Let X be an infinite-dimensional separable real or complex Banach space and A a closed standard operator algebra on X. Then every local automorphism of A is an automorphism. The assumptions of infinitedimensionality, separability, and closeness are all indispensable.

Let X be a compact Hausdorff space and let H be a separable Hilbert space. We prove that the group of all order automorphisms of the C∗-algebra C(X)⊗ B(H) is algebraically reflexive.

We answer a question of E. Kirchberg (personal communication): does the relative commutant of a separable C*-algebra in its ultrapower depend on the choice of the ultrafilter? All algebras and all subalgebras in this note are C*-algebras and C*subalgebras, respectively, and all ultrafilters are nonprincipal ultrafilters on N. Our C*-terminology is standard (see e.g., [2]). In the following U ra...

Let C be a unital AH-algebra and let A be a unital separable simple C∗-algebra with tracial rank zero. Suppose that φ1, φ2 : C → A are two unital monomorphisms. We show that there is a continuous path of unitaries {ut : t ∈ [0,∞)} of A such that lim t→∞ u∗tφ1(a)ut = φ2(a) for all a ∈ C if and only if [φ1] = [φ2] in KK(C,A), τ ◦ φ1 = τ ◦ φ2 for all τ ∈ T (A) and the rotation map η̃φ1,φ2 associate...

A depth two extension A | B is shown to be weak depth two over its double centralizer V A (V A (B)) if this is separable over B. We introduce a notion of codepth two coalgebra homomorphism g : C → D, dual to a depth two algebra homomorphism. It is shown that the endomorphism ring of bicomodule endomorphisms End D C D forms a right bialgebroid over the centralizer subalgebra g * : D * → C * of t...

Let A be a unital separable C *-algebra, and D a K1-injective strongly self-absorbing C *-algebra. We show that if A is D-absorbing, then the crossed product of A by a compact second countable group or by Z or by R is D-absorbing as well, assuming the action satisfying a Rokhlin property. In the case of a compact Rokhlin action we prove a similar statement about approximate divisibility.

We introduce stabilised property $\Gamma$, a $\mathrm {C}^*$-algebraic variant of $\Gamma$ which is invariant under stable isomorphism. then show that simple separable nuclear {C}^*$-algebras with and {Cu}(A) \cong \mathrm {Cu}(A \otimes \mathcal {Z})$ absorb the Jiang-Su algebra $\mathcal {Z}$ tensorially.

The main purpose of this work is to introduce the class of the monadic dynamic algebras (dynamic algebras with one quantifier). Similarly to a theorem of Kozen we establish that every separable monadic dynamic algebra is isomorphic to a monadic (possibly non-standard) Kripke structure. We also classify the simple (monadic) dynamic algebras. Moreover, in the dynamic duality theory, we analyze th...

We show that finitely generated subhomogeneous C∗-algebras have finite decomposition rank. As a consequence, any separable ASH C∗-algebra can be written as an inductive limit of subhomogeneous C∗-algebras each of which has finite decomposition rank. It then follows from work of H. Lin and of the second named author that the class of simple unital ASH algebras which have real rank zero and absor...

for all a, x in 21. I t is clear that associative algebras are alternative. The most famous examples of alternative algebras which are not associative are the so-called Cayley-Dickson algebras of order 8 over $. Let S be an algebra of order 2 over % which is either a separable quadratic field over 5 or the direct sum 5 ©3There is one automorphism z—>z of S (over %) which is not the identity aut...