Mathematical Foundations of Quantum Mechanics 89(3)A(B+C)=AB+BC;
(4) (B+C)A⊃BA+CA;
(5)A⊂BandB⊂CimplyA⊂C;
(6)A⊂BandB⊂AimplyA=B;
(7)AB⊂BAimpliesA(D(B))⊂D(B)ifD(A)=H;
(8)AB=BAimpliesD(B)=A−^1 (D(B)) ifD(A)=H(so,A(D(B)) =
D(B)isAis surjective).To go on, we define some abstract algebraic structures naturally arising in the
space of operators on a Hilbert space.
Definition 2.2.8. LetAbe an associative complex algebraA.
(1)Ais aBanach algebraif it is a Banach space such that||ab||≤||a||||b||
for a, b∈A.AunitalBanach algebra is a Banach algebra with unit
multiplicative element 11 , satisfying|| 11 ||=1.
(2)Ais a (unital)∗-algebraif it is an (unital) algebra equipped with an anti
linear mapAa→a∗∈A,calledinvolution, such that(a∗)∗=aand
(ab)∗=b∗a∗fora, b∈A.
(3)Ais a (unital)C∗-algebraif it is a (unital) Banach algebraAwhich is
also a∗-algebra and||a∗a||=||a||^2 fora ∈A.A∗-homomorphism
from the∗-algebraAto the the∗-algebraBis an algebra homomorphism
preserving the involutions (and the unities if both present). A bijective
∗-homomorphism is called∗-isomorphism.Exercise 2.2.9.Prove that 11 ∗=1 1 in a unital∗-algebra and that||a∗||=||a||if
a∈AwhenAis aC∗-algebra.
Solution. From 1 1 a=a 1 1=aand the definition of∗, we immediately have
a∗ 11 ∗=1 1 ∗a∗=a∗. Since (b∗)∗=b,wehavefoundthatb 11 ∗=1 1 ∗b=bfor every
b∈A. Uniqueness of the unit implies 1 1 ∗=11. Regarding the second property,
||a||^2 =||a∗a||≤||a∗||||a||so that||a||≤||a∗||. Everywhere replacingafora∗and
using (a∗)∗, we also obtain||a∗||≤||a||,sothat||a∗||=||a||.
We remind the reader that a linear mapA:X→Y,whereX andY are
normed complex vector spaces with resp. norms||·||Xand||·||Y,issaidtobe
boundedif
||Ax||Y≤b||x||X for someb∈[0,+∞)andallx∈X. (2.28)As is well known[8; 5], it turns out that the following proposition holds.