructed rigorously. We explore two related possibilities, namely an extended notion of algebras of observables, and the use of singular states on the algebra of bounded operators. As applications we construct the essentially unique infinite analogue of maximally entangled states, and the singular state used heuristically in the fundamental paper of Einstein, Rosen and Podolsky.

Infinitely entangled states

We prove that weak amenability of a locally compact group imposes a strong condition on its amenable closed normal subgroups. This extends non weak amenability results of Haagerup (1988) and Ozawa--Popa (2010). A von Neumann algebra analogue is also obtained.

Theory of Operator Algebras II

Citations

Examples of factors which have no Cartan subalgebras

An ergodic theorem for quantum diagonal measures

Infinitely entangled states

Examples of groups which are not weakly amenable

Isomorphisms of ordered structures of abelian C⁎-subalgebras of C⁎-algebras

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Citations

Examples of factors which have no Cartan subalgebras

An ergodic theorem for quantum diagonal measures

Infinitely entangled states

Examples of groups which are not weakly amenable

Isomorphisms of ordered structures of abelian C⁎-subalgebras of C⁎-algebras

Related Papers (5)

Fundamentals of the Theory of Operator Algebras

Operator algebras and quantum statistical mechanics

C-Algebras and Their Automorphism Groups

C[*]-algebras and W[*]-algebras

Completely Bounded Maps and Operator Algebras

C[]-algebras and W[]-algebras