Isometries of absolute order unit spaces
- PDF / 327,859 Bytes
- 15 Pages / 439.37 x 666.142 pts Page_size
- 97 Downloads / 170 Views
Positivity
Isometries of absolute order unit spaces Anil Kumar Karn1
· Amit Kumar1
Received: 13 June 2019 / Accepted: 12 December 2019 © Springer Nature Switzerland AG 2019
Abstract We prove that a unital, bijective linear map between absolute order unit spaces is an isometry if and only if it is absolute value preserving. We deduce that, on (unital) J B-algebras, such maps are precisely Jordan isomorphisms. Next, we introduce the notions of absolutely matrix ordered spaces and absolute matrix order unit spaces and prove that a unital, bijective ∗-linear map between absolute matrix order unit spaces is a complete isometry if, and only if, it is completely absolute value preserving. We obtain that on (unital) C∗ -algebras such maps are precisely C∗ -algebra isomorphisms. Keywords Absolutely ordered space · Absolute oder unit space · Isometry · Absolute value preserving maps · Absolute matrix order unit space Mathematics Subject Classification Primary 46B40; Secondary 46L05 · 46L30
1 Introduction In [12], Kakutani proved that an abstract M-space is precisely a concrete C(K , R) space for a suitable compact and Hausdorff space K . In [6], Gelfand and Naimark proved that an abstract (unital) commutative C∗ -algebra is precisely a concrete C(K , C) space for a suitable compact and Hausdorff space K . Thus Gelfand-Naimark theorem for commutative C∗ -algebras, in the light of Kakutani theorem, yields that the self-adjoint part of a commutative C∗ -algebra is, in particular, a vector lattice.
The Amit Kumar was financially supported by the Senior Research Fellowship of the University Grants Commission of India.
B
Anil Kumar Karn [email protected] Amit Kumar [email protected]
1
School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, District - Khurda, P.O. - Jatni, Bhubaneswar, Odisha 752050, India
123
A. K. Karn, A. Kumar
However, Kadison’s anti-lattice theorem suggest that the self-adjoint part of a general C∗ -algebra may not be a vector lattice [10]. Following Kadison’s functional representation of the self-adjoint part of a unital C∗ -algebra A as the space of continuous affine functions on the state space S(A) of A, it becomes evident that the order structure of a C∗ -algebra is rich with many properties. The works of Kadison, Effros, Størmer and Pedersen, besides many others, highlight various aspects of order structure of a C∗ -algebra and encourages us to expect a ‘noncommutative vector lattice’ or a ‘near lattice’ structure in it. The monogragh [19] (and references therein), for example, is a good source of information for this purpose. Keeping this point of view, the first author also worked in this direction [13–15]. In [16], he introduced the notion of absolutely ordered spaces and that of absolute order unit spaces. The self-adjoint parts of unital C∗ -algebras and (unital) M-spaces are examples of absolute order unit spaces. It was shown that under an additional condition (see [15, Theorem 4.12]) an absolutely ordered space turns out to be a vector la
Data Loading...
