Related papers: Regular inclusions of simple unital $C^*$-algebras

We show that the angle between intermediate $C^*$-subalgebras of an inclusion of simple $C^*$-algebras with finite Watatani index is stable. The notion of angle is instrumental in providing a bound for the cardinality of the lattice of…

We study regular irreducible inclusions $B\subset A$ of simple unital $C^*$-algebras admitting a conditional expectation. We introduce a generalized notion of quasi-basis extending Watatani's framework and show that such inclusions admit a…

Motivated by [2] and [5], the notions of interior and exterior angles between a pair of compatible intermediate W*-subalgebras of an inclusion of W*-algebras with a normal conditional expectation with finite probabilistic index are…

We study uniform perturbations of intermediate C*-subalgebras of inclusions of simple C*-algebras. If a unital simple C*-algebra has a simple C*-subalgebra of finite index, then sufficiently close simple intermediate C*-subalgebras are…

We introduce the category of C*-discrete inclusions of C*-algebras $A\subset B$ with a faithful conditional expectation $E:B\twoheadrightarrow A$. This class includes many examples such as finite Watatani index inclusions, and also abundant…

Let $1 \in A \subset B$ be an inclusion of unital C*-algebras of index-finite type and depth 2. Suppose that $A$ is infinite dimensional simple with $tsr(A) = 1$ and SP-property. Then $tsr(B) \leq 2$. As a corollary when $A$ is a simple…

We examine inclusions of $C^*$-algebras of the form $A^H \subseteq A \rtimes_{r} G$, where $G$ and $H$ are groups acting on a unital simple $C^*$-algebra $A$ by outer automorphisms and $H$ is finite. It follows from a theorem of Izumi that…

Let B be a unital C*-algebra, let A be a unital subalgebra, and let E be a conditional expectation from B to A with index-finite type and a quasi-basis of n elements. Then the topological stable rank satisfies \tsr (B) \leq \tsr (A) + n -…

We prove that if a conditional expectation from a simple $C^*$-algebra onto its $C^*$-subalgebra satisfies the Pimsner-Popa inequality, there exists a quasi-basis. As an application, we establish the Galois correspondence for outer actions…

Analogous to subfactor theory, employing Watatani's notions of index and $C^*$-basic construction of certain inclusions of $C^*$-algebras, (a) we develop a Fourier theory (consisting of Fourier transforms, rotation maps and shift operators)…

We show that for a large class of actions $\Gamma \curvearrowright \mathcal{A}$ of $C^*$-simple groups $\Gamma$ on unital $C^*$-algebras $\mathcal{A}$, including any non-faithful action of a hyperbolic group with trivial amenable radical,…

Given any irreducible inclusion $\mB \subset \mA$ of unital $C^*$-algebras with a finite-index conditional expectation $E: \mA \to \mB$, we show that the set of $E$-compatible intermediate $C^*$-subalgebras is finite, thereby generalizing a…

We investigate the normal subgroups of the groups of invertibles and unitaries in the connected component of the identity. By relating normal subgroups to closed two-sided ideals we obtain a "sandwich condition" describing all the closed…

For an inclusion of C*-algebras $D\subseteq A$ with $D$ abelian, we show that when $n\in A$ normalizes $D$, $n^*n$ and $nn^*$ commute with $D$. As a corollary, when $D$ is a regular MASA in $A$, every approximate unit for $D$ is also an…

We provide a complete classification for regular subalgebras $B \subset M$ of injective factors satisfying a natural relative commutant condition. We show that such subalgebras are classified by their associated amenable discrete measured…

We study pairs (C,D) of unital C*-algebras where D is a regular abelian C*-subalgebra of C. When D is a MASA in C, we prove the existence and uniqueness of a completely positive unital map E of C into the injective envelope I(D) of D whose…

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…

Let $P \subset A$ be a inclusion of unital C*-algebras and $E\colon A \to P$ be a conditional expectation of index finite type. We introduce a Rokhlin property for $E$ and discuss about $\mathcal{D}$-absorbing proeprty, where $\mathcal{D}$…

We show that the commutator subgroup of the group of unitaries connected to the identity in a simple unital C*-algebra is simple modulo its center. We then go on to investigate the role of regularity properties in the structure of the…

Let A be a unital C*-algebra. We shall introduce involutive A-A-equivalence bimodules and prove that any C*-algebra containing A with Watatani index 2 is constructed by an involutive A-A-equivalence bimodule.