Theory of operator algebras 1 by M. Takesaki

By M. Takesaki

The unifying subject is the Banach area duality for operator algebras, permitting readers to acknowledge the affinity among operator algebras and degree thought on in the community compact areas.

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J The inner product is often denoted as (x, y) or ⟨x, y⟩ . 2. THE GEOMETRIC SIGNIFICANCE OF THE DOT PRODUCT 31 Notice how you put the conjugate on the entries of the vector y. It makes no difference if the vectors happen to be real vectors but with complex vectors you must do it this way. The reason for this is that when you take the inner product of a vector with itself, you want to get the square of the length of the vector, a positive number. Placing the conjugate on the components of y in the above definition assures this will take place.

It remains to consider when equality occurs. If either vector equals zero, then that vector equals zero times the other vector and the claim about when equality occurs is verified. Therefore, it can be assumed both vectors are nonzero. 5 implies one of the vectors must be a multiple of the other. Say b = αa. If α < 0 then equality cannot occur in the first inequality because in this case 2 2 (a · b) = α |a| < 0 < |α| |a| = |a · b| Therefore, α ≥ 0. 2. THE GEOMETRIC SIGNIFICANCE OF THE DOT PRODUCT 27 so |a| = |a − b + b| ≤ |a − b| + |b| .

Such lines are called skew lines. You might consider whether there exist two skew lines, each of which is the intersection of a pair of planes selected from a set of exactly three planes such that there is no point of intersection between the three planes. You can also see that if you tilt one of the planes you could obtain every pair of planes having a nonempty intersection in a line and yet there may be no point in the intersection of all three. It could happen also that the three planes could intersect in a single point as shown in the following picture.

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