By M. L. Agranovskii

This booklet reports translation-invariant functionality areas and algebras on homogeneous manifolds. The crucial subject is the connection among the homogeneous constitution of a manifold and the category of translation-invariant functionality areas and algebras at the manifold. Agranovskiĭ obtains classifications of translation-invariant areas and algebras of features on semisimple and nilpotent Lie teams, Riemann symmetric areas, and bounded symmetric domain names. while such classifications are attainable, they lead in lots of circumstances to new characterizations of the classical functionality areas, from the perspective in their staff of admissible alterations of variable. The algebra of holomorphic services performs an important function in those classifications whilst a homogeneous advanced or CR-structure exists at the manifold. This ends up in new characterizations of holomorphic features and their boundary values for one- and multidimensional advanced domains.

Readership: experts in useful research, harmonic research, functionality thought of 1 and a number of other advanced variables, and Lie crew illustration.

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I. SPACES AND ALGEBRAS ON NONCOMPACT MANIFOLDS 20 (7) E8 . In this case one has also - Id E W, since the root system contains the orthogonal basis obtained by adding the vector e1 + + e8 to the set of vectors corresponding to E7 . (8) F4. The basis vectors e1 , e2 , e3, e4 are roots; therefore, - Id E W . (9) G2 . The vectors e1 , /e2 are roots; therefore, - Id E W. Thus the condition - Id _= W does not hold for symmetric spaces with root systems of types An , D2n_ I (n > 1) , E6. Here is a list of irreducible symmetric spaces of noncompact type with these root systems (see [28]): SL(n+1,C)/SU(n+1), n> 1 (An); SO(4n + 2, C)/SO(4n + 2) (D2n+1); C E6 /E6 (E6) ; SL(n, 1[8)/SO(n), n>1 (An) ; SU*(2n)/Sp(n), n > 2 (An-1); SOo(2n + 1 , 2n + 1)/(SO(2n + 1) x SO(2n + 1)) El E IV (*) (D2n+1) ; (E6); (A2).

Then the Beurling spectra a(AH) and a(AH) coincide. PROOF. It follows directly from (2') that AK is the uniform closure of the set CO(H) n v ([A n LK(X)]o) . 4, the latter is contained in AH ; hence, AK C AH and a(AH) C a(AH). Suppose the spectra do not coincide, and let A E a(AH)\a(AH) . There exists a neighborhood U of X whose closure does not intersect a(AH) . Since the spectrum of a convolution is contained in the intersection of the spectra of the II. TRANSLATION INVARIANT SPACES ON LIE GROUPS 28 factors, we may use convolution to construct a function g0 E C°(H) n V(A°) whose Beurling spectrum does not intersect a(AH) .

Suppose that A n L' (G) is uniformly dense in A and that the cyclic linear span of the K-bi-invariant functions that belong to An L1(G) is dense in A fl LI (G) in the LI -norm. Then either A = {0} or A = Co (G/Z) , where Z is a subgroup of the center of G. We say that function a f is K-bi-invariant if f (kI xk2) = f (x), k1 , k2 E K COROLLARY. Let G be a connected semisimple real Lie group with trivial center and f a nonzero K-bi-invariant function belonging to L' (G). Then every element of C0(G) can be uniformly approximated by polynomials in translates of f .