Completely bounded norms of right module maps

TL;DR

This paper investigates the completely bounded norms of right module maps on matrix algebras, revealing when these norms equal their operator norms and providing sharp bounds for specific cases, with implications for infinite-dimensional operator algebras.

Contribution

It characterizes when right module maps have equal cb-norm and operator norm, and determines sharp bounds for the supremum of cb-norms in various matrix size cases.

Findings

For n=2, cb-norm equals operator norm for right D_n-module maps.

Explicit bounds for the supremum of cb-norms, C(m,n), are established in certain cases.

Existence of bounded right D-module maps on compact operators that are not completely bounded.

Abstract

It is well-known that if T is a D_m-D_n bimodule map on the m by n complex matrices, then T is a Schur multiplier and $\|T\|_{cb}=\|T\|$. If n=2 and T is merely assumed to be a right D_2-module map, then we show that $\|T\|_{cb}=\|T\|$. However, this property fails if m>1 and n>2. For m>1 and n=3,4 or $n\geq m^2$, we give examples of maps T attaining the supremum C(m,n)=\sup \|T\|_{cb} taken over the contractive, right D_n-module maps on M_{m,n}, we show that C(m,m^2)=\sqrt{m} and succeed in finding sharp results for C(m,n) in certain other cases. As a consequence, if H is an infinite-dimensional Hilbert space and D is a masa in B(H), then there is a bounded right D-module map on the compact operators K(H) which is not completely bounded.