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∗-Algebras in Several Complex Variables
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
Recall that an associative complex Banach *-algebra A is called a C*-algebra if ||a*a|| = ||a||2 for all a ∈ A. The Banach *-algebra ℬ(H) of all bounded operators on a complex Hilbert space H is a C* -algebra, and so is every operator-norm closed *-subalgebra. Conversely, by the Gelfand-Naimark theorem, every C*-algebra can be realized in this way. The compact operatorsK(H) form a C* -subalgebra of ℬ(H).
Banach Algebras
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
C∗-algebras are Banach algebras of special interest. Their importance among Banach algebras is comparable to that of Hilbert spaces among other Banach spaces.
Banach Algebras
Published in Hugo D. Junghenn, Principles of Analysis, 2018
( $ \boldsymbol{(} $ b) $ \boldsymbol{)} $ If H $ \mathcal{H} $ is a Hilbert space, then B(H) $ \mathcal{B}(\mathcal{H}) $ is a C∗ $ C^* $ -algebra, where involution is the adjoint operation. The spaces B00(H) $ \mathcal{B}_{00}(\mathcal{H}) $ , B0(H) $ \mathcal{B}_0(\mathcal{H}) $ , B1(H) $ \mathcal{B}_1(\mathcal{H}) $ , and B2(H) $ \mathcal{B}_2(\mathcal{H}) $ are ideals of B(H) $ \mathcal{B}(\mathcal{H}) $ .
General system of cubic–quartic functional equations in quasi-β-normed spaces
Published in International Journal of General Systems, 2022
Some concrete and basic examples of -algebras are as follows. The set with the usual addition and product with one of the following involutions: The set , the space of continuous functions on a compact Hausdorff space X equipped with the pointwise addition and product of functions and scalar multiplication by constants. Moreover, the norm defined by and the involution is , for all .The prototypical example of a -algebra is the algebra of bounded (equivalently continuous) linear operators defined on a complex Hilbert space H; here, denotes the adjoint operator of the operator . In fact, every -algebra , is -isomorphic to a norm-closed adjoint closed subalgebra of for a suitable Hilbert space H; this is the content of the Gelfand–Naimark theorem.