The vanishing interior of the spectrum

CHAPTER 2 BANACH ALGEBRAS The 10 sections of this chapter can be split into four groups. The general theory of Banach algebras is represented in Secs. 1.2 and 2.2. The problems discussed here are basically related to the structural behavior of the spectrum of an element in an abstract Banach algebra. In Secso 3.2 and 4.2 we have decided to place the conjecture regarding the convolution measure algebra M(G) on a locally compact Abelian group G. The algebra M(G) is very interesting from several points of view, including the spectral one. The basis of the investigation of M(G) has been given in the well-known paper of Wiener and Pitt [Duke Math. J., ~, No. 2, pp. 420-436 (1938)]. Recently, the algebra M(G) has been subjected to intense investigation. However, the majority of the works have disclosed various pathologies in the structure of this algebra and the number of "positive" achievements is small. J. L. Taylor has computed the cohomology of the space of maximal ideals of the algebra M(G) (see [2] in the list of references of Sec. 3.2), while G. Brown and W. Moran have described the structure semigroups of important subalgebras in M(G) [Acta Math., 132, Nos. I-2, 77-~09 (1974)]. Recent progress is connected with the solution of I. Glicksberg's problem: When is ~ * M a closed ideal in M? [see B. Host and F. Parreau, C. R. Acad. Sci. Paris, 285, pp. AI5-A17 (1977)]. The question of the description of the Shilov boundary, discussed in Sec. 3.2, is undoubtedly one of the central questions of this theory; it has been posed by Taylor and awaits its solution for quite many years. In Sec. 4.2 one considers the problem of describing the homomorphisms of the L-subalgebras of M(G) in the spirit of the well-known Cohen--Rudin theorem. The following series of questions (5.2-8.2) is connected with the investigation of more visible but not less enigmatic algebras. This is the algebra H~(V) of all holomorphic and uniformly bounded functions in the domain V,Vc~, and the disc-algebra A. We mention that i n in Sec. 5.2 one expresses the interesting conjecture about the axiomatic description of the algebra H~(~) in the category of uniform algebras. In Sec. 6.2 one formulates 11 problems regarding the algebra H~(V). As mentioned in Sec. 6.2, the corona problem has not been solved yet for an arbitrary domain V. We would like to complete the literature to Sec. 6.2 by some references. M. F. Behrens [Trans. Am. Math. Soc., 161, pp. 359-379 (1971)] have shown that it is sufficient to solve the corona problem only in the class of domains of a special type and for some of them it has been proved that the algebra H~(V) has no corona. For recent results on this theme, see W. M. Deeb and D. R. Wilken [Trans. Am. Math. Soc., 231, No. I, pp. 107-115 (1977)] and W. M. Deeb [Trans. Am. Math. Soc., 231, No. I, pp. 101106 (1977)].

2111

1.2.

THE VANISHING INTERIOR OF THE SPECTRUM~

Let A and B be complex unital Banach algebras and let S e A c ~ that

~(~)D~(~)

and 8 ~ ( ~ ( ~ )

is its boundary.

, where OA(X)

singular

~t~o

zero divisor

be a sequence

in

series algebra A = { ~ Z ~ : ~ I a ~ [ ~ < ~ } algebra under the norm i[ ~ radius r.

Shilov

Shilov

[I] has proved that if ~ ,

in A (i.e., % -- x i s n o t i n v e r t i b l e i n a n y s u p e r a l g e b r a B o f A )

-- x is an approximate (n + ~ ) . Let

is the spectrum of x relative to A and ~OA(X)

Taking A to be the unital Banach algebra generated by x, in this context

we say that x is nontrivial if i ~ A ( ~ ) ~ . manently

; then it is well known

~+

(AZD) of A, i.e., if ~ A , I I ~ = J

,~=~,~[~I~

if and only if

m ~ O; t h e n

denotes a sequence of complex numbers)

which is generated by z, and OA(Z)

[I] shows that for an appropriate

is per-

, such that yn(h -- x) § 0

w i t h ~o = 1 a n d ~n+m 4 ~n~m, n ,

({~I

~-~

choice of the sequence

the power

is a Banach

is a disk of a [&~Ir~v0 ~

that 0 < r0 < r and if ~ . ~ [ ~ ; ~ $ ~ $ ~ ] , ~ - ~ is an AZD in A. Thus in every superalgebra annulus is contained in OB(X) and we say that OA(X) has a nonvanishing interior.

such B this

If A is a uniform algebra then it is easy to show that for each nontrivial element x we can construct a superalgebra B such that ~6~(~)=~. If T is a subnormal operator on a Hilbert space (i.e., T has a normal extension in a largerHilbert space) then the algebra which it generates is a uniform algebra [2]; hence the same is true of T. Shilov's theorem has been extended by Arens [3] to commutative Banach algebras which are not necessarily singly generated and Bollobas [4] has shown that it is not, in general, possible to construct a superalgebra B of a Banach algebra A in which all the elements which are not AZD's in A become simultaneously invertible. Questions.

Let A be generated by the nontrivial

element x such that % -- x is an AZD in

A if and only if ~ e ~ ( ~ ) . Can one construct a superalgebra B such that OB(X) = ~OA(X) , i.e., superalgebra B in which int oA(X) vanishes simultaneously? If x is a nontrivial element of a C*-algebra A does there exist a superalgebra B of A x such that OB(X) = ~OAx(X) , where Ax is the unital Banach algebra generated by x in A? LITERATURE I 9

2. 3. .

G. E. Shilov, "On normed rings with one generator," Mat. Sb., 21(63), 25-47 (1947). J. Bram, "Subnormal operators," Duke Math. J., 22, 75-94 (1955). R. Arens, "Inverse producing extensions of normed algebras," Trans. Am. Math. Soc., 88, 536-548 (1958). B. Bollobas, "Adjoining inverses to commutative Banach algebras," Trans. Am. Math. Soc., 181, 165-174 (1973).

tG. J. MURPHY and T. T. WEST.

2112

CITED

39 Trinity College, Dublin 2, Ireland.