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Universal arrows to forgetful functors from categories of topological algebra

Published online by Cambridge University Press:  17 April 2009

Vladimir G. Pestov
Vladimir G. Pestov
Affiliation:
Department of Mathematics Victoria, University of Wellington, PO Box 600 Wellington, New Zealandvladimir.pestov@vuw.ac.nz
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We survey the present trends in theory of universal arrows to forgetful functors from various categories of topological algebra and functional analysis to categories of topology and topological algebra. Among them are free topological groups, free locally convex spaces, free Banach-Lie algebras, and more. An accent is put on the relationship of those constructions with other areas of mathematics and their possible applications. A number of open problems is discussed; some of them belong to universal arrow theory, and other may become amenable to the methods of this theory.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 48 , Issue 2 , October 1993 , pp. 209 - 249
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Ado, I.D., ‘The representation of Lie algebras by matrices’, in Amer. Math. Soc. Transl. 9 (Amer. Math. Soc., Providence, R.I., 1962), pp. 308–327.Google Scholar
[2]Antonovskiĭ, M.Ja., Boltyanskiĭ, V.G. and Sarymsakov, T.A., Metric spaces over semi-fields, Trudy Tashkent. Cos. Univ. 191, (in Russian), 1961.Google Scholar
[3]Arens, R., ‘A generalization of normed rings’, Pacif. J. Math. 2 (1952), 455–471.Google Scholar
[4]Arens, R. and Eells, J., ‘On embedding uniform and topological spaces’, Pacific J. Math. 6 (1956), 397–403.CrossRefGoogle Scholar
[5]Arhangel'skiĭ, A.V., ‘Mappings related to topological groups’, Soviet Math. Dokl. 9 (1968), 1011–1015.Google Scholar
[6]Arhangel'skiĭ, A.V., Topological spaces and continuous mappings. Remarks on topological groups, (in Russian) (Moscow University Press, Moscow, 1969).Google Scholar
[7]Arhangel'skiĭ, A.V., ‘Relations among invariants of topological groups and their subspaces’, Russian Math. Surveys 35 (1980), 1–23.Google Scholar
[8]Arhangel'skiĭ, A.V., ‘Any topological group is a quotient group of a zero-dimensional topological group’, Soviet Math. Dokl. 23 (1981), 615–618.Google Scholar
[9]Arhangel'skiĭ, A.V., ‘Classes of topological groups’, Russian Math. Surveys 36 (1981), 151–174.Google Scholar
[10]Arhangel'skiĭ, A.V., ‘Theorem on τ-approximation and functional duality’, Math. Notes 31 (1982), 421–432.Google Scholar
[11]Arhangel'skiĭ, A.V., ‘On linear homeomorphisms of function spaces’, Soviet Math. Dokl. 264 (1982), 1289–1292.Google Scholar
[12]Arhangel'skiĭ, A.V., ‘Algebraic objects generated by topological structures’, (in Russian), in Advances in Science and Technology. Algebra, Topology, Geometry 25 (VINITI, Moscow, 1987), pp. 141–198.Google Scholar
[13]Arhangel'skiĭ, A.V., ‘A survey of C p-theory’, Questions and Answers in General Topology 5 (1987), 1–109.Google Scholar
[14]Arnautov, V.I., Vodinchar, M.I. and Mikhalev, A.V., Introduction to theory of topological rings and moduli, (in Russian) (Štiinca, Kishinev, 1981).Google Scholar
[15]Bartocci, C., Bruzzo, U. and Ruipérez, D. Hernández, The Geometry of supermanifolds (Kluwer Academic Publishing, Dordrecht, 1991).Google Scholar
[16]Bartocci, C., Bruzzo, U., Ruipérez, D. Hernández, and Pestov, V.G., Foundations of supermanifold theory: the axiomatic approach (Dipartimento di Matematica, Università di Genova, January 1992). Preprint no. 194 (to appear in: Diff. Geom. and its Appl.)Google Scholar
[17]Berezin, F.A., Introduction to superanalysis (D.Reidel Publ. Co, Dordrecht, Boston MA, 1987).CrossRefGoogle Scholar
[18]Berger, S.M., ‘On deductive varieties of locally convex spaces’, Comment. Math. Univ. Carolin. 29 (1988), 465–475.Google Scholar
[19]Blackadar, B., ‘Shape theory for C* algebras’, Math. Scand. 56 (1985), 249–275.Google Scholar
[20]Blecher, O.P. and Paulsen, V.I., ‘Tensor products of operator spaces’, J. Funct. Anal. 99 (1991), 262–292.CrossRefGoogle Scholar
[21]Borceaux, F. and van den Bossche, G., ‘An essay on noncommutative topology’, Topology Appl. 31 (1989), 203–223.CrossRefGoogle Scholar
[22]Bourbaki, N., Lie groups and Lie algebras, Chapters I-III (Springer-Verlag, Berlin, Heidelberg, New York, 1989).Google Scholar
[23]Brown, R., ‘Some non-projective subgroups of free topological groups’, Proc. Amer. Math. Soc. 52 (1975), 433–441.Google Scholar
[24]Brown, R. and Hardy, J.P.L., ‘Subgroups of free topological groups and free topological products of topological groups’, J. London Math. Soc. 10 (1975), 431–440.CrossRefGoogle Scholar
[25]Burov, Yu. A., ‘Mutual decompositions of weak topological bases of a topological vector space’, (in Russian), Uspekhi Mat. Nauk 30 (1984), no 5, 237–238.Google Scholar
[26]Cartan, E., La topologie des groupes de Lie (Hermann, Paris, 1936).Google Scholar
[27]Comfort, W.W. and van Mill, J., ‘On the existence of free topological groups’, Topology Appl. 29 (1988), 245–269.Google Scholar
[28]Corson, H.H., ‘The weak topology of a Banach space’, Trans. Amer. Math. Soc. 101 (1961), 1–15.CrossRefGoogle Scholar
[29]de la Harpe, P., ‘Les extensions de gl(E) par un noyau de dimension finie sont triviales’, J. Funct. Anal. 33 (1979), 362–373.Google Scholar
[30]DeWitt, B.S., Supermanifolds (Cambridge University Press, London, 1984).Google Scholar
[31]Diestiel, J., Morris, S.A. and Saxon, S.A., ‘Varieties of linear topological spaces’, Trans. Amer. Math. Soc. 172 (1972), 207–230.CrossRefGoogle Scholar
[32]Dixmier, J., Enveloping algebras (North-Holland Publishing Co., Amsterdam, New York, Oxford, 1977).Google Scholar
[33]Drinfeld, V.G., ‘Quantum groups’, in Proceedings, International Congress of Mathematicians 1 (Berkeley, 1986), pp. 798–820.Google Scholar
[34]Effros, E.G., ‘Advances in quantized functional analysis’, in Proceedings, International Congress of Mathematicians (Berkeley, 1986), pp. 906–916.Google Scholar
[35]Exel, R. and Loring, T.A., ‘Finite-dimensional representations of free product C* algebras’, Internat. J. Math. (to appear).Google Scholar
[36]Fay, T., Ordman, E. and Smith-Thomas, B.V., ‘Free topological group over the rationals’, Gen. Topol. Appl. 10 (1979), 33–47.CrossRefGoogle Scholar
[37]Flood, J., Free topological vector spaces, Ph.D. thesis (Australian National University, Canberra, 1975).Google Scholar
[38]Flood, J., ‘Free locally convex spaces’, Dissertationes Math. CCXXI (PWN, Warczawa, 1984).Google Scholar
[39]Gelbaum, B.R., ‘Free topological groups’, Proc. Amer. Math. Soc. 12 (1961), 737–743.Google Scholar
[40]Goodearl, K.M. and Menai, P., ‘Free and residually finite dimensional C*-algebras’, J. Fund. Anal. 90 (1990), 391–410.Google Scholar
[41]Graev, M.I., ‘Free topological groups’, in Amer. Math. Soc. Transl. Ser. One 8 (Amer. Math. Soc., Providence, R.I., 1962), pp. 305–364.Google Scholar
[42]Graev, M.I., ‘Theory of topological groups I’, (in Russian), Uspekhi Mat. Nauk 5 (1950), no 2–56.Google Scholar
[43]Graev, M.I., ‘On free products of topological groups’, (in Russian), Izv. Akad. Nauk SSSR. Ser. mat. 14 (1950), 343–354.Google Scholar
[44]Gul'ko, S.P., ‘On uniform homeomorphisms of spaces of continuous functions’, (in Russian), Trudy Mat. Inst. Steklov 193 (1990).Google Scholar
[45]Gul'ko, S.P. and Khmyleva, T.E., ‘Compactness is not preserved by the relation of t−equivalence’, Math. Notes 39 (1986), 484–488.Google Scholar
[46]Guran, I I., ‘On topological groups, close to being Lindelöf’, Soviet Math. Dokl. 23 (1981), 173–175.Google Scholar
[47]Hardy, J.P.L., Topological groupoids: coverings and universal constructions, Ph.D. thesis (University College of North Wales, Bangor, Wales, 1974).Google Scholar
[48]Hardy, J.P.L., Morris, S.A. and Thompson, H.B., ‘Applications of the Stone-Čech compactification to free topological groups’, Proc. Amer. Math. Soc. 55 (1976), 160–164.Google Scholar
[49]Helemskiĭ, A.Ya., Banach and Polynormed Algebras. General Theory, Representations, Homology, (in Russian) (Nauka, Moscow, 1989).Google Scholar
[50]Hofmann, K.H., ‘An essay on free compact groups’, in Categorical aspects of topology and analysis, Lecture Notes in Math. 915 (Springer-Verlag, Berlin, Heidelberg, New York, 1982), pp. 171–197.Google Scholar
[51]Hofmann, K.H. and Morris, S.A., ‘Free compact groups I: Free compact abelian groups’, Topology Appl. 23 (1986), 41–64.CrossRefGoogle Scholar
[52]Hofmann, K.H. and Morris, S.A., ‘Free compact groups II: The center’, Topology Appl. 28 (1988), 215–231.Google Scholar
[53]Hofmann, K.H. and Morris, S.A., ‘Free compact groups III: Free semisimple compact groups’, in Categorical Topology, (Adàmek, J. and MacLane, S., Editors) (World Scientific, Singapore, 1989), pp. 208–219.Google Scholar
[54]Hofmann, K.H. and Morris, S.A., ‘Free compact groups IV: Splitting the component and the structure of the commutator group’, J. Pure Appl. Algebra 70 (1991), 89–96.Google Scholar
[55]Hofmann, K.H. and Morris, S.A., ‘Free compact groups V: Remarks on projectivity’, in Category Theory at Work, (Herrlich, E. and Porst, H.-E., Editors) (Heldermann-Verlag, Berlin, 1991), pp. 177–198.Google Scholar
[56]Hunt, D.C. and Morris, S.A., ‘Free subgroups of free topological groups’, in Proc. Second Internat. Cont. Theory of Groups (Canberra 1973), Lecture Notes in Math. 37 (Springer-Verlag, Berlin, Heidelberg, New York, 1974), pp. 377–387.Google Scholar
[57]Jadczyk, A. and Pilch, K., ‘Classical limit of CAR and self-duality in the infinite-dimensional Grassmann algebra’, in Quantum theory of particles and fields, (Jancewicz, B. and Lukierski, J., Editors) (World Scientific, Singapore, 1983).Google Scholar
[58]Jaffe, A., Lesniewski, A. and Osterwalder, K., ‘Quantum K−theory. I. The Chern character’, Comm. Math. Phys. 118 (1988), 1–14.CrossRefGoogle Scholar
[59]Johnstone, P.T., Topos theory.(Academic Press, London, New York, San Fransisco, 1977).Google Scholar
[60]Joiner, C., ‘Free topological groups and dimension’, Trans. Amer. Math. Soc. 220 (1976), 401–418.Google Scholar
[61]Junnila, H.J.K., ‘Stratifiable pre-images of topological spaces’, in Topology II (Proc. Fourth Colloq. Budapest, 1978), Colloq. Math. Soc. János Bolyai. 23 (North Holland, Amsterdam, New York, 1980), pp. 689–703.Google Scholar
[62]Kac, V., Infinite dimensional lie algebras, Second ed. (Cambridge University Press, Cambridge, 1985).Google Scholar
[63]Kac, V., ‘Constructing groups associated to infinite-dimensional Lie algebras’, in Infinite-Dimensional groups with applications, (Kac, V., Editor) 4, MSRI Pubi. (Springer-Verlag, Berlin, Heidelberg, New York, 1985), pp. 167–216.Google Scholar
[64]Kakutani, S., ‘Free topological groups and infinite direct products of topological groups’, Proc. Imp. Acad. Tokyo 20 (1944), 595–598.Google Scholar
[65]Kaplansky, I., Lie algebras and locally compact groups (Chicago University Press, Chicago, 1971).Google Scholar
[66]Kats, G.I., ‘Isomorphic mapping of topological groups into a direct product of groups satisfying the first axiom of countability’, (in Russian), Uspekhi Mat. Nauk 8 (1953), 107–113.Google Scholar
[67]Katz, E., ‘Free topological groups and principal fiber bundles’, Duke Math. J., 42 (1975), 83–90.Google Scholar
[68]Katz, E. and Morris, S.A., ‘Free products of topological groups with amalgamation’, Pacific J. Math. 119 (1985), 169–180.Google Scholar
[69]Katz, E. and Morris, S.A., ‘Free abelian topological groups on countable CW-complexes’, Bull. Austral. Math. Soc. 41 (1990), 451–456.Google Scholar
[70]Katz, E., Morris, S.A. and Nickolas, P., ‘A free subgroup of the free abelian topological group on the unit interval’, Bull. London Math. Soc. 14 (1982), 392–402.Google Scholar
[71]Katz, E., Morris, S.A. and Nickolas, P., ‘Free abelian topological groups on spheres’, Quart. J. Math. Oxford Ser. 2 35 (1984), 173–181.CrossRefGoogle Scholar
[72]Katz, E., Morris, S.A. and Nickolas, P., ‘Free subgroups of free abelian topological groups’, Math. Proc. Cambridge Philos. Soc. 100 (1986), 347–353.Google Scholar
[73]Katz, E., Morris, S.A. and Nickolas, P., ‘Free abelian topological groups and adjunction spaces’, J. Pure Appl. Algebra 68 (1990), 209–214.Google Scholar
[74]Katz, E., Morris, S.A. and Nickolas, P., ‘Characterization of bases of subgroups of free topological groups’, J. London Math. Soc. (2) 27 (1983), 421–426.Google Scholar
[75]Klimek, S. and Lesniewski, A., ‘Pfaffians on Banach spaces’, J. Funct. Anal. 102 (1991), 314–330.Google Scholar
[76]Kobayashi, O., Yoshioka, A., Maeda, Y. and Omori, H., ‘The theory of infinite-dimensional Lie groups and its applications’, Acta Appl. Math. 3 (1985), 71–106.Google Scholar
[77]Kobayashi, Y., and Nagamachi, Sh., ‘Usage of infinite-dimensional nuclear algebras in superanalysis’, Lett. Math Phys. 14 (1987), 15–23.Google Scholar
[78]Leĭderman, A.G., Morris, S.A. and Pestov, V.G., ‘The free topological abelian group and the free locally convex space on the unit interval’, in preparation.Google Scholar
[79]Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces sequence space Vol. 1 (Springer-Verlag, Berlin, Heidelberg, New York, 1977).Google Scholar
[80]Lane, S. Mac, Categories for the working mathematician, Graduate Texts in Mathematics 5 (Springer-Verlag, Berlin, Heidelberg, New York, 1971).Google Scholar
[81]Mack, J., Morris, S.A. and Ordman, E.T., ‘Free topological groups and the projective dimension of a locally compact abelian group’, Proc. Amer. Math. Soc. 40 (1973), 399–402.Google Scholar
[82]Mal'cev, A.I., ‘Free topological algebras’, Amer. Math. Soc. Transl. (2) 17 (1961), 173–200.Google Scholar
[83]Manin, Yu.I., Yu. I., Gauge field theory and complex geometry, Grundlehren Math. Wiss. 289 (Springer-Verlag, Berlin, Heidelberg,. New York, 1988).Google Scholar
[84]Manin, Yu.I., Quantum Groups and Non-Commutative Differential Geometry, 1988. Montréal University preprint CRM-1561.Google Scholar
[85]Markov, A.A., ‘On free topological groups’, (in Russian), Doklady Akad. Nauk SSSR 31 (1941), 299–301.Google Scholar
[86]Markov, A.A., ‘Three papers on topological groups’, in Amer. Math. Soc. Tranel. 30 (Amer. Math. Soc, Providence R.I., 1950).Google Scholar
[87]Michael, E., ‘Multiplicatively-convex topological algebras’, Mem. Amer. Math. Soc. 11 (1952).Google Scholar
[88]Michael, E., ‘A short proof of the Arens-Eells embedding theorem’, Proc. Amer. Math. Soc. 15 (1964), 415–416.Google Scholar
[89]Milnor, J., ‘Remarks on infinite-dimensional Lie groups’, in Relativité, groupes et topologie II. Les Houches, Session XL (Eisevier Sci. Pubi., Amsterdam, 1984), pp. 1007–1058.Google Scholar
[90]Milyutin, A.A., ‘Isomorphism of the spaces of continuous functions over compact sets of the cardinality of the continuum’, in Function Theory, Functional Analysis, and Applications 2 (Kharkov State University, Kharkov, 1966), pp. 150–156.Google Scholar
[91]Morris, S.A., ‘Varieties of topological groups’, Bull. Austral. Math. Soc. 1 (1969), 145–160.Google Scholar
[92]Morris, S.A., ‘Varieties of topological groups. II’, Bull. Austral. Math. Soc. 2 (1970), 1–13.Google Scholar
[93]Morris, S.A., ‘Quotient groups of topological groups with no small subgroups’, Proc. Amer. Math. Soc. 31 (1972), 625–626.CrossRefGoogle Scholar
[94]Morris, S.A., Pontryagin duality and the structure of locally compact abelian groups (Cambridge University Press, Cambridge, London, New York, Melbourne, 1977).CrossRefGoogle Scholar
[95]Morris, S.A., ‘Varieties of topological groups. A survey’, Colloq. Math. 46 (1982), 147–165.CrossRefGoogle Scholar
[96]Morris, S.A., ‘Free abelian topological groups’, in Categorical Topology, Proc. Conference Toledo, Ohio, 1983 (Heldermann-Verlag, Berlin, 1984), pp. 375–391.Google Scholar
[97]Morris, S.A., Ordman, E.T. and Thompson, H.B., ‘The topology of free products of topological groups’, in Proc. Second Internat. Conf. Theory of Groups (Canberra 1973), Lecture Notes in Math. 372 (Springer-Verlag, Berlin, Heidelberg, New York, 1974), pp. 504–515.CrossRefGoogle Scholar
[98]Morris, S.A. and Pestov, V.G., ‘Open subgroups of free abelian topological groups’, Math. Proc. Comb. Phil. Soc.Google Scholar
[99]Morris, S.A. and Thompson, H.B., ‘Invariant metrics on free topological groups’, Bull. Austral. Math. Soc. 9 (1973), 83–88.Google Scholar
[100]Morris, S.A. and Thompson, H.B., ‘Free topological groups with no small subgroups’, Proc. Amer. Math. Soc. 46 (1974), 431–437.Google Scholar
[101]Mycielski, J., ‘On the extension of equalities in connected topological groups’, Fund. Math. 44 (1957), 300–302.CrossRefGoogle Scholar
[102]Nagata, J., ‘On lattices of functions on topological spaces’, Osaka J. Math. 1 (1949), 166–181.Google Scholar
[103]Nakayama, T., ‘A note on free topological groups’, Proc. Imp. Acad. Tokyo 19 (1943), 471–475.Google Scholar
[104]Nickolas, P., Free topological groups and free products of topological groups, Ph.D. thesis (School of Mathematics, University of New South Wales, Kensington, N.S.W., 1976).Google Scholar
[105]Nickolas, P., ‘Subgroups of the free topological group on [0,1]’, J. London Math. Soc. (2) 12 (1976), 199–205.Google Scholar
[106]Nickolas, P., ‘Reflexivity of topological groups’, Proc. Amer. Math. Soc. 65 (1977), 137–141.Google Scholar
[107]Nickolas, P., ‘A Kurosh subgroup theorem for topological groups’, Proc. London Math. Soc. (3) 42 (1981), 461–477.CrossRefGoogle Scholar
[108]Noble, N., ‘K-Groups and duality’, Trans. Amer. Math. Soc. 151 (1970), 551–561.Google Scholar
[109]Nummela, E., ‘The completion of a topological group’, Bull. Austral. Math. Soc. 21 (1980), 407–417.Google Scholar
[110]Nummela, E., ‘Uniform free topological groups and Samuel compactifications’, Topology Appl. 13 (1982), 77–83.Google Scholar
[111]Okunev, O.G., ‘A method for constructing examples of M−equivalent spaces’, Topology Appl. 36 (1990), 157–170.Google Scholar
[112]Ordman, E.T. and Smith-Thomas, B.V., ‘Sequential conditions and free topological groups’, Proc. Amer. Math. Soc. 79 (1980), 319–326.Google Scholar
[113]Pavlovskiĭ, D.S., ‘On spaces of continuous functions’, Soviet Math. Dokl. 22 (1980), 34–37.Google Scholar
[114]Pestov, V.G., ‘On the structure and embeddings of topological groups’, Manuscript deposited at VINITI (Moscow) on April 13, 1981, No. 1495–81 Dep., (in Russian).Google Scholar
[115]Pestov, V.G., ‘Some properties of free topological groups’, Moscow Univ. Math. Bull. 37 (1982), 46–49.Google Scholar
[116]Pestov, V.G., ‘The coincidence of the dimensions dim of 1-equivalent topological spaces’, Soviet Math. Dokl. 26 (1982), 380–382.Google Scholar
[117]Pestov, V.G., Topological groups and algebraic envelopes of topological spaces, (in Russian) (Moscow State University, Moscow, 1983).Google Scholar
[118]Pestov, V.G., ‘Some topological properties preserved by the relation of M-equivalence’, Russian Math. Surveys 39 (1984), 223–224.Google Scholar
[119]Pestov, V.G., ‘To the theory of free topological groups: free groups, extensions, and compact coverability’, (in Russian) Manuscript deposited at VINITI (Moscow) on April 1, 1985, no. 2207–85Dep.,.Google Scholar
[120]Pestov, V.G., ‘Neighbourhoods of unity in free topological groups’, Moscow Univ. Math. Bull. 40 (1985), 8–12.Google Scholar
[121]Pestov, V.G., ‘Free topological abelian groups and the Pontryagin duality’, Moscow Univ. Math. Bull. 41 (1986), 1–4.Google Scholar
[122]Pestov, V.G., ‘Free Banach spaces and representations of topological groups’, Functional Anal. Appl. 20 (1986), 70–72.CrossRefGoogle Scholar
[123]Pestov, V.G., ‘A note on group topologizations’, (in Russian), in Topology structures and their mappings (Latvian University Press, Riga, 1987), pp. 93–95.Google Scholar
[124]Pestov, V.G., ‘The category of Tychonoff spaces embeds into the category of topological groups as a subcategory’, (in Russian), Bull. Acad. Sci. Georgian SSR 128 (1987), 257–259.Google Scholar
[125]Pestov, V.G., ‘On unconditionally closed sets and a conjecture of A.A. Markov’, Siberian Math. J. 29 (1988), 260–266.CrossRefGoogle Scholar
[126]Pestov, V.G., ‘Fermeture nonstandard des algèbres et groupes de Lie banachiques’, C.R. Acad. Sci. Parie Ser.1 Math. 306 (1988), 643–645.Google Scholar
[127]Pestov, V.G., ‘Even sectors of Lie superalgebras as locally convex Lie algebras’, J. Math. Phys. 32 (1991), 24–32.Google Scholar
[128]Pestov, V.G., ‘Ground algebras for superanalysis’, Rep. Math. Phys. 29 (1991), 275–287.Google Scholar
[129]Pestov, V.G., ‘General construction of Banach-Grassmann algebras’, Atti Accad. Naz. Lincei Rend. (9) 3 (1992), 223–231.Google Scholar
[130]Pestov, V.G., ‘Free Banach-Lie algebras, couniversal Banach-Lie groups, and more’, Pacific J. Math 157 (1993), 137–144.Google Scholar
[131]Pestov, V.G., ‘Enlargable Banach-Lie algebras and free topological groups’, Bull. Austral. Math. Soc. 48, 23–32.Google Scholar
[132]Pestov, V.G., ‘An analytic structure emerging in presence of infinitely many odd coordinates’, New Zealand J. Maths (to appear).Google Scholar
[133]Pestov, V.G., ‘A note on quantum unitary groups (Woronowicz's approach)’. Research report RP-92–102 (Department of Mathematics, Victoria University of Wellington 1992)Google Scholar
[134]Phillips, N.C., ‘Inverse limits of C* algebras and applications’, in Operator algebras and applications, vol 1, (Evans, and Takesaki, , Editors), London Mathematical Society Lecture Notes 135 (Cambridge Univ. Press, New York, 1988), pp. 127–185.Google Scholar
[135]Porst, H.E., ‘Towards a new construction of free topological groups’, Manuscript.Google Scholar
[136]Postnikov, M.M., Lie Groups and Lie Algebras, Lectures in Geometry. Semester V, Translated from Russian by Shokurov, V. (Mir, Moscow, 1986).Google Scholar
[137]Protasov, I.V., ‘Local theorems for topological groups’, Math. USSR Izvestija 15 (1980), 625–633.Google Scholar
[138]Protasov, I.V., ‘Varieties of topological algebras’, Siberian Math. J. 25 (1984), 125–134.Google Scholar
[139]Protasov, I.V., ‘Minimal varieties of topological groups’, Dokl. Akad. Nauk Ukrain. SSR. A (1988, no. 3), 14–15.Google Scholar
[140]Protasov, I.V. and Sidorchuk, A.D., ‘On varieties of topological algebraic systems’, Soviet Math. Dokl. 23 (1981), 184–187.Google Scholar
[141]Raĭkov, D.A., ‘On the completion of topological groups’, (in Russian), Izv. AN SSSR. Ser. Mat. 9 (1945), 513–518.Google Scholar
[142]Raĭkov, D.A., ‘Free locally convex spaces for uniform spaces’, (in Russian), Mat. Sb. (N.S.) 63 (1964), 582–590.Google Scholar
[143]Remus, D., ‘A short solution of Markov's problem on connected group topologies’, Proc. Amer. Math. Soc. 110 (1990), 1109–1110.Google Scholar
[144]Reshetikhin, N.Yu., Takhtadzhyan, L.A. and Faddeev, L.D., ‘Quantization of Lie groups and Lie algebras’, Leningrad Math. J. 1 (1990), 193–225.Google Scholar
[145]Robinson, A., ‘Germs’, in Applications of model theory to algebra, analysis and probability. (Holt, Rinehart and Winston, New York, 1969), pp. 138–149.Google Scholar
[146]Roelcke, W. and Dierolf, S., Uniform structures in topological groups and their quotients (McGraw–Hill, New York, 1981).Google Scholar
[147]Rosso, M., ‘Comparaison des groupes SU(2) quantiques de Drinfeld et de Woronowicz’, C.R. Acad. Sci. Paris, Série 1304 (1987), 323–326.Google Scholar
[148]Ruan, Z. Z., ‘Subspaces of C*-algebras’, J. Funct. Anal., 76 (1988), 217–230.Google Scholar
[149]Samuel, P., ‘On universal mappings and free topological groups’, Bull. Amer. Math. Soc. 54 (1948), 591–598.Google Scholar
[150]Schaefer, H.H., Topological vector spaces (The Macmillan Co, New York, London, 1966).Google Scholar
[151]Shakhmatov, D.B., ‘Zerodimensionality of free topological groups and topological groups with noncoinciding dimensions’, Bull. Acad. Pol. Sci. 37 (1989), 497–506.Google Scholar
[152]Sipacheva, O.V., ‘Description of topology of free topological groups without using uniform structure’, (in Russian), in General topology. Mappings of topological spaces. (MGU, Warszawa, 1986), pp. 122–129.Google Scholar
[153]Sipacheva, O.V., ‘Zero-dimensionality and completeness in free topological groups I, II’, Serdica 15 (1989), 119–140; 141–154.Google Scholar
[154]Sipacheva, O.V. and Uspenskiĭ, V.V., ‘Free topological groups with no small subgroups, and Graev metrics’, Moscow Univ. Math. Bull. 42 (1987), 24–29.Google Scholar
[155]Świerczkowski, S., ‘Embedding theorems for local analytic groups’, Acta Math. 114 (1965), 207–235.Google Scholar
[156]Świerczkowski, S., ‘Cohomology of local group extensions’, Trans. Amer. Math. Soc. 128 (1967), 291–320.Google Scholar
[157]Świerczkowski, S., ‘The path-functor on Banach-Lie algebras’, Indag. Math. 33 (1971), 235–239.Google Scholar
[158]Taylor, W., ‘Varieties of topological algebras’, J. Austral. Math. Soc. Ser A 23 (1977), 207–241.Google Scholar
[159]Thompson, H.B., ‘A remark on free topological groups with no small subgroups’, J. Austral. Math. Soc. 18 (1974), 482–484.Google Scholar
[160]Tkachenko, M.G., ‘On the topology of free groups over compacta’, (in Russian), in Seminar on general topology, (Alexandroff, P.S., Editor) (Moscow University Press, Moscow, 1983), pp. 89–106.Google Scholar
[161]Tkachenko, M.G., ‘On the Souslin property in free topological groups over compacta’, Math. Notes 34 (1983), 790–793.Google Scholar
[162]Tkachenko, M.G., ‘On completeness of free abelian topological groups’, Soviet Math. Dokl 27 (1983), 341–345.Google Scholar
[163]Tkachenko, M.G., ‘On topologies of free groups’, Czechoslovak Math J. 33 (1984), 57–69.Google Scholar
[164]Tkachenko, M.G., ‘On completeness of topological groups’, Siberian Math. J. 25 (1984), 122–131.Google Scholar
[165]Tkachuk, V.V., ‘On a method of constructing examples of M−equivalent spaces’, Russian Math. Surveys 38 (1983), 135–136.Google Scholar
[166]Tkachuk, V.V., ‘Homeomorphisms of free topological groups do not preserve compactness’, Math. Notes 42 (1987), 752–756.Google Scholar
[167]Uspenskiĭ, V.V., ‘A topological group generated by a Lindelöf Σ-space has the Souslin property’, Soviet Math. Dokl. 26 (1982), 166–169.Google Scholar
[168]Uspenskiĭ, V.V., ‘On the topology of free locally convex space’, Soviet Math. Dokl. 27 (1983), 780–785.Google Scholar
[169]Uspenskiĭ, V.V., ‘On subgroups of free topological groups’, Soviet Math. Dokl. 32 (1985), 847–849.Google Scholar
[170]Uspenskiĭ, V.V., ‘A universal topological group with countable base’, Funct. Anal. Appl. 20 (1986), 160–161.Google Scholar
[171]Uspenskiĭ, V.V., ‘The free topological groups of metrizable spaces’, (in Russian), Izv. Akad. Nauk SSSR 54 (1990), 1295–1319.Google Scholar
[172]van Est, W.T. and Świerczkowski, S., ‘The path functor and faithful representability of Banach Lie algebras’, J. Austral. Math. Soc. 16 (1973), 471–482.Google Scholar
[173]Valov, V.M. and Pasynkov, B.A., ‘Free groups of topological spaces’, (in Russian), C.R. Acad. Bulgare Sci. 34 (1981), 1049–1052.Google Scholar
[174]van Mill, J., Infinite-dimensional topology (North-Holland Publ. Co., Amsterdam, New York, Oxford, Tokyo, 1989).Google Scholar
[175]Wheeler, R.F., ‘Weak and pointwise compactness in the space of bounded continuous functions’, Trans. Amer. Math. Soc. 266 (1981), 515–530.Google Scholar
[176]Woronowicz, S.L., ‘Compact matrix pseudogroups’, Comm. Math. Phys. 111 (1987), 613–665.Google Scholar
[177]Zambakhidze, L.G., ‘On the interrelations between peripherical properties of Tychonoff spaces, their powers, exponents, free groups, and semigroups’, Russian Math. Surveys 34 (1979), 256–260.Google Scholar
[178]Zambakhidze, L.G., ‘On interrelations between dimensions of free bases of free topological groups’, (in Russian), Bull. Acad. Sci. Georgian SSR 97 (1980), 569–572.Google Scholar
[179]Zarichnyĭ, M.M., ‘Free topological groups of absolute neighbourhood retracts and infinite-dimensional manifolds’, Soviet Math. Dokl. 26 (1982), 367–371.Google Scholar
[180]Zarichnyĭ, M.M., ‘Preservation of ANR(3H)-spaces and infinite-dimensional manifolds by certain covariant functors’, Soviet Math. Dokl. 28 (1983), 105–109.Google Scholar