Conjugations on Hilbert $C^{\ast }$-modules

Abstract

This work initiates a systematic development of conjugation theory within the framework of Hilbert $C^{\ast }$-modules, extending classical complex symmetric operator theory beyond the Hilbert space setting. We first explore the foundational structure of $\ast$-conjugate-automorphisms $♯$ on $C^{\ast }$-algebras and characterize their existence through isomorphisms with opposite algebras. We show that the set ${\mathbb{L}}_c(\mathcal{E})$ of all adjointable conjugate-linear operators on a Hilbert $C^{\ast }$-module $\mathcal{E}$ possesses the structure of a Hilbert $\mathbb{L}(\mathcal{E})$-module. We also introduce the key notion of $♯$-conjugation on a Hilbert $C^{\ast }$-module and fully characterize such modules in terms of real forms and real $C^{\ast }$-algebras. Moreover, we investigate the geometric properties of $♯$-conjugate duals. A central achievement is the development of polar decomposition theory for semiregular linear (conjugate-linear) operators acting on a Hilbert $C^{\ast }$-module equipped with a $♯$-conjugation $C$, thereby extending the celebrated Garcia--Putinar theorem [Trans. Amer. Math. Soc. 359 (2007), 3913--3931]. As a consequence, we show that any $C$-symmetric unitary operator factors as a product of two $♯$-conjugations. A comparative analysis of alternative definitions shows that requiring the naive adjoint condition $⟨Cx,y⟩=⟨x,Cy{⟩}^{\ast }$ would force the underlying $C^{\ast }$-algebra to be essentially commutative, thus justifying the adopted approach. We supply various examples and counterexamples that extensively illustrate our results.

33 pages