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NONCOMMUTATIVE COMPLEX
ANALYSIS Leonid Vaksman Introduction Beginning with the works of Murray and von Neumann on operator algebras [5],
the construction of noncommutative analogues of the
most fundamental mathematical theories has been invariably attracting the
attention of experts. Among the widely known examples are the theory of
C*-algebras, operator K-theory, noncommutative
differential geometry, and quantum group theory. These are the noncommutative analogues of important branches of general
topology, algebraic topology, differential geometry, and group theory,
respectively. We
will be interested in noncommutative complex
analysis. In 1934, Lavrentiev obtained an important
result in the theory of approximation in the complex domain. He proved that
every continuous function on a compact subset with empty interior, K, of the complex
plane C, can be uniformly approximated by polynomials. New approaches to this
and similar theorems were found in the 50s, which lead to the birth of the
theory of uniform algebras, the latter being closely related to complex
analysis [3]. The first substantial results of noncommutative
complex analysis were obtained by Arveson in The next important step was made in the mid 90s. Noncommutative
analogues of bounded symmetric domains were found [20] in the framework of the
theory of quantum groups [11], which later lead to a noncommutative
analogue of function theory in such domains. Let us note that bounded symmetric domains [6] invariably attract the
attention of geometers, algebraists and analysts, because they serve as a
source of exactly solvable problems in complex analysis, noncommutative
harmonic analysis and classical mathematical physics. The simplest
bounded symmetric domain is the unit disc D = {z ∈ C | |z| < 1}.
Its
quantum analogue was introduced by Klimek and Lesniewski [12]. For pedagogical reasons, one should begin
the study of the quantum theory of bounded symmetric domains with the quantum
disc, cf. the second chapter of the book ``Quantum Bounded Symmetric Domains,''
available online. Chapter III of the book contains a background of general
theory of quantum bounded symmetric domains and the results on Harish-Chandra
modules, function spaces on quantum symmetric domains, canonical embeddings,
covariant differential calculi, and invariant differential operators. A wide array of literature is available on quantum bounded symmetric domains,
for instance, the set of lecture notes [23] and the papers of Bershtein, Kolesnik, Proskurin, Shklyarov, Sinel'schikov, Stolin, Turowska, Vaksman, Zhang [8, 10, 9,
21, 13, 17, 18, 1, 15, 22, 16, 19]. The work [2] is especially notable. It uses
methods of the theory of unitary dilatations of Szökefalvi-Nagy
and Foiaş [4] to describe the Shilov-Arveson boundary of a quantum ball, introduced by Pusz and Woronowicz [14]. These
articles are simpler than the third chapter of the book, since they either omit
the proofs, or discuss only one of Cartan's series of
irreducible bounded symmetric domains [6, p. 387]. However, the results and
proofs of the third chapter of the book are applicable to all quantum bounded
symmetric domains at once. I am deeply grateful to my students Bershtein,
Kolesnik, Korogodsky, Shklyarov; coauthors Soibelman
and Stolin; and colleagues Jakobsen,
Koelink, Klimyk, Kolb, Samoilenko, Turowska, Schmüdgen, Zhang, for numerous helpful discussions. A special role in my life was played by Vladimir Drinfeld.
In the mid 80s he taught me the basics of the theory of quantum groups and helped
me return to mathematics after an involuntary break I had to take from it for
many years. Now, after decades, Volodya's help and support
are still very important to me. REFERENCES 3. T. Gamelin.
Uniform Algebras, Mir, 4. S. Nagy, C. Foiaş,
Harmonic Analysis of operators in a Hilbert space, Mir, 5. J. von Neumann. Selected Works in
Functional Analysis, Nauka, 6. S. Helgason, Differential Geometry and Symmetric Spaces,
Mir, 7. W.
Arveson, Subalgebras
of C*-algebras, Acta Math., 123 (1969), 141-224. 8. O. Bershtein, Degenerate principal series of quantum
Harish-Chandra modules, J. Math.
Phys., 45, (2004), No 10, 3800-3827. 9. O. Bershtein, Ye. Kolesnik, L. Vaksman, On a q-analog of the Wallach-Okounkov
formula, Lett.
Math. Phys., 78 (2006), 97-109. 10.
O. Bershtein, A. Stolin, L. Vaksman, Spherical
principal series of quantum Harish-Chandra modules, Mathematical Physics, Analysis, and Geometry, 3 (2007), No 2, 157-169. 11.
J. C. Jantzen, Lectures on Quantum Groups, 12.
S. Klimek, A. Lesniewski, A two-parameter quantum deformation of the unit
disc, J. Funct.
Anal., 115 (1993), 1-23. 13.
D. Proskurin, L. Turowska, On the C*-algebra associated with Pol(Mat2,2)q, Methods Funct. Anal. and Topology, 7 (2001),
88-92. 14. 15.
D. Shklyarov, S. Sinel'shchikov, L. Vaksman, A q-analogue
of the Berezin quantization method, Lett. Math. Phys., 49 (1999), 253-261. 16.
D. Shklyarov, S. Sinel'shchikov, L. Vaksman, Geometric
realizations for some series of representations of the quantum group SU2,2,
, Mathematical Physics, Analysis, and
Geometry, 8 (2001), No 1, 90-110. 17.
D. Shklyarov, G. Zhang, A q-analogue of the Berezin transform on
the unit ball, J. Math. Phys., 44
(2003), No 9, 4344-4373. 18.
D. Shklyarov, G. Zhang, Covariant q-differential operators and unitary highest weight
representations for Uq sun,n,
J. Math. Phys., 46 (2005),
No 6, ~062307 1-24. 19.
S. Sinel'shchikov, A. Stolin, L. Vaksman, A quantum
analogue of the Bernstein functor, Journal of Lie theory, 17 (2007), No
1, 73-89. 20.
S. Sinel'shchikov, L. Vaksman, On q-analogues of bounded symmetric domains and Dolbeault complexes, Mathematical Physics, Analysis, and Geometry, 1 (1998), No 1, 75-100. |