A Kowalski–Słodkowski theorem for 2-local $$^*$$ ∗ -homomorphisms on von Neumann algebras

Abstract

It is established that every (not necessarily linear) 2-local *homomorphism from a von Neumann algebra into a C * -algebra is linear and a * -homomorphism. In the setting of (not necessarily linear) 2-local * -homomorphism from a compact C * -algebra we prove that the same conclusion remains valid. We also prove that every 2-local Jordan *homomorphism from a JBW * -algebra into a JB * -algebra is linear and a Jordan * -homomorphism.1991 Mathematics Subject Classification. Primary 47B49 46L40 (46L57 47B47 47D25 47A 15A99 16S50 47L30).

Weak-2-local symmetric maps on C⁎-algebras

“…The main difficulty in the proof of this result is to show that every such a mapping is linear. It is also remarked at the introduction of [7], that "it would be of great interest to explore if the same conclusion remains true for general C * -algebras". Surprisingly, we can give now a very simple proof of the general result as a consequence of our previous corollary and the results in [27].…”

“…The main result in [7] proves that every 2-local * -homomorphism from a von Neumann algebra into a C * -algebra is linear and a * -homomorphism. The main difficulty in the proof of this result is to show that every such a mapping is linear.…”

“…Burgos, F.J. Fernández-Polo, J. Garcés and the second author of this note in [7]. The main result in the just quoted paper proves that every (not necessarily linear) 2-local * -homomorphism from a von Neumann algebra or from a compact C * -algebra into another C * -algebra is linear and a * -homomorphism; • The same authors mentioned in the previous point show in [8] that every 2-local triple homomorphism from a JBW * -triple into a JB * -triple is linear and a triple homomorphism; • Sh.…”

Weak-2-local isometries on uniform algebras and Lipschitz algebras

“…For 1 ≤ p < ∞ and p = 2, Al-Halees and Fleming [1] showed that ℓ p is 2-iso-reflexive. In the setting of B(H), C * -algebras and JB * -triples there exists a extensive literature on different classes of (weak-)2-local of maps (see, for example, [2,3,4,10,11,13,14,19,20,23,31,35,39,40] and [43]).…”

2-Local Automorphisms on AW ∗-Algebras

“…In [6] it was established that every 2-local * -homomorphism from a von Neumann algebra into a C * -algebra is a linear * -homomorphism. These authors also proved that every 2-local Jordan * -homomorphism from a JBW*-algebra into a JB*-algebra is a Jordan *-homomorphism.…”