Geodesic
Theory of representations of a topological (non-Banach) involutory algebra
Teoretičeskaâ i matematičeskaâ fizika, Tome 2 (1970) no. 2, pp. 153-168 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A definition is given of the $^*$-representation of a specified topological (non-Banach) involutory algebra. The concepts of a symmetric representation, a representation adjoint to a specified one, and a self-conjugate representation, are introduced. It is demonstrated that the conjugation operation on a representation exhibits properties similar to the properties of the ordinary operation of conjugation of linear operators in Hilbert space. Various forms of algebraic similarity between representations, such as isomorphism, similarity, and unitary equivalence, are defined and explored. The concepts of broad and narrow commutants of a specified symmetric representation are introduced, and it is shown that the triviality of a broad commutant is equivalent to the purity of the generating functional. It is also demonstrated that self-conjugate representations exhibit a simpler algebraic structure than do simply symmetric representations. Specifically, the concepts of broad and narrow commutants merge in the case of a self-conjugate representation. 
@article{TMF_1970_2_2_a0,
     author = {A. N. Vasil'ev},
     title = {Theory of representations of a~topological {(non-Banach)} involutory algebra},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {153--168},
     year = {1970},
     volume = {2},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1970_2_2_a0/}
}
TY  - JOUR
AU  - A. N. Vasil'ev
TI  - Theory of representations of a topological (non-Banach) involutory algebra
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1970
SP  - 153
EP  - 168
VL  - 2
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_1970_2_2_a0/
LA  - ru
ID  - TMF_1970_2_2_a0
ER  - 
%0 Journal Article
%A A. N. Vasil'ev
%T Theory of representations of a topological (non-Banach) involutory algebra
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1970
%P 153-168
%V 2
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_1970_2_2_a0/
%G ru
%F TMF_1970_2_2_a0
A. N. Vasil'ev. Theory of representations of a topological (non-Banach) involutory algebra. Teoretičeskaâ i matematičeskaâ fizika, Tome 2 (1970) no. 2, pp. 153-168. http://geodesic.mathdoc.fr/item/TMF_1970_2_2_a0/

[1] I. M. Gelfand, N. Ya. Vilenkin, Obobschennye funktsii, t. 4, gl. 1, Fizmatgiz, 1961 | MR

[2] M. A. Naimark, Normirovannye koltsa, gl. 4, Gostekhizdat, 1956 | MR

[3] A. S. Wightman, Phys. Rev., 101 (1956), 860 | DOI | MR | Zbl

[4] H. J. Borchers, Nuovo Cim., 24 (1962), 214 | DOI | MR | Zbl

[5] R. Striter, A. Vaitman, RST, spin i statistika i vse takoe, «Nauka», 1966

[6] I. M. Gelfand, N. Ya. Vilenkin, Obobschennye funktsii, t. 4, gl. 3, Fizmatgiz, 1961 | MR

[7] N. I. Akhiezer, I. M. Glazman, Teoriya lineinykh operatorov, gl. 3, Gostekhizdat, 1961

[8] H. J. Borchers, Commun. math. Phys., 1 (1965), 49 | DOI | MR | Zbl

[9] A. N. Vassilev, Commun. math. Phys. (to appear)

[10] P. Ruelle, Helv. Phys. Acta, 35 (1962), 1 | MR

[11] M. A. Naimark, Normirovannye koltsa, gl. 7, Gostekhizdat, 1956 | MR