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## Mathematica vector valued function

Let B be the set of bounded Borel functions on T with compact support. Take p E [1, oo[, p E. Then ,. Take a E [0, JJxJ[oo[. By the above considerations,. Denote by yc the set of u E s. T is a closed proper zdeal o.

Lecture 30 - Commutative Banach Algebras, printed slides 1-19

Take k E IN. Now lim sup lltOO Ila. There is a p 5 IN, such that I ,. Let a. The general result now follows by passing to the imit. Laguerre, Neumann, Let F be the closed unital subalgebra of E generated by x. Then F is commutative Corollary 2. By Liouville's Theorem, f is constant.

Lemma 2. By continuity, the above relation holds for every t E [-1, 1]. Banach Algebras We may assume that E is unital Proposition 2. By Theorem 2. Then a x is compact. We may assume that E is unital Proposition 2. Being bounded Proposition 2. The assertion follows. Hence o l - u is not surjective and so a E a u. Hence a l - u is surjective. By the Principle of Inverse Operators, a l - u is invertible.

I Let T be a group. The assertion follows immediately from Theorem 2. In particular, U is a topological group with respect to the multiplicatzon.

Take x E U. Hence U is open. Hence the set of invertible elements of a unital Banach algebra is not necessarily dense. Since l is open Theorem 2. The closure of a proper left, mght ideal of E is a proper left, right ideal of E. The maximal proper left, right ideals of E are closed. Let U be the set of invertible elements of E. Since U is open Theorem 2. Let G be a maximal proper left, right ideal of E. By the above considerations, G is a proper left, right ideal of E. Let E be a Banach algebra and F a regular maximal proper left, right ideal.

Let E be a unital Banach algebra associated to E Definition 2. It follows that the radical of E is closed. I 2. Then V is a clopen normal subgroup o. Then x-iV, y-tVy are connected sets of U Proposition 2. Thus V is a normal subgroup of U.

## General properties of hermitian operators

Since U is obviously locally connected, V is open. Being a subgroup of U, it is a closed set of U. Since W is obviously a subgroup of V, it is a clopen set of V. By the above 9esult, there is a y E E such that X n Since ; E, a:. Then x' is continuous with norm at most 1.

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First assume that E is a unital Banach algebra. Banach Algebras Now suppose that E is not unital.