If a nuclear separable C*-algebra A can be approximated by C*-subalgebras satisfying the UCT, then A satisfies the UCT. It is also shown that the validity of the UCT for all separable nuclear C*-algebras is equivalent to a certain finite dimensional approximation property.

In this paper, the exact sequences of fuzzy homology are constructed under certain condition. Using these results, the theorem of universal coefficients for fuzzy homology is proved.

that holds in the same generality as the universal coefficient theorem of Rosenberg and Schochet. There are advantages, in some circumstances, to using HomΛ(K(A),K(B)) in place of KK(A,B). These advantages derive from the fact that K(A) can be equipped with order and scale structures similar to those on K0(A). With this additional structure, the “Λ−module” K(A) becomes a powerful invariant of C...

We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra A generated by an irreducible representation of such a group has decomposition rank at most 3. If, in addition, A satisfies the universal coefficient theorem, another string of deep results shows it is classifiable by its Ellio...

We give a classification theorem for unital separable simple nuclear C∗-algebras with tracial topological rank zero which satisfy the Universal Coefficient Theorem. We prove that if A and B are two such C∗-algebras and (K0(A),K0(A)+, [1A], K1(A)) = (K0(B), K0(B)+, [1B ], K1(B)), then A = B.

We compare different algebraic structures in twisted equivariant K-theory for proper actions of discrete groups. After the construction a module structure over untwisted K-theory, we prove completion Theorem Atiyah–Segal type K-theory. Using universal coefficient theorem, cocompletion Borel K-homology

We define the filtrated K-theory of a C∗-algebra over a finite topological space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of filtrated K-theory. For finite spaces with totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences f...

We give a classification theorem for unital separable nuclear simple C∗-algebras with tracial rank no more than one. Let A and B be two unital separable simple nuclear C∗-algebras with TR(A), TR(B) ≤ 1 which satisfy the universal coefficient theorem. We show that A ∼= B if and only if (K0(A),K0(A)+, [1A], K1(A), T (A)) ∼= (K0(B), K0(B)+, [1B ], K1(B), T (B)).

Let A, B be separable simple unital tracially AF C*-algebras. Assuming that A is exact and satisfies the Universal Coefficient Theorem (UCT) in KK-theory, we prove the existence, and uniqueness modulo approximately inner automorphisms, of nuclear ∗-homomorphisms from A to B with prescribed K-theory data. This implies the AF-embeddability of separable exact residually finite dimensional C*-algeb...

We re-examine holographic versions of the c-theorem and entanglement entropy in the context of higher curvature gravity and the AdS/CFT correspondence. We select the gravity theories by tuning the gravitational couplings to eliminate nonunitary operators in the boundary theory and demonstrate that all of these theories obey a holographic c-theorem. In cases where the dual CFT is even-dimensiona...