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 of finite groups (and more generally finite dimensional Kac algebras) on simple $C^*$-algebras. We also introduce the notion of sectors for stable simple $C^*$-algebras and purely infinite simple $C^*$-algebras in the Cuntz standard form, and discuss several applications, including their relationship to $K$-theory.
We shall introduce the notion of the Picard group for an inclusion of $C^*$-algebras. We shall also study its basic properties and the relation between the Picard group for an inclusion of $C^*$-algebras and the ordinary Picard group. Furthermore, we shall give some examples of the Picard groups for unital inclusions of unital $C^*$-algebras.
We consider two saturated Fell bundles over a countable discrete group, whose unit fibers are $σ$-unital $C^*$-algebras. Then by taking the reduced cross-sectional $C^*$-algebras, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions of $C^*$-algebras is irreducible, that is, the relative commutant of one of the unit fiber algebras, which is a $σ$-unital $C^*$-algebra, in the multiplier $C^*$-algebra of the reduced cross-sectional $C^*$-algebra is trivial. We show that the two sa…
The literature contains interesting examples of inclusions of simple C$^*$-algebras with the property that all intermediate C$^*$-algebras likewise are simple. In this article we take up a systematic study of such inclusions, which we refer to as being C$^*$-irreducible by the analogy that all intermediate von Neumann algebras of an inclusion of factors are again factors precisely when the given inclusion is irreducible. We give an intrinsic characterization of when an inclusion of C$^*$-algebras is C$^*$-irreducible, and use this to revisit known and exhibit new C$^*$-irreducible inclusions a…
Let $A$ be a unital $C^*$-algebra and $X$ an invulutive $A-A$-equivalence bimodule. Let $A\subset C_X$ be the unital inclusion of unital $C^*$-algebras induced by $X$. We suppose that $A' \cap C_X =\mathbf{C} 1$. We shall compute the Picard group of the unital inclusion $A\subset C_X$.
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 $\mathcal{D}$ is a separable, unital, strongly self-absorbing C*-algebra. In this paper we consider permanent properties for strongly self-absorbing property under inclusions of unital C*-algebras with a finite Watatani index.
We consider two twisted actions of a countable discrete group on $σ$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions is irreducible, that is, the relative commutant of one of the $σ$-unital $C^*$-algebra in the multiplier $C^*$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.
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.
We introduce notions of the Rohlin property and the approximate representability for inclusions of unital $C^*$-algebras. We investigate a dual relation between the Rohlin property and the approximate representability. We prove that a number of classes of unital $C^*$-algebras are closed under inclusions with the Rohlin property, including: AF algebras, AI algebras, AT algebras, and related classes characterized by direct limit decomposition using semiprojective building blocks. $C^*$-algebras with stable rank one. $C^*$-algebras with real rank zero.
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.
We prove that the notion of relative property (T) (or rigidity) for inclusions of finite von Neumann algebras defined in [Po1] is equivalent to a weaker property, in which no ``continuity constants'' are required. The proof is by contradiction and uses infinite products of completely positive maps, regarded as correspondences.
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 C*-algebra with $\tsr(A) = 1$ and SP-property and $α$ an action of a finite group $G$ on $\Aut(A)$, $\tsr(A \rtimes_αG) \leq 2$.
In this note we study inclusions of second quantization algebras, namely inclusions of von Neumann algebras on the Fock space of a separable complex Hilbert space H, generated by the Weyl unitaries with test functions in closed, real linear subspaces of H. We show that the class of irreducible inclusions of standard second quantization algebras is non empty, and that they are depth two inclusions, namely the third relative commutant of the Jones' tower is a factor. When the smaller vector space has codimension n into the bigger, we prove that the corresponding inclusion of second quantization …
We consider cones in a Hilbert space associated to two von Neumann algebras and determine when one algebra is included in the other. If a cone is assocated to a von Neumann algebra, the Jordan structure is naturally recovered from it and we can characterize projections of the given von Neumann algebra with the structure in some special situations.
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 $A^H \subseteq A$ is $C^*$-irreducible, in the sense that all intermediate $C^*$-algebras are simple. We show that $A^H \subseteq A \rtimes_{r} G$ is $C^*$-irreducible for all $G$ and $H$ as above if and only if $G$ and $H$ have trivial intersection in the outer automorphisms of $A$, and we give a Galois type classification of all intermediate $C^*$-algebras…
Let $A$ and $B$ be unital $C^*$-algebras and let $H$ be a finite dimensional $C^*$-Hopf algebra. Let $H^0$ be its dual $C^*$-Hopf algebra. Let $(ρ, u)$ and $(σ, v)$ be twisted coactions of $H^0$ on $A$ and $B$, respectively. In this paper, we shall show the following theorem: We suppose that the unital inclusions $A\subset A\rtimes_{ρ, u}H$ and $B\subset B\rtimes_{σ, v}H$ are strongly Morita equivalent. If $A'\cap (A\rtimes_{ρ, u}H)=\BC1$, then there is a $C^*$-Hopf algebra automorphism $λ^0$ of $H^0$ such that the twisted coaction $(ρ, u)$ is strongly Morita equivalent to the twisted coaction…
Let $P \subset A$ be an inclusion of separable unital C*-algebras with finite Watatani index. Suppose that $E \colon A \rightarrow P$ has the Rokhlin property, that is, there is a projection $e \in A' \cap A^\infty$ such that $E^\infty(e) = ({\rm Index}E)^{-1}1$. We show that if $A$ has nuclear dimension $n$, then $P$ has nuclear dimension less than or equal to $n$. In particular, if an action $α$ of a finite group $G$ on $A$ has the Rokhlin property, then the nuclear dimension of the crossed product algebra $A \rtimes_αG$ is less than or equal to that of $A$.
Let $1 \in A \subset B$ be an inclusion of C*-algebras of C*-index-finite type with depth 2. We try to compute topological stable rank of $B$ ($= \tsr(B)$) when $A$ has topological stable rank one. We show that $\tsr(B) \leq 2$ when $A$ is a tsr boundedly divisible algebra, in particular, $A$ is a C*-minimal tensor product $UHF \otimes D$ with $\tsr(D) = 1$. When $G$ is a finite group and $α$ is an action of $G$ on UHF, we know that a crossed product algebra $UHF \rtimes_αG$ has topological stable rank less than or equal to two. These results are affirmative datum to a generalization of a ques…
In this paper we define $L^{2}$-homology and $L^{2}$-Betti numbers for tracial *-algebras $A$ with respect to a von Neumann subalgebra $B$. When $B$ is reduced to the field of complex numbers we recover the $L^{2}$-Betti numbers of $A$ as defined by A. Connes and D. Shlyakhtenko, but we will show that taking into account the role of the von Neumann subalgebra yields to a number of advantages like, for instance, a much better behavior with respect to compression and directed sums. Our main result is that $L^{2}$-homology and $L^{2}$-Betti numbers of discrete measured groupoids and equivalence r…
In this paper we will introduce the tracial Rokhlin property for an inclusion of separable simple unital C*-algebras $P \subset A$ with finite index in the sense of Watatani, and prove theorems of the following type. Suppose that $A$ belongs to a class of C*-algebras characterized by some structural property, such as tracial rank zero in the sense of Lin. Then $P$ belongs to the same class. The classes we consider include:(1) Simple C*-algebras with real rank zero or stable rank one, (2) Simple C*-algebras with tracial rank zero or tracial rank less than or equal to one, (3) Simple C*-algebras…
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 - 1. As an application, we show that if a unital inclusion A \subset B of C*-algebras has index-finite type and finite depth, and A is simple with stable rank one and Property (SP), then B has cancellation. In particular, if A is a simple unital C*-algebra with stable rank one and Property (SP), and a finite group G acts on A, then the crossed product has cance…
In recent joint work with V. Jones and D. Shlyakhtenko, we have given a diagrammatic description of Popa's symmetric enveloping inclusion for planar algebra subfactors. In this paper we give a diagrammatic construction of the associated Jones tower, in the case that the planar algebra is finite-depth. We then use this construction to describe the planar algebra of the symmetric enveloping inclusion, which is known to be isomorphic to the planar algebra of Ocneanu's asymptotic inclusion by a result of Popa. As an application we give a planar algebraic computation of the (reduced) fusion algebra…
Given a finite-index and finite-depth subfactor, we define the notion of \textit{quantum double inclusion} - a certain unital inclusion of von Neumann algebras constructed from the given subfactor - which is closely related to that of Ocneanu's asymptotic inclusion. We show that the quantum double inclusion when applied to the Kac algebra subfactor $R^H \subset R$ produces Drinfeld double of $H$ where $H$ is a finite-dimensional Kac algebra acting outerly on the hyperfinite $II_1$ factor $R$ and $R^H$ denotes the fixed-point subalgebra. More precisely, quantum double inclusion of $R^H \subset …
Let $P \subset A$ be an inclusion of $σ$-unital C*-algebras with a finite index in the sense of Izumi. Then we introduce the Rokhlin property for a conditional expectation $E$ from $A$ onto $P$ and show that if $A$ is simple and satisfies any of the property $(1) \sim (12)$ listed in the below, and $E$ has the Rokhlin property, then so does $P$. (1) Simplicity;(2) Nuclearity;(3) C*-algebras that absorb a given strongly self-absorbing C*-algebra $\mathcal{D}$; (4)C*-algebras of stable rank one; (5) C*-algebras of real rank zero;(6) C*-algebras of nuclear dimension at most $n$, where $n \in Z^+$…
In \cite{Sde2018} we defined the notion of \textit{quantum double inclusion} associated to a finite-index and finite-depth subfactor and studied the quantum double inclusion associated to the Kac algebra subfactor $R^H \subset R$ where $H$ is a finite-dimensional Kac algebra acting outerly on the hyperfinite $II_1$ factor $R$ and $R^H$ denotes the fixed-point subalgebra. In this article we analyse quantum double inclusions associated to the family of Kac algebra subfactors given by $\{ R^H \subset R \rtimes \underbrace{H \rtimes H^* \rtimes \cdots}_{\text{$m$ times}} : m \geq 1 \}$. For each $…
We give a formula for the stable rank of inclusions of unital Banach algebras in the sense of finite Watatani index. As an application we show that the stable rank of $\ell^1$-algebras of Disk algebras by any action of finite groups is 2.
In this work we give a generalization of matched pairs of (finite) groups to describe a general class of depth two inclusions of factor von Neumann algebras and the C*-quantum groupoids associated with, using double groupoids.
We prove that if an amenable operator algebra is nearly contained in a complemented dual operator algebra, then it can be embedded inside this dual operator algebra via a similarity. The proof relies on a B.E. Johnson Theorem on approximately multiplicative maps.
Let $A$ and $B$ be $σ$-unital $C^*$-algebras and $X$ and $Y$ an $A-A$-equivalence bimodule and a $B-B$-equivalence bimodule, respectively. Also, let $A\rtimes_X \mathbb{Z}$ and $B\rtimes_Y \mathbb{Z}$ be the crossed products of $A$ and $B$ by $X$ and $Y$, respectively. Furthermore, let $A\subset A\rtimes_X \mathbb{Z}$ and $B\subset B\rtimes_Y \mathbb{Z}$ be the inclusions of $C^*$-algebras induced by $X$ and $Y$, respectively. We suppose that $A' \cap M(A\rtimes_X \mathbb{Z})=\mathbb{C} 1$. In this paper we shall show that the inclusions $A\subset A\rtimes_X \mathbb{Z}$ and $B\subset B\rtimes_…
Given an action of a compact quantum group on a unital C*-algebra, one can amplify the action with an adjoint representation of the quantum group on a finite dimensional matrix algebra, and consider the resulting inclusion of fixed point algebras. We show that this inclusion is a finite index inclusion of C*-algebras when the quantum group acts freely. We show that two natural definitions for a quantum group to act freely, namely the Ellwood condition and the saturatedness condition, are equivalent.
A $C^*$-algebra is said to have the LP property if the linear span of projections is dense in a given algebra. In the first part of this paper, we show that an AH algebra $A = \underrightarrow{\lim}(A_i,ϕ_i)$ has the LP property if and only if every real-valued continuous function on the spectrum of $A_i$ (as an element of $A_i$ via the non-unital embedding) belongs to the closure of the linear span of projections in $A$. As a consequence, a diagonal AH-algebra has the LP property if it has small eigenvalue variation. The second contribution of this paper is that for an inclusion of unital $C^…
We define the local multiplier module of a Hilbert module in analogy to the local multiplier algebra for $C^*$-algebras. We use properties of the local multiplier module to lift non-closed actions on $C^*$-algebras by Hilbert bimodules to closed actions on local multiplier algebras, and descend known results on such closed actions down to their unclosed counterparts. We define aperiodic dynamical inclusions and characterise them as crossed products by inverse semigroup actions. We describe the slice structure for such inclusions and show that all slices arise as linear combinations of slices a…
After a summary on module algebra actions of C^*-weak Hopf algebras we outline the proof of a reconstruction theorem stating that every finite index depth 2 inclusion N < M of unital C^*-algebras with finite dimensional centers is isomorphic to the invariant subalgebra inclusion M^A < M with respect to a regular weak Hopf algebra action. The proof uses the language of C^*-2-categories.
We show that indecomposable weak Kac algebras are free over their Cartan subalgebras and prove a duality theorem for their actions. Using this result, for any biconnected weak Kac algebra we construct a minimal action on the hyperfinite II_1 factor. The corresponding crossed product inclusion of II_1 factors has depth 2 and an integer index. Its first relative commutant is, in general, non-trivial, so we derive some arithmetic properties of weak Kac algebras from considering reduced subfactors.
We consider compact group actions on C*- and W*- algebras. We prove results that relate the duality property of the action (as defined in the Introduction) with other relevant properties of the system such as the relative commutant of the fixed point algebras being trivial (called the irreducibility of the inclusion) and also to the Galois correspondence between invariant C*-subalgebras containing the fixed point algebra and the class of closed normal subgroups of the compact group.
We study the problem of extending a state on an abelian $C^*$- subalgebra to a tracial state on the ambient $C^*$-algebra. We propose an approach that is well-suited to the case of regular inclusions, in which there is a large supply of normalizers of the subalgebra. Conditional expectations onto the subalgebra give natural extensions of a state to the ambient $C^*$-algebra; we prove that these extensions are tracial states if and only if certain invariance properties of both the state and conditional expectations are satisfied. In the example of a groupoid $C^*$-algebra, these invariance prop…
We study uniform perturbations of crossed product C$^*$-algebras by amenable groups. Given a unital inclusion of C$^*$-algebras $C\subseteq D$ and sufficiently close separable intermediate C$^*$-subalgebras $A$, $B$ for this inclusion with a conditional expectation from $D$ onto $B$, if $A=C\rtimes G$ with $G$ discrete amenable, then $A$ and $B$ are isomorphic. Furthermore, if $C\subseteq D$ is irreducible, then $A=B$.
When a unital \ca $A$ has topological stable rank one (write $\tsr(A) = 1$), we know that $\tsr(pAp) \leq 1$ for a non-zero projection $p \in A$. When, however, $\tsr(A) \geq 2$, it is generally faluse. We prove that if a unital C*-algebra $A$ has a simple unital C*-subalgebra $D$ of $A$ with common unit such that $D$ has \PSP and $\sup_{p\in P(D)}\tsr(pAp) < \infty$, then $\tsr(A) \leq 2.$ As an application let $A$ be a simple unital \ca with $\tsr(A) = 1$ and \PSP, $\{G_k\}_{k=1}^n$ finite groups, $\af_k$ actions from $G_k$ to ${\rm Aut}((...((A\times_{\af_1}G_1)\times_{\af_2} G_2)...)\times…
In a former article, in collaboration with Jean-Michel Vallin, we have constructed two "quantum groupo\"ıds" dual to each other, from a depth 2 inclusion of von Neumann algebras $M_0\subset M_1$, in such a way that the canonical Jones'tower associated to the inclusion can be described as a tower of successive crossed-products by these two structures. We are now investigating in greater details these structures in the presence of an appropriate modular theory on the basis $M'_0\cap M_1$, and we show how these examples fit with Lesieur's "measured quantum groupo\"ıds".
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).
The groupoid normalisers of a unital inclusion $B\subseteq M$ of von Neumann algebras consist of the set $\mathcal{GN}_M(B)$ of partial isometries $v\in M$ with $vBv^*\subseteq B$ and $v^*Bv\subseteq B$. Given two unital inclusions $B_i\subseteq M_i$ of von Neumann algebras, we examine groupoid normalisers for the tensor product inclusion $B_1\ \overline{\otimes}\ B_2\subseteq M_1\ \overline{\otimes}\ M_2$ establishing the formula $$ \mathcal{GN}_{M_1\,\overline{\otimes}\,M_2}(B_1\ \overline{\otimes}\ B_2)''=\mathcal{GN}_{M_1}(B_1)''\ \overline{\otimes}\ \mathcal{GN}_{M_2}(B_2)'' $$ when one i…
