In this paper an automorphism of a unital C*-algebra is said to be /locally inner/ if on any element it agrees with some inner automorphism. We make a fairly complete study of local innerness in von Neumann algebras, incorporating comparison with the pointwise innerness of Haagerup-Stormer. On some von Neumann algebras, including all with separable predual, a locally inner automorphism must be inner. But a transfinitely recursive construction demonstrates that this is not true in general. As an application, we show that the diagonal sum descends to a well-defined map on the automorphism orbits of a unital C*-algebra if and only if all its automorphisms are locally inner.
We show that an E_0-semigroup acting on the algebra of all adjointable operators on a Hilbert module, which has a spatial product system, arises as the the restriction of a semigroup inner automorphisms.
We obtain some results that relate the Connes spectrum with the innerness of automorphisms in the fixed point local multiplier algebra. The results are variants and/or extensions of corresponding results of Olesen, Pedersen and Stormer [6], [7], [9] and Ara and Mathieu [1] to the case of local multiplier algebras.
Let A be a simple unital AT algebra of real rank zero and Inn(A) the group of inner automorphisms of A. In the previous paper we have shown that the natural map of the group of approximately inner automorphisms into Ext(K_1(A),K_0(A)) oplus Ext(K_0(A),K_1(A)) is surjective; the kernel of this map includes the subgroup of automorphisms which are homotopic to Inn(A). In this paper we consider the quotient of the group of approximately inner automorphisms by the smaller normal subgroup AInn(A) which consists of asymptotically inner automorphisms and describe it as OrderExt(K_1(A),K_0(A)) oplus Ex…
We study distinguished subalgebras and automorphisms of boundary quotients arising from algebraic dynamical systems $(G,P,θ)$. Our work includes a complete solution to the problem of extending Bogolubov automorphisms from the Cuntz algebra in $2 \leq p<\infty$ generators to the $p$-adic ring $C^*$-algebra. For the case where $P$ is abelian and $C^*(G)$ is a maximal abelian subalgebra, we establish a picture for the automorphisms of the boundary quotient that fix $C^*(G)$ pointwise. This allows us to show that they form a maximal abelian subgroup of the entire automorphism group. The picture al…
We prove that it is relatively consistent with the usual axioms of mathematics that all automorphisms of the Calkin algebra are inner. Together with a 2006 Phillips--Weaver construction of an outer automorphism using the Continuum Hypothesis, this gives a complete solution to a 1977 problem of Brown-Douglas-Fillmore. We also give a simpler and self-contained proof of the Phillips--Weaver result.
We introduce a class of automorphisms of compact quantum groups which may be thought of as inner automorphisms and explore the behaviour of normal subgroups of compact quantum groups under these automorphisms. We also define the notion of center of a compact quantum group and compute the center for several examples. We briefly touch upon the commutator subgroup of a compact quantum group and discuss how its relation with the center can be different from the classical case.
Given a von Neumann algebra $M$ we consider the central extension $E(M)$ of $M.$ For type I von Neumann algebras $E(M)$ coincides with the algebra $LS(M)$ of all locally measurable operators affiliated with $M.$ In this case we show that an arbitrary automorphism $T$ of $E(M)$ can be decomposed as $T=T_a\circ T_ϕ,$ where $T_a(x)=axa^{-1}$ is an inner automorphism implemented by an element $a\in E(M),$ and $T_ϕ$ is a special automorphism generated by an automorphism $ϕ$ of the center of $E(M).$ In particular if $M$ is of type I$_\infty$ then every band preserving automorphism of $E(M)$ is inner…
We obtain two characterizations of the bi-inner Hopf *-automorphisms of a finite-dimensional Hopf C*-algebra, by means of an analysis of the structure of convolution products in this class of Hopf C*-algebra.
We show that a factor $M$ is full if and only if the $C^*$-algebra generated by its left and right regular representations contains the compact operators. We prove that the bicentralizer flow of a type $\mathrm{III}_1$ factor is always ergodic. As a consequence, for any type $\mathrm{III}_1$ factor $M$ and any $λ\in ]0,1]$, there exists an irreducible AFD type $\mathrm{III}_λ$ subfactor with expectation in $M$. Moreover, any type $\mathrm{III}_1$ factor $M$ which satisfies $M \cong M \otimes R_λ$ for some $λ\in ]0,1[$ has trivial bicentralizer. Finally, we give a counter-example to the charact…
It is established that every derivation continuous with respect to the local measure topology acting on the *-algebra $LS(\mathcal{M})$ of all locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$ is necessary inner. If $\mathcal{M}$ is a properly infinite von Neumann algebra, then every derivation on $LS(\mathcal{M})$ is inner. In addition, it is proved that any derivation on $\mathcal{M}$ with values in Banach $\mathcal{M}$-bimodule of locally measurable operators is inner.
We study the topological structure of the automorphism groups of compact quantum groups showing that, in parallel to a classical result due to Iwasawa, the connected component of identity of the automorphism group and of the "inner" automorphism group coincide. For compact matrix quantum groups, which can be thought of as quantum analogues of compact Lie groups, we prove that the inner automorphism group is a compact Lie group and the outer automorphism group is discrete. Applications of this to the study of group actions on compact quantum groups are highlighted. We end with the construction …
Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple \CA with tracial rank zero. Suppose that $ϕ_1, ϕ_2: C\to A$ are two unital monomorphisms. We show that there is a continuous path of unitaries $\{u_t: t\in [0, \infty)\}$ of $A$ such that $$ \lim_{t\to\infty}u_t^*ϕ_1(a)u_t=ϕ_2(a)\tforal a\in C $$ if and only if $[ϕ_1]=[ϕ_2]$ in $KK(C,A),$ $τ\circ ϕ_1=τ\circ ϕ_2$ for all $τ\in T(A)$ and the rotation map ${\tildeη}_{ϕ_1,ϕ_2}$ associated with $ϕ_1$ and $ϕ_2$ is zero. In particular, an automorphism $\af$ on a unital separable simple \CA $A$ in ${\cal N}$ with tracial rank zero…
WZW models are abstract conformal field theories with an infinite dimensional symmetry which accounts for their integrability, and at the same time they have a sigma model description of closed string propagation on group manifolds which, in turn, endows the models with an intuitive geometric meaning. We exploit this dual algebraic and geometric property of WZW models to construct an explicit example of a field-dependent reflection matrix for open-strings in Nappi-Witten model. Demanding the momentum outflow at the boundary to be zero determines a certain combination of the left and right chir…
In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of II_1 factors. Here are some sample results: (1) an automorphism is approximately inner if and only if its ultrapower is aleph_0-locally inner; (2) the ultrapower of an outer automorphism is always outer; (3) for unital *-homomorphisms from a separable nuclear C*-algebra into an ultrapower of a II_1 factor, equality of the induced traces implies unitary equivalence. All statements are proved using operator algebraic techniques, but in the…
For a locally compact quantum group $\mathbb{G}$ we define its center, $\mathscr{Z}(\mathbb{G})$, and its quantum group of inner automorphisms, $\mathrm{Inn}(\mathbb{G})$. We show that one obtains a natural isomorphism between $\mathrm{Inn}(\mathbb{G})$ and $\mathbb{G}/\!\mathscr{Z}(\mathbb{G})$, we characterize normal quantum subgroups of a compact quantum group as those left invariant by the action of the quantum group of inner automorphisms and discuss several examples.
The Proper Forcing Axiom implies all automorphisms of every Calkin algebra associated with an infinite-dimensional complex Hilbert space and the ideal of compact operators are inner. As a means of the proof we introduce the notion of Polish $ω_1$-trees and cohrerent families of Polish spaces and prove some uniformization results.
We make a detailed study of locally inner actions on C*-algebras whose primitive ideal spaces have locally compact Hausdorff complete regularizations. We suppose that $G$ has a representation group and compactly generated abelianization $G_{ab}$. Then if the complete regularization of $\Prim(A)$ is $X$, we show that the collection of exterior equivalence classes of locally inner actions of $G$ on $A$ is parameterized by the group $\E_G(X)$ of exterior equivalence classes of $C_0(X)-actions of $G$ on $C_0(X,\K)$. Furthermore, we exhibit a group isomorphism of $\E_G(X)$ with the direct sum $H^1(…
Let A be an exact C^*-algebra, let G be a locally compact group, and let (A,G,α) be a C*-dynamical system. Each automorphism α_g induces a spatial automorphism Ad_{\lamba_g} on the reduced crossed product A\times_αG. In this paper we examine the question, first raised by E. Stormer, of when the topological entropies of α_g and Ad_{α_g} coincide. This had been answered by N. Brown for the particular case of discrete abelian groups. Using different methods, we extend his result to a wider class of groups called locally [FIA]^-. This class includes all abelian groups, both discrete and continuous…
In the present paper derivations and *-automorphisms of algebras of unbounded operators over the ring of measurable functions are investigated and it is shown that all L^0-linear derivations and L^{0}-linear *-automorphisms are inner. Moreover, it is proved that each L^0-linear automorphism of the algebra of all linear operators on a bo-dense submodule of a Kaplansky-Hilbert module over the ring of measurable functions is spatial.
We investigate the structure of the outer automorphism group of the Cuntz algebra and the closely related problem of conjugacy of MASAa in O_n. In particular, we exhibit an uncountable family of MASAs, conjugate to the standard MASA D_n via Bogolubov automorphisms, that are not inner conjugate to D_n.
