scholarly journals Certain Unitary Representations of the Infinite Symmetric Group, I

1987 ◽  
Vol 105 ◽  
pp. 121-128 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
Symmetric Group ◽  
Regular Representation ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Type I ◽  
Discrete Topology ◽  
Type Ii ◽  
Infinite Symmetric Group ◽  
Von Neumann ◽  
Group I

Let X be the set of all natural numbers and let be the group of all finite permutations of X. The group equipped with the discrete topology, is called the infinite symmetric group. It was discussed in F. J. Murray and J. von Neumann as a concrete example of an ICC-group, which is a discrete group with infinite conjugacy classes. It is proved that the regular representation of an ICC-group is a factor representation of type II1. The infinite symmetric group is, therefore, a group not of type I. This may be the reason why its unitary representations have not been investigated satisfactorily. In fact, only few results are known. For instance, all indecomposable central positive definite functions on , which are related to factor representations of type IIl, were given by E. Thoma. Later on, A. M. Vershik and S. V. Kerov obtained the same result by a different method in and gave a realization of the representations of type II1 in. Concerning irreducible representations, A. Lieberman and G. I. Ol’shanskii obtained a characterization of a certain family of countably many irreducible representations by introducing a particular topology in However, irreducible representations have been studied not so actively as factor representations.

scholarly journals Certain unitary representations of the infinite symmetric group, II

1987 ◽  
Vol 106 ◽  
pp. 143-162 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
Symmetric Group ◽  
Discrete Group ◽  
Inductive Limit ◽  
Symmetric Groups ◽  
Discrete Groups ◽  
Type I ◽  
Infinite Symmetric Group ◽  
Locally Finite ◽  
Distinctive Features ◽  
Infinite Dimensional

The infinite symmetric group is the discrete group of all finite permutations of the set X of all natural numbers. Among discrete groups, it has distinctive features from the viewpoint of representation theory and harmonic analysis. First, it is one of the most typical ICC-groups as well as free groups and known to be a group of non-type I. Secondly, it is a locally finite group, namely, the inductive limit of usual symmetric groups . Furthermore it is contained in infinite dimensional classical groups GL(ξ), O(ξ) and U(ξ) and their representation theories are related each other.

Introduction to Von Neumann Algebras and Continuous Geometry

1960 ◽  
Vol 3 (3) ◽  
pp. 273-288 ◽  
Author(s):  
Israel Halperin
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Linear Operators ◽  
Mathematical Work ◽  
Type I ◽  
Type Ii ◽  
Von Neumann ◽  
The Past ◽  
Projection Geometry ◽  
Continuous Geometry

What is a von Neumann algebra? What is a factor (i) of type I, (ii) of type II, (iii) of type III? What is a projection geometry? And finally, what is a continuous geometry?The questions recall some of the most brilliant mathematical work of the past 30 years, work which was done by John von Neumann, partly in collaboration with F. J. Murray, and which grew out of von Neumann1 s analysis of linear operators in Hilbert space.

scholarly journals Infinite symmetric group, pseudomanifolds, and combinatorial cobordism-like structures

2018 ◽  
Vol 10 (03) ◽  
pp. 605-625 ◽  
Author(s):  
Alexander A. Gaifullin ◽  
Yury A. Neretin
Keyword(s):  
Symmetric Group ◽  
Hilbert Spaces ◽  
Double Coset ◽  
Product Formula ◽  
Linear Operators ◽  
The Other ◽  
Unitary Representations ◽  
Infinite Symmetric Group ◽  
Bounded Linear ◽  
Coset Spaces

We consider a category [Formula: see text] whose morphisms are [Formula: see text]-dimensional pseudomanifolds equipped with certain additional structures (coloring and labeling of some cells), multiplication of morphisms is similar to a concatenation of cobordisms. On the other hand, we consider the product [Formula: see text] of [Formula: see text] copies of infinite symmetric group. We construct a correspondence between the sets of morphisms of [Formula: see text] and double coset spaces of [Formula: see text] with respect to certain subgroups. We show that unitary representations of [Formula: see text] produce functors from the category of [Formula: see text] to the category of Hilbert spaces and bounded linear operators.

scholarly journals MOTERŲ MIOKARDO INFARKTO KLINIKINĖS SIMPTOMATIKOS IR BAIGČIŲ YPATUMAI

2015 ◽  
Vol 25 (1) ◽  
pp. 44-49 ◽  
Author(s):  
Žaneta Petrulionienė ◽  
Pranas Šerpytis ◽  
Dovilė Jančauskaitė ◽  
Urtė Gargalskaitė ◽  
Brigita Brazauskaitė ◽  
...  
Keyword(s):  
Myocardial Infarction ◽  
Acute Myocardial Infarction ◽  
Heart Rhythm ◽  
Type I ◽  
Type Ii ◽  
Group I ◽  
Primary Arterial Hypertension ◽  
St Elevation ◽  
Atrial Fibrilation ◽  
Group Ii

Objective. The aim of the present study was to compare differences of symptoms, comorbidities, risk factors and outcomes in younger (up to 55 years-old) and older (over 55 years-old) women with acute myocardial infarction. Materials and methods. In this retrospective study we analised 473 cases of women with acute myocardial infarction treated in 2012. Patients were divided into two groups according to their age: group I (up to 55 years) and group II (older than 55 years). The first group included 37 patients while the second group - 436 patients. Results. The average age of patients was 72,3 ± 11,07 m. Myocardial infarction with ST elevation were diagnosed to 54,3%, of wich Q+ 70,4%, Q- 29,6% (p 0,001), non-ST elevation 45,7%, of wich Q+ 6,5%, Q- 93,5% (p 0,001), no statistically significant difference was observed between the groups. 73% women in group I had primary arterial hypertension (I grade 2,7%, II 59,5%, III 10,8%), in the second group - 92,7% (I grade 0,7%, II 86%, III 6%), (p 0,001). Moreover, 13,5% patients in group I had diabetes (5,4% of type I, 8,1% of type II), in the group II 23,9% (0,3% of type I, 23,6% of type II), (p 0,001). Heart rhythm disorders were more often in women over 55 years-old (32,8%), (p=0,006). Among these patients, persistent atrial fibrilation were identified in 27,2%, permanent atrial fibrilation in 18,4%, ventricular fibrilation in 4,1% patients (all p 0,001). 5,4% women in younger group had previously experienced myocardial infarction while in older group - 20% (p=0,03). The spread of the pain to left hand was more common in the group of younger patients (27%) (p=0,047). Futhermore, in group I 18,9% felt weakness, while in group II - 38,5% (p=0,018). Fatal outcomes were observed in 6,8% patients, all of them were older than 55 years. Among patients with fatal outcomes Killip IV were found to 78,1% patients (p 0,001). Conclusions. Older women (≥ 55 years-old) treated for acute myocardial infarction more often had the grade II of primary arterial hypertension, heart rhythm disorder, previously experienced mycardial infarction and felt weakness. Younger patients (below 55 years old) had type I diabetes more often and were characterized by pain spreading to the left arm. Determined Killip IV leaded to increased lethality.

The relationship between plaque cap morphology and access technique in lower extremity chronic total occlusion

Vascular ◽  
2018 ◽  
Vol 27 (2) ◽  
pp. 135-143
Author(s):  
Saygin Turkyilmaz ◽  
Ali Aycan Kavala
Keyword(s):  
Lower Extremity ◽  
Chronic Total Occlusion ◽  
Univariate Analysis ◽  
Type I ◽  
Type Ii ◽  
Total Occlusion ◽  
Type Iii ◽  
Type Iv ◽  
Antegrade Approach ◽  
Group I

Objectives To evaluate access success according to plaque cap morphology in subjects with lower limb chronic total occlusion. Methods A retrospective study was performed for a three-year period. Subjects with lower extremity chronic total occlusion (Rutherford category 3–6, ischaemia) were included in the study. Cap morphology was classified according to The chronic total occlusion crossing approach based on plaque cap morphology (CTOP) classification system. When describing the classification by a traditional antegrade approach, Types I, II, III and IV were defined as follows: Type I: concave proximal and distal caps; Type II: concave proximal and convex distal caps; Type III: convex proximal and concave distal caps; Type IV: convex proximal and distal caps. For the study, the data on demographics, access type, and direction crossed, access conversion, crossing success, crossing location, extravascular ultrasound guidance, catheter used, subjects, and localization of were recorded. The effect of cap morphology on crossing strategy and success was evaluated. Results A total of 110 subjects were enrolled in this study. The type of chronic total occlusion was determined by angiography in 100% of the subjects. The number of the subjects according to CTOP morphology for Types I, II, III and IV were 22 (20%), 39 (35.5%), 23 (20.9%) and 26 (23.6%), respectively. Superficial femoral artery, popliteal, anterior tibial, posterior tibial localizations did not differ among the CTOP types ( p = 0.649, p = 0.831, p = 0.923 and p = 0.903, respectively). Among the pre-operation parameters, lesion length was the only one that is significantly shorter in Type I (14.23 ± 1.93 cm) subjects when compared with Types II (21.77 ± 3.78 cm), III (21.17 ± 2.31 cm) and IV (19.85 ± 3.29 cm) subjects ( p < 0.001, for all comparisons). Antegrade access was significantly higher in group I than in group III. Planned dual access was also significantly lower in CTOP Type I than in CTOP Types II, III and IV. Antegrade crossed direction was significantly higher in CTOP Type I than in CTOP Types II, III and IV ( p = 0.001, for all comparisons). True lumen crossing was significantly higher in CTOP Type I than in CTOP Type II ( p = 0.002). In univariate analysis, chronic total occlusion Type IV was the only significant factor for antegrade crossing ( p = 0.001). Multivariate analysis demonstrated that chronic total occlusion Type IV (OR = 0.09, p = 0.001) was an independent risk factor for antegrade crossing. The odds of antegrade crossing for chronic total occlusion Type IV was 0.190 times that of chronic total occlusion Types I–II–III combined (OR (95% CI): 0.190 (0.070, 0.519), p = 0.001). Conclusions CTOP Type I accesses with an antegrade access, and Type IV accesses with a retrograde strategy. Type II and Type III CTOP will need planned dual access in order to prevent device bending and subintimal access.

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