We study maps of positive operators of the Schatten -classes (), which preserve the -norms of convex combinations, that is, . They are exactly those carrying the form for a unitary or antiunitary . In the case , we have the same conclusion whenever it just holds for all the positive Hilbert-Schmidt class operators of norm . Some examples are demonstrated.

We state a conjecture for the formulas of the depth 4 low-weight rotational eigenvectors and their corresponding eigenvalues for the 3^G subfactor planar algebras. We prove the conjecture in the case when |G| is odd. To do so, we find an action of G on the reduced subfactor planar algebra at f^{(2)}, which is obtained from shading the planar algebra of the even half. We also show that this reduced subfactor planar algebra is a Yang-Baxter planar algebra.

In this note, we will discuss how to relate an operator ideal on Banach spaces to the sequential structures it defines. Concrete examples of ideals of compact, weakly compact, completely continuous, Banach-Saks and weakly Banach-Saks operators will be demonstrated.

Let E and E be two Hilbert E -modules over E -algebras E and E , respectively. Let E be a surjective linear isometry from E onto E and E a map from E into E . We will prove in this paper that if the E -algebras E and E are commutative, then E preserves the inner products and E is a module map, ie, there exists a E -isomorphism E between the E -algebras such that <div class="gsh_dspfr"> E

In this paper, we prove that into isometries and disjointness preserving linear maps from<i>C</i>₀(<i>X</i>) into<i>C</i>₀(<i>Y</i>) are essentially weighted composition operators<i>Tf</i>=<i>h</i><i>f</i> for some continuous map and some continuous scalar-valued function<i>h</i>.