A generalization of free action

On the Simplicity of some Cuntz-Pimsner Algebras

“…standard proof in von Neumann algebras [15,Proposition 3.6.7] (which applies here, too), v à is taken to be any extreme point in the weak-* compact, convex set K X f$ P E yy j f $ 1 k$kgX Since f 9u$u à f $ for all unitaries u P A yy and all $ P E yy Y K is preserved under the maps $ U 3 9u$u à Y u unitary in A yy X Further, these maps clearly carry extreme points of K to extreme points of K. Thus 9uv à u à is an extreme point of K for every unitary in A yy X But then, the uniqueness part of the polar decomposition implies that there is only one extreme point in K. Whence, 9uv à u à v à for all unitaries u P A yy Y and so v à P F 1 X This, in turn, implies that for all a P A yy and all unitaries u P A yy Y we have jf jau f v à au f 9uv à a f v à ua jf juaY which means that jf j is tracial. For the converse, simply note that if jf j is tracial on A yy while v à P F 1 Y then for all $ P E yy and all unitaries u P A yy Y f 9u$u à jf jv9u$u à jf ju à v9u$ jf jv$ f $X Suppose M is a von Neumann algebra and that is an automorphism of MX In [9], Kallman termed to be free, or freely acting, in case the only solution b to the equation ab ba, for all a P MY is b 0X If one views M as a correspondence E over itself, with 9 , then is free precisely when F 1 vanishes. (We don't make any distinction, at this point, between analysis in E and analysis in E yy X Thus, Theorem 15 says that our Condition F is a correspondence-theoretic way of saying that all the powers of an autoon the simplicity of some cuntz-pimsner algebrasmorphism are freely acting.…”

“…In fact, when the correspondence E is AY with the left action of A on A given by an automorphism 9, then Condition F is equivalent to Kallman's notion of free action [9], considered in the second dual of AX We will prove this in Theorem 34 in Section 3.…”

On Automorphisms of Prime Rings with Involution

“…a(x) = axa~x for all x G R), then a is a dependent element of a. If 0 is the only dependent element of a then a is said to be freely acting on R. Dependent elements were first introduced by Choda, Kasahara and Nakamoto [8] for automorphisms of C*-algebras in the process of generalization of the notion of free action of automorphisms of von Neumann algebras (due to von Neumann and Murray [15,16], see also Kallman [9]) to C-algebras. Several other authors have also studied dependent elements in operator algebras (see e.g.…”

Inner product modules arising from compact automorphism groups of von Neumann algebras

“…In [2], R. Kallman shows that every outer automorphism of an infinite conjugacy class group induces an outer automorphism of the corresponding II xfactor, and points out that there exist infinite-conjugacy-class groups all of whose automorphisms are inner. The example he mentions is the semidirect product D -(2* x Q (where Q is the additive group of rationals, ß* is the multiplicative group of nonzero rationals, and multiplication in D is defined by (ax, bx) (a2, b2) = (axa2, axb2 + bx) for ax, a2 E Q*, bx, b2 E Q).…”