# zbMATH — the first resource for mathematics

Partial representations and partial group algebras. (English) Zbl 0954.20004
Let $$G$$ be a finite group and $$K$$ a field. The partial group $$K$$-algebra $$K_{par}(G)$$ is the universal $$K$$-algebra with unit $$1$$ generated by the symbols $$[g]$$, $$g\in G$$, with relations $$[e]=1$$, $$[s^{-1}][s][t]=[s^{-1}][st]$$, $$[s][t][t^{-1}]=[st][t^{-1}]$$. The representations of $$K_{par}(G)$$ are in one-to-one correspondence with the so called partial representations of $$G$$.
The authors investigate the structure of $$K_{par}(G)$$, and they prove that if $$G_1$$ and $$G_2$$ are finite Abelian groups and the characteristic does not divide the order of $$G_1$$, then $$K_{par}(G_1)$$ is isomorphic to $$K_{par}(G_2)$$ if and only if $$G_1$$ is isomorphic to $$G_2$$.

##### MSC:
 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 16S34 Group rings
Full Text:
##### References:
 [1] Brown, L.G.; Douglas, R.G.; Fillmore, P.A., Extensions of C*-algebras and K-homology, Ann. math., 105, 265-324, (1977) · Zbl 0376.46036 [2] Cuntz, J., Simple C*-algebras generated by isometries, Comm. math. phys., 57, 173-185, (1977) · Zbl 0399.46045 [3] Cuntz, J., K-theory for certain C*-algebras, Ann. math., 113, 181-197, (1981) · Zbl 0437.46060 [4] Cuntz, J.; Krieger, W., A class of C*-algebras and topological Markov chains, Invent. math., 56, 251-268, (1980) · Zbl 0434.46045 [5] Curtis, C.W.; Reiner, I., Methods of representation theory, (1981), Wiley New York [6] Dade, E.C., Deux groupes finis distinct ayant la Même algèbre de group sur tout corps, Math. Z., 119, 345-348, (1971) · Zbl 0201.03303 [7] Douglas, R.G., Banach algebras techniques in the theory of Toeplitz operators, Conf. board of math. sc., regional conf. ser. in math., 15, (1973), Am. Math. Soc Providence [8] Douglas, R.G., On the C*-algebra of a one-parameter semigroup of isometries, Acta math., 128, 143-151, (1972) · Zbl 0232.46063 [9] Exel, R., Partial actions of groups and actions of semigroups, Proc. amer. math. soc., 126, 3481-3494, (1998) · Zbl 0910.46041 [10] Exel, R., Amenability for Fell bundles, J. reine angew. math., 492, 41-73, (1997) · Zbl 0881.46046 [11] Exel, R., Partial representations and amenable Fell bundles over free groups, Pacific J. math., 192, 39-64, (2000) · Zbl 1030.46069 [12] Exel, R.; Laca, M., Cuntz – krieger algebras for infinite matrices, J. reine angew. math., 512, 119-172, (1999) · Zbl 0932.47053 [13] Kumjian, A.; Pask, D.; Raeburn, I.; Renault, J., Graphs, groupoids and cuntz – krieger algebras, J. funct. anal., 144, 505-541, (1997) · Zbl 0929.46055 [14] Nica, A., C*-algebras generated by isometries and wiener – hopf operators, J. operator theory, 27, 1-37, (1991) [15] Passman, D.S., The algebraic structure of group rings, (1977), Interscience New York · Zbl 0366.16003 [16] Pimsner, M.; Voiculescu, D., Exact sequences for K-groups and ext-groups of certain cross-product C*-algebras, J. operator theory, 4, 93-118, (1980) · Zbl 0474.46059 [17] Quigg, J.C.; Raeburn, I., Characterizations of crossed products by partial actions, J. operator theory, 37, 311-340, (1997) · Zbl 0890.46048 [18] Renault, J., A groupoid approach to C*-algebras, Lecture notes in mathematics, 793, (1980), Springer-Verlag Berlin/New York [19] Roggenkamp, K.W.; Taylor, M.J., Group rings and class groups, (1992), Birkäuser Basel [20] Roseblum, M., A concrete spectral theory for self-adjoint Toeplitz operators, Amer. J. math., 87, 709-718, (1965) · Zbl 0135.16701 [21] Rottländer, A., Nachweis der existenz nicht-isomorpher gruppen von gleicher situation der untergruppen, Math. Z., 28, 641-653, (1928) · JFM 54.0144.01 [22] Sandling, R., The isomorphic problem for group rings, a survey, Springer lecture notes in math., 1141, (1985), Springer-Verlag Berlin/New York, p. 239-255 [23] Sehgal, S.K., Units in integral group rings, (1993), Longman Haslow · Zbl 0803.16022 [24] Serre, J.P., Linear representations of finite groups, Graduate texts in mathematics, 42, (1977), Springer-Verlag Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.