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First-degree relatives of patients with psychotic disorder have higher levels of polygenic risk (PRS) for schizophrenia and higher levels of intermediate phenotypes.
We conducted, using two different samples for discovery (n = 336 controls and 649 siblings of patients with psychotic disorder) and replication (n = 1208 controls and 1106 siblings), an analysis of association between PRS on the one hand and psychopathological and cognitive intermediate phenotypes of schizophrenia on the other in a sample at average genetic risk (healthy controls) and a sample at higher than average risk (healthy siblings of patients). Two subthreshold psychosis phenotypes, as well as a standardised measure of cognitive ability, based on a short version of the WAIS-III short form, were used. In addition, a measure of jumping to conclusion bias (replication sample only) was tested for association with PRS.
In both discovery and replication sample, evidence for an association between PRS and subthreshold psychosis phenotypes was observed in the relatives of patients, whereas in the controls no association was observed. Jumping to conclusion bias was similarly only associated with PRS in the sibling group. Cognitive ability was weakly negatively and non-significantly associated with PRS in both the sibling and the control group.
The degree of endophenotypic expression of schizophrenia polygenic risk depends on having a sibling with psychotic disorder, suggestive of underlying gene–environment interaction. Cognitive biases may better index genetic risk of disorder than traditional measures of neurocognition, which instead may reflect the population distribution of cognitive ability impacting the prognosis of psychotic disorder.
This first systematic account of the basic theory of normed algebras, without assuming associativity, includes many new and unpublished results and is sure to become a central resource for researchers and graduate students in the field. This second volume revisits JB*-triples, covers Zel'manov's celebrated work in Jordan theory, proves the unit-free variant of the Vidav–Palmer theorem, and develops the representation theory of alternative C*-algebras and non-commutative JB*-algebras. This completes the work begun in the first volume, which introduced these algebras and discussed the so-called non-associative Gelfand–Naimark and Vidav–Palmer theorems. This book interweaves pure algebra, geometry of normed spaces, and infinite-dimensional complex analysis. Novel proofs are presented in complete detail at a level accessible to graduate students. The book contains a wealth of historical comments, background material, examples, and an extensive bibliography.
This first systematic account of the basic theory of normed algebras, without assuming associativity, includes many new and unpublished results and is sure to become a central resource for researchers and graduate students in the field. This first volume focuses on the non-associative generalizations of (associative) C*-algebras provided by the so-called non-associative Gelfand–Naimark and Vidav–Palmer theorems, which give rise to alternative C*-algebras and non-commutative JB*-algebras, respectively. The relationship between non-commutative JB*-algebras and JB*-triples is also fully discussed. The second volume covers Zel'manov's celebrated work in Jordan theory to derive classification theorems for non-commutative JB*-algebras and JB*-triples, as well as other topics. The book interweaves pure algebra, geometry of normed spaces, and complex analysis, and includes a wealth of historical comments, background material, examples and exercises. The authors also provide an extensive bibliography.