{'fraction_non_alphanumeric': 0.03285420944558522, 'fraction_numerical': 0.0, 'mean_word_length': 5.587837837837838, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Gelfand–Naimark Theorems for Ordered *-Algebras

The Gelfand-Naimark theorems provide important insight into the structure of general and of commutative C*-algebras. It is shown that these can be generalized to certain ordered *-algebras. More precisely, for Archimedean ordered *-algebras dominated by a sequence of positive elements, a faithful representation as operators is constructed. Similarly, for commutative such algebras, a faithful representation as complex-valued functions is constructed if an additional necessary regularity condition is fulfilled. These results generalize the Gelfand-Naimark representation theorems to classes of *-algebras that are much larger than C*-algebras, and especially contain *-algebras of unbounded operators. Representations of positive linear functionals by integrals are shortly discussed as well. On the way, a new construction of an essentially known representation of certain Riesz spaces by real-valued functions is given.