Homomorphisms on algebras of Lipschitz functions
Volume 199 / 2010
Studia Mathematica 199 (2010), 95-106
MSC: Primary 46H10, 47B48; Secondary 47L10.
DOI: 10.4064/sm199-1-6
Abstract
We characterize a class of $*$-homomorphisms on ${\rm Lip}_*(X, \mathcal{B}(\mathcal{H})),$ a non-commutative Banach $*$-algebra of Lipschitz functions on a compact metric space and with values in $\mathcal{B}(\mathcal{H}).$ We show that the zero map is the only multiplicative $\ast $-preserving linear functional on ${\rm Lip}_*(X, \mathcal{B}(\mathcal{H})).$ We also establish the algebraic reflexivity property of a class of $*$-isomorphisms on ${\rm Lip}_*(X, \mathcal{B}(\mathcal{H})).$
